1 Introduction
    1.1 What is ALGLIB
    1.2 ALGLIB license
    1.3 Documentation license
    1.4 Reference Manual and User Guide
    1.5 Acknowledgements
2 ALGLIB structure
    2.1 Packages
    2.2 Subpackages
    2.3 Open Source and Commercial versions
3 Compatibility
    3.1 CPU
    3.2 OS
    3.3 Compiler
    3.4 Optimization settings
4 Compiling ALGLIB
    4.1 Adding to your project
    4.2 Configuring for your compiler
    4.3 Utilizing SIMD
        4.3.1 Overview
        4.3.2 Achieving best performance
        4.3.3 Achieving best portability
        4.3.4 Disabling SIMD instruction sets at compile time
    4.4 Utilizing SMP
    4.5 Examples (free and commercial editions)
        4.5.1 Introduction
        4.5.2 Compiling under Windows
        4.5.3 Compiling under Linux
5 Using ALGLIB
    5.1 Thread-safety
    5.2 Global definitions
    5.3 Datatypes
    5.4 Constants
    5.5 Functions
    5.6 Working with vectors and matrices
    5.7 Using functions: 'expert' and 'friendly' interfaces
    5.8 Handling errors
    5.9 Working with Level 1 BLAS functions
    5.10 Reading data from CSV files
6 Working with commercial version
    6.1 Benefits of commercial version
    6.2 Working with SIMD support (Intel/AMD users)
    6.3 Using multithreading
        6.3.1 General information
        6.3.2 Compiling ALGLIB in the multithreaded mode
        6.3.3 Two kinds of parallelism
        6.3.4 Activating parallelism
        6.3.5 Controlling cores count
        6.3.6 More on parallel callbacks
        6.3.7 SMT (CMT/hyper-threading) issues
    6.4 Linking with Intel MKL
        6.4.1 Using lightweight Intel MKL supplied by ALGLIB Project
        6.4.2 Using your own installation of Intel MKL
7 Advanced topics
    7.1 Using Red Zones to find memory access violations
    7.2 Replacing stdlib rand() as an entropy source with OpenSSL implementation
    7.3 Exception-free mode
    7.4 Partial compilation
    7.5 Testing ALGLIB
8 ALGLIB packages and subpackages
    8.1 AlglibMisc package
    8.2 DataAnalysis package
    8.3 DiffEquations package
    8.4 FastTransforms package
    8.5 Integration package
    8.6 Interpolation package
    8.7 LinAlg package
    8.8 Optimization package
    8.9 Solvers package
    8.10 SpecialFunctions package
    8.11 Statistics package

1 Introduction

1.1 What is ALGLIB

ALGLIB is a cross-platform numerical analysis and data mining library. It supports several programming languages (C++, C#, Java, Delphi, VB.NET, Python) and several operating systems (Windows, *nix family).

ALGLIB features include:

Data analysis (classification/regression, including neural networks)
Optimization and nonlinear solvers
Interpolation and linear/nonlinear least-squares fitting
Linear algebra (direct algorithms, EVD/SVD), direct and iterative linear solvers, Fast Fourier Transform and many other algorithms (numerical integration, ODEs, statistics, special functions)

ALGLIB Project (the company behind ALGLIB) delivers to you several editions of ALGLIB:

ALGLIB Free Edition - full functionality but limited performance and license
ALGLIB Commercial Edition - high-performance version of ALGLIB with business-friendly license

Free Edition is a serial version without multithreading support and with a limited set of SIMD optimizations. Commercial Edition is a heavily optimized version of ALGLIB. It supports multithreading, it is extensively optimized, and (on Intel platforms) - our commercial users may enjoy a precompiled version of ALGLIB which internally calls Intel MKL to accelerate low-level tasks. We obtained a license from Intel corp., which allows us to integrate Intel MKL into ALGLIB, so you don't have to get a separate license from Intel.

1.2 ALGLIB license

ALGLIB Free Edition is distributed under license which favors non-commercial usage, but is not well suited for commercial applications:

ALGLIB for C++ and ALGLIB for C# are distributed under GPL 2+, GPL license version 2 or at your option any later version. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses
ALGLIB for Delphi and ALGLIB for CPython are distributed under ALGLIB Personal and Academic Use License Agreement.

ALGLIB Commercial Edition is distributed under license which is friendly to commercial users. A copy of the commercial license can be found at http://www.alglib.net/commercial.php.

1.3 Documentation license

This reference manual is licensed under BSD-like documentation license:

Redistribution and use of this document (ALGLIB Reference Manual) with or without modification, are permitted provided that such redistributions will retain the above copyright notice, this condition and the following disclaimer as the first (or last) lines of this file.

THIS DOCUMENTATION IS PROVIDED BY THE ALGLIB PROJECT "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE ALGLIB PROJECT BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS DOCUMENTATION, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

1.4 Reference Manual and User Guide

ALGLIB Project provides two sources of information: ALGLIB Reference Manual (this document) and ALGLIB User Guide.

ALGLIB Reference Manual contains full description of all publicly accessible ALGLIB units accompanied with examples. Reference Manual is focused on the source code: it documents units, functions, structures and so on. If you want to know what unit YYY can do or what subroutines unit ZZZ contains, the Reference Manual is a place to go. Free software needs free documentation - that's why the ALGLIB Reference Manual is licensed under BSD-like documentation license.

Additionally to the Reference Manual we provide you User Guide. User Guide is focused on more general questions: how fast ALGLIB is? how reliable is it? what are the strong and weak sides of the algorithms used? We aim to make ALGLIB User Guide an important source of information both about ALGLIB and numerical analysis algorithms in general. We want it to be a book about algorithms, not just software documentation. And we want it to be unique - that's why the ALGLIB User Guide is distributed under a less-permissive personal-use-only license.

1.5 Acknowledgements

ALGLIB was not possible without contribution of the following open source projects:

We also want to thank developers of the Intel's evelopment center for their help during MKL integration.

2 ALGLIB structure

2.1 Packages

ALGLIB is a C++ interface to the computational core written in C. Both C library and C++ wrapper are automatically generated by code generation tools developed within ALGLIB project. ALGLIB includes 11 packages and 5 support units:

The packages are:

alglibmisc.cpp - contains different algorithms which are hard to classify
dataanalysis.cpp - contains data mining algorithms
diffequations.cpp - contains differential equation solvers
fasttransforms.cpp - contains FFT and other related algorithms
integration.cpp - contains numerical integration algorithms
interpolation.cpp - contains interpolation algorithms
linalg.cpp - contains linear algebra algorithms
optimization.cpp - contains optimization algorithms
solvers.cpp - contains linear and nonlinear solvers
specialfunctions.cpp - contains special functions
statistics.cpp - statistics

The support units are:

alglibinternal.cpp - contains internal functions which are used by other packages, but not exposed to the external world
ap.cpp - contains publicly accessible vector/matrix classes, most important and general functions and other "basic" functionality
kernels_sse2.cpp - contains SSE2-accelerated implementations of performance-critical functions
kernels_avx2.cpp - contains AVX2-accelerated implementations of performance-critical functions
kernels_fma.cpp - contains FMA-accelerated implementations of performance-critical functions

One package may rely on other ones, but we have tried to reduce number of dependencies. Every package relies on ap.cpp and many packages rely on alglibinternal.cpp and kernels. But many packages require only these five to work, and many other packages need significantly less than 13 packages. For example, statistics.cpp requires five files mentioned above and only one additional package - specialfunctions.cpp.

2.2 Subpackages

There is one more concept to learn - subpackages. If you look at the list of ALGLIB packages, you will see that each package includes several subpackages. For example, linalg.cpp includes trfac, svd, evd and other subpackages. These subpackages do no exist as separate files, namespaces or other entities. They are just subsets of one large unit which provide significantly different functionality. They have separate documentation sections, but if you want to use svd subpackage, you have to include linalg.h, not svd.h.

2.3 Open Source and Commercial versions

ALGLIB comes in two versions - open source (GPL-licensed) and commercial (closed source) one. Both versions have same functionality, i.e. may solve same set of problems. However, commercial version differs from the open source one in following aspects:

License. Commercial ALGLIB is licensed under non-copyleft license which is friendly to commercial users.
Performance. Many algorithms in the commercial ALGLIB are multi-threaded. Open source ALGLIB is single-threaded and can not efficiently use modern multicore CPU's.
In addition to that, Commercial Edition can be linked with proprietary high-performance kernels (Intel MKL).

This documentation applies to both versions of ALGLIB. Detailed description of commercial version can be found below.

3 Compatibility

3.1 CPU

ALGLIB is compatible with any CPU which:

supports double precision arithmetics
complies with IEEE 754 floating point standard (especially in its handling of IEEE special values)
either big-endian or little-endian (but not mixed-endian)

Most mainstream CPU's (in particular, x86, x86_64, ARM) satisfy these requirements.

As for Intel architecture, ALGLIB works with both FPU-based and SIMD-based implementations of the floating point math.

3.2 OS

ALGLIB for C++ (both open source and commercial versions) can be compiled in OS-agnostic mode (no OS-specific preprocessor definitions), when it is compatible with any OS which supports C++98 standard library. In particular, it will work under any POSIX-compatible OS and under Windows.

If you want to use multithreaded capabilities of the commercial version of ALGLIB, you should compile it in OS-specific mode by #defining either AE_OS=AE_WINDOWS or AE_OS=AE_POSIX at compile time, depending on OS being used. Former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family, while latter means almost any POSIX-compatible OS. It applies only to commercial version of ALGLIB. Open source version is always OS-agnostic, even in the presence of OS-specific definitions.

3.3 Compiler

ALGLIB is compatible with any C++ compiler which:

supports 32-bit and 64-bit signed integer datatypes
emits code which handles comparisons with IEEE special values without raising an exception. We don't require that x/0 will return INF. But at least we must be able to compare double precision value with infinity or NAN without raising exception.

All modern compilers (MSVC, GCC, Clang, ICC) satisfy these requirements.

However, some very old compilers (ten years old version of Borland C++ Builder, for example) may emit code which does not correctly work with IEEE special values. If you use one of these old compilers, we recommend you to run ALGLIB test suite to ensure that library works.

3.4 Optimization settings

ALGLIB is compatible with any kind of optimizing compiler as long as:

volatile modifier is correctly handled (i.e. compiler does not optimize volatile reads/writes)
optimized code correctly handles IEEE special values

Generally, all kinds of optimization that were marked by compiler vendor as "safe" are possible. For example, ALGLIB can be compiled:

under MSVC: with /O1, /O2, /Og, /Os, /Ox, /Ot, /Oy, /fp:precise, /fp:except, /fp:strict
under GCC: with -O1, -O2, -O3, -Os

From the other side, following "unsafe" optimizations will break ALGLIB:

under MSVC: /fp:fast
under GCC: -Ofast, -ffast-math

4 Compiling ALGLIB

4.1 Adding to your project

Adding ALGLIB to your project is easy - just pick packages you need and... add them to your project! Under most used compilers (GCC, MSVC) it will work without any additional settings. In other cases you will need to define several preprocessor definitions (this topic will be detailed below), but everything will still be simple.

By "adding to your project" we mean that you should a) compile .cpp files with the rest of your project, and b) include .h files you need. Do not include .cpp files - these files must be compiled separately, not as part of some larger source file. The only files you should include are .h files, stored in the /src folder of the ALGLIB distribution.

As you see, ALGLIB has no project files or makefiles. Why? There are several reasons:

first, because many ALGLIB users don't need separate static library (which will be created by invoking makefile) - they prefer to integrate source code in their projects.
second, because we want ALGLIB to be usable in any programming environment, whether it is Visual Studio, GNU build system or something else. The best solution is to write package which doesn't depend on any particular programming environment.

In any case, compiling ALGLIB is so simple that even without project file you can do it in several minutes.

4.2 Configuring for your compiler

If you use modern versions of MSVC, GCC, Clang or ICC, you don't need to configure ALGLIB at all. But if you use outdated versions of these compilers (or something else), then you may need to tune definitions of several data types:

alglib_impl::ae_int32_t - signed integer which is 32 bits wide
alglib_impl::ae_int64_t - signed integer which is 64 bits wide
alglib_impl::ae_uint64_t - unsigned integer which is 64 bits wide
alglib_impl::ae_int_t - signed integer which has same width as pointer

ALGLIB tries to autodetect your compiler and to define these types in compiler-specific manner:

ae_int32_t is defined as int, because this type is 32 bits wide in all modern compilers.
ae_int64_t is defined as _int64 (MSVC) or as signed long long (GCC, Sun Studio).
ae_uint64_t is defined as unsigned _int64 (MSVC) or as unsigned long long (GCC, Sun Studio).
ae_int_t is defined as ptrdiff_t.

In most cases, it is enough. But if anything goes wrong, you have several options:

if your compiler provides stdint.h, you can define AE_HAVE_STDINT conditional symbol
alternatively, you can manually define AE_INT32_T and/or AE_INT64_T and/or AE_UINT64_T and/or AE_INT_T symbols. Just assign datatype name to them, and ALGLIB will automatically use your definition. You may define all or just one/two types (those which are not detected automatically).

4.3 Utilizing SIMD

4.3.1 Overview

ALGLIB has two-layered structure: some set of basic performance-critical primitives is implemented using optimized code (kernels_sse2.cpp, kernels_avx2.cpp and kernels_fma.cpp), and the rest of the library is built on top of these primitives. By default, ALGLIB uses generic C code to implement these primitives (matrix multiplication, decompositions, etc.). This code works on any CPU architecture that has IEEE-754 floating point support (including exotic DSPs).

However, much better performance may be achieved by utilizing SIMD-capable high-performance kernels included into the ALGLIB. These kernels are implemented using C/C++ SIMD intrinsics (as opposed to the assembly language) and provide good performance combined with benefits of using high-level language like C/C++.

As of the ALGLIB version 3.18, same set of SIMD kernels is included in both Free and Commercial editions.

Commercial Edition may be linked with Intel MKL which replaces some of ALGLIB's own SIMD kernels. Intel MKL is written in the assembly language and has much better performance than the C/C++ SIMD code included into the ALGLIB. This feature is absent in the Free Edition.

Presently ALGLIB supports only x86/x64 SIMD, including SSE2, AVX2 and FMA instruction sets. This support can be activated by compiling ALGLIB with AE_CPU=AE_INTEL macro symbol being #defined at the global level. Sections below discuss how to compile ALGLIB in order to get the fastest executable possible or, alternatively, to get the most portable one.

4.3.2 Achieving best performance

Obviously, the fastest executable can be obtained by compiling entire ALGLIB (both low level kernels and the code relying on these kernels) using the best SIMD instruction set available, e.g. FMA. In this case compiler may use SIMD to accelerate both SIMD-capable kernels and generic C code, e.g. use SIMD registers to pass parameters to function, perform quick memcpy() and so on.

In order to do so, two steps are necessary:

#define AE_CPU=AE_INTEL preprocessor symbol at the global level
compile all ALGLIB sources with SSE2/AVX2/FMA compiler switches

The former tells ALGLIB that it may use C/C++ SIMD intrinsics to accelerate computations. Depending on the compiler being used, the latter step can be done as follows:

under MSVC: add /arch:AVX2 compiler switch (which activates both AVX2 and FMA)
for the GCC/Clang/ICC family of compilers: add -mavx2 -mfma pair of options

The downside of this approach is that your entire application becomes less portable. Of course, ALGLIB includes a CPU dispatcher which checks at runtime which SIMD instruction sets are available at your CPU. If your CPU lacks FMA, ALGLIB won't call functions from kernels_fma.cpp. Similarly, if you don't have AVX2, kernels_avx2.cpp will be ignored. The problem is that because entire ALGLIB was compiled with SIMD support, a compiler may insert SIMD instructions on its own to accelerate even generic C parts.

The point above applies in some degree to all modern compilers, although the behavior may change from version to version and may depend on the optimization settings. In some cases (usually with optimization turned off) compilers generate completely portable code even in the presence of SIMD support. As a counterexample, some MSVC versions are known to replace generic C code like a+b*c by the fused multiply-adds when the code is compiled with /arch:AVX2 - which, by the way, violates the language specification.

4.3.3 Achieving best portability

If one needs a SIMD-capable application with portability guarantees, one has to:

#define AE_CPU=AE_INTEL preprocessor symbol at the global level
compile kernels_fma.cpp with SSE2/AVX2/FMA support enabled at the compiler level
compile kernels_avx2.cpp with SSE2/AVX2 support enabled
compile kernels_sse2.cpp with SSE2 support enabled
compile the rest of the library using no special compiler settings

ALGLIB guarantees that it won't call functions from kernels_fma.cpp without checking for FMA support first, won't access kernels_avx2.cpp without checking for AVX2 and so on. Thus, when the library is compiled this way, it will have guaranteed portability across x86/x64 systems.

4.3.4 Disabling SIMD instruction sets at compile time

By default, ALGLIB includes SIMD kernels for all x64 SIMD instruction sets present: SSE2, AVX2 and FMA. Of course, it has a CPU dispatcher which detects CPU support for SIMD and will choose proper SIMD kernel or generic C fallback. However, in some cases SIMD instruction sets should be deactivated at compile time. Most often it happens when one has to compile ALGLIB using some really outdated compiler which does not support post-SSE SIMD intrinsics.

It is possible to selectively disable a SIMD instruction set of your choice by combining AE_CPU=AE_INTEL #definition with one of the following #defines:

AE_NO_FMA to skip compilation of the FMA-capable kernels
AE_NO_AVX2 to skip compilation of the AVX2/FMA-capable kernels
AE_NO_SSE2 to skip compilation of all SSE2/AVX2/FMA SIMD kernels

4.4 Utilizing SMP

If you want to use multithreaded capabilities of the Commercial Edition, you should compile it in the OS-specific mode by #defining either AE_OS=AE_WINDOWS, AE_OS=AE_POSIX or AE_OS=AE_LINUX (POSIX with Linux-specific extensions) at compile time, depending on the OS being used. The former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family, while the latter means almost any POSIX-compatible OS (or any OS from the Linux family). Only threading-related functions are used by ALGLIB: pthreads under POSIX systems, threading subset of WinAPI under Windows.

The paragraph above applies only to the Commercial Edition. The open source version is always OS-agnostic (does not use functions beyond C/C++ standard library) even in the presence of OS-specific definitions.

4.5 Examples (free and commercial editions)

4.5.1 Introduction

In this section we'll consider different compilation scenarios for free and commercial versions of ALGLIB - from simple platform-agnostic compilation to compiling/linking with MKL extensions.

We assume that you unpacked ALGLIB distribution in the current directory and saved here demo.cpp file, whose code is given below. Thus, in the current directory you should have exactly one file (demo.cpp) and exactly one subdirectory (alglib-cpp folder with ALGLIB distribution).

4.5.2 Compiling under Windows

File listing below contains the very basic program which uses ALGLIB to perform matrix-matrix multiplication. After that program evaluates performance of GEMM (function being called) and prints result to console. We'll show how performance of this program continually increases as we add more and more sophisticated compiler options.

demo.cpp (WINDOWS EXAMPLE)

#include <stdio.h>
#include <windows.h>
#include "LinAlg.h"

double counter()
{
    return 0.001*GetTickCount();
}

int main()
{
    alglib::real_2d_array a, b, c;
    int n = 2000;
    int i, j;
    double timeneeded, flops;
    
    // Initialize arrays
    a.setlength(n, n);
    b.setlength(n, n);
    c.setlength(n, n);
    for(i=0; i<n; i++)
        for(j=0; j<n; j++)
        {
            a[i][j] = alglib::randomreal()-0.5;
            b[i][j] = alglib::randomreal()-0.5;
            c[i][j] = 0.0;
        }
    
    // Set global threading settings (applied to all ALGLIB functions);
    // default is to perform serial computations, unless parallel execution
    // is activated. Parallel execution tries to utilize all cores; this
    // behavior can be changed with alglib::setnworkers() call.
    alglib::setglobalthreading(alglib::parallel);
    
    // Perform matrix-matrix product.
    flops = 2*pow((double)n, (double)3);
    timeneeded = counter();
    alglib::rmatrixgemm(
        n, n, n,
        1.0,
        a, 0, 0, 0,
        b, 0, 0, 1,
        0.0,
        c, 0, 0);
    timeneeded = counter()-timeneeded;
    
    // Evaluate performance
    printf("Performance is %.1f GFLOPS\n", (double)(1.0E-9*flops/timeneeded));
    
    return 0;
}

Examples below cover Windows compilation from command line with MSVC. It is very straightforward to adapt them to compilation from MSVC IDE - or to another compilers. We assume that you already called %VCINSTALLDIR%\bin\amd64\vcvars64.bat batch file which loads 64-bit build environment (or its 32-bit counterpart). We also assume that current directory is clean before example is executed (i.e. it has ONLY demo.cpp file and alglib-cpp folder). We used 3.2 GHz 4-core CPU for this test.

First example covers platform-agnostic compilation without optimization settings - the most simple way to compile ALGLIB. This step is same in both open source and commercial editions. However, in platform-agnostic mode ALGLIB is unable to use all performance related features present in commercial edition.

We starts from copying all cpp and h files to current directory, then we will compile them along with demo.cpp. In this and following examples we will omit compiler output for the sake of simplicity.

OS-agnostic mode, no compiler optimizations

> copy alglib-cpp\src\*.* .
> cl /I. /EHsc /Fedemo.exe *.cpp
> demo.exe
Performance is 0.7 GFLOPS

Well, 0.7 GFLOPS is not very impressing for a 3.2GHz CPU... Let's add /Ox to compiler parameters.

OS-agnostic mode, /Ox optimization

> cl /I. /EHsc /Fedemo.exe /Ox *.cpp
> demo.exe
Performance is 0.9 GFLOPS

Still not impressed. Let's turn on optimizations for x86 architecture: define AE_CPU=AE_INTEL, add /arch:AVX2. This option provides some speed-up in both free and commercial editions of ALGLIB.

OS-agnostic mode, ALGLIB knows it is x86/x64

> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /arch:AVX2 *.cpp
> demo.exe
Performance is 4.5 GFLOPS

It is good, but we have 4 cores - and only one of them was used. Defining AE_OS=AE_WINDOWS allows ALGLIB to use Windows threads to parallelize execution of some functions. Starting from this moment, our example applies only to Commercial Edition.

ALGLIB knows it is Windows on x86/x64 CPU (COMMERCIAL EDITION)

> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /arch:AVX2 /DAE_OS=AE_WINDOWS *.cpp
> demo.exe
Performance is 16.0 GFLOPS

Not bad. And now we are ready to the final test - linking with MKL extensions.

Linking with MKL extensions differs a bit from standard way of linking with ALGLIB. ALGLIB itself is compiled with one more preprocessor definition: we define AE_MKL symbol. We also link ALGLIB with appropriate (32-bit or 64-bit) alglib???_??mkl.lib static library, which is an import library for special lightweight MKL distribution, shipped with ALGLIB. We also should copy to current directory appropriate alglib???_??mkl.dll binary file which contains Intel MKL.

Linking with MKL extensions (COMMERCIAL EDITION)

> copy alglib-cpp\addons-mkl\alglib*64mkl.lib .
> copy alglib-cpp\addons-mkl\alglib*64mkl.dll .
> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /DAE_OS=AE_WINDOWS /DAE_MKL *.cpp alglib*64mkl.lib
> demo.exe
Performance is 33.1 GFLOPS

From 0.7 GFLOPS to 33.1 GFLOPS - you may see that commercial version of ALGLIB is really worth it!

4.5.3 Compiling under Linux

demo.cpp (LINUX EXAMPLE)

#include <stdio.h>
#include <sys/time.h>
#include "LinAlg.h"

double counter()
{
    struct timeval now;
    alglib_impl::ae_int64_t r, v;
    gettimeofday(&now, NULL);
    v = now.tv_sec;
    r = v*1000;
    v = now.tv_usec/1000;
    r = r+v;
    return 0.001*r;
}

int main()
{
    alglib::real_2d_array a, b, c;
    int n = 2000;
    int i, j;
    double timeneeded, flops;
    
    // Initialize arrays
    a.setlength(n, n);
    b.setlength(n, n);
    c.setlength(n, n);
    for(i=0; i<n; i++)
        for(j=0; j<n; j++)
        {
            a[i][j] = alglib::randomreal()-0.5;
            b[i][j] = alglib::randomreal()-0.5;
            c[i][j] = 0.0;
        }
    
    // Set global threading settings (applied to all ALGLIB functions);
    // default is to perform serial computations, unless parallel execution
    // is activated. Parallel execution tries to utilize all cores; this
    // behavior can be changed with alglib::setnworkers() call.
    alglib::setglobalthreading(alglib::parallel);
    
    // Perform matrix-matrix product.
    flops = 2*pow((double)n, (double)3);
    timeneeded = counter();
    alglib::rmatrixgemm(
        n, n, n,
        1.0,
        a, 0, 0, 0,
        b, 0, 0, 1,
        0.0,
        c, 0, 0);
    timeneeded = counter()-timeneeded;
    
    // Evaluate performance
    printf("Performance is %.1f GFLOPS\n", (double)(1.0E-9*flops/timeneeded));
    
    return 0;
}

Examples below cover x64 Linux compilation from command line with GCC. We assume that current directory is clean before example is executed (i.e. it has ONLY demo.cpp file and alglib-cpp folder). We used 2.3 GHz 2-core Skylake CPU with 2x Hyperthreading enabled for this test.

OS-agnostic mode, no compiler optimizations

> cp alglib-cpp/src/* .
> g++ -I. -o demo.out *.cpp
> ./demo.out
Performance is 0.9 GFLOPS

Let's add -O3 to compiler parameters.

OS-agnostic mode, -O3 optimization

> g++ -I. -o demo.out -O3 *.cpp
> ./demo.out
Performance is 2.8 GFLOPS

Better, but not impressed. Let's turn on optimizations for x86 architecture: define AE_CPU=AE_INTEL, add -mavx2 -mfma. This option provides some speed-up in both free and commercial editions of ALGLIB.

OS-agnostic mode, ALGLIB knows it is x86/x64

> g++ -I. -o demo.out -O3 -DAE_CPU=AE_INTEL -mavx2 -mfma *.cpp
> ./demo.out
Performance is 5.0 GFLOPS

It is good, but we have 4 cores (in fact, 2 cores - it is 2-way hyperthreaded system) and only one of them was used. Defining AE_OS=AE_POSIX allows ALGLIB to use POSIX threads to parallelize execution of some functions. You should also specify -pthread flag to link with pthreads standard library. Starting from this moment, our example applies only to Commercial Edition.

ALGLIB knows it is POSIX OS on x86/x64 CPU (COMMERCIAL EDITION)

> g++ -I. -o demo.out -O3 -DAE_CPU=AE_INTEL -mavx2 -mfma -DAE_OS=AE_POSIX -pthread *.cpp
> ./demo.out
Performance is 9.0 GFLOPS

Not bad. You may notice that performance growth was ~2x, not 4x. The reason is that we tested ALGLIB on hyperthreaded system: although we have 4 logical cores, they share computational resources of just 2 physical cores. And now we are ready to the final test - linking with MKL extensions.

We should note that on typical Linux system shared libraries are not loaded from current directory by default. Either you install them into one of the system directories, or use some way to tell linker/loader that you want to load shared library from some specific directory. For our example we choose to update LD_LIBRARY_PATH environment variable.

Linking with MKL extensions (COMMERCIAL EDITION, relevant for ALGLIB 3.13)

> cp alglib-cpp/addons-mkl/libalglib*64mkl.so .
> ls *.so
libalglib313_64mkl.so
> g++ -I. -o demo.out -O3 -DAE_CPU=AE_INTEL -DAE_OS=AE_POSIX -pthread -DAE_MKL -L. *.cpp -lalglib313_64mkl
> LD_LIBRARY_PATH=.
> export LD_LIBRARY_PATH
> ./demo.out
Performance is 33.8 GFLOPS

Final result: from 0.9 GFLOPS to 33.8 GFLOPS!

5 Using ALGLIB

5.1 Thread-safety

Both open source and commercial versions of ALGLIB are 100% thread-safe as long as different user threads work with different instances of objects/arrays. Thread-safety is guaranteed by having no global shared variables.

However, any kind of sharing ALGLIB objects/arrays between different threads is potentially hazardous. Even when this object is seemingly used in read-only mode!

Say, you use ALGLIB neural network NET to process two input vectors X0 and X1, and get two output vectors Y0 and Y1. You may decide that neural network is used in read-only mode which does not change state of NET, because output is written to distinct arrays Y. Thus, you may want to process these vectors from parallel threads.

But it is not read-only operation, even if it looks like that! Neural network object NET allocates internal temporary buffers, which are modified by neural processing functions. Thus, sharing one instance of neural network between two threads is thread-unsafe!

5.2 Global definitions

ALGLIB defines several conditional symbols (all start with "AE_" which means "ALGLIB environment") and two namespaces: alglib_impl (contains computational core) and alglib (contains C++ interface).

Although this manual mentions both alglib_impl and alglib namespaces, only alglib namespace should be used by you. It contains user-friendly C++ interface with automatic memory management, exception handling and all other nice features. alglib_impl is less user-friendly, is less documented, and it is too easy to crash your system or cause memory leak if you use it directly.

5.3 Datatypes

ALGLIB (ap.h header) defines several "basic" datatypes (types which are used by all packages) and many package-specific datatypes. "Basic" datatypes are:

alglib::ae_int_t - signed integer type used by library
alglib::complex - double precision complex datatype, safer replacement for std::complex
alglib::ap_error - exception which is thrown by library
boolean_1d_array - 1-dimensional boolean array
integer_1d_array - 1-dimensional integer array
real_1d_array - 1-dimensional real (double precision) array
complex_1d_array - 1-dimensional complex array
boolean_2d_array - 2-dimensional boolean array
integer_2d_array - 2-dimensional integer array
real_2d_array - 2-dimensional real (double precision) array
complex_2d_array - 2-dimensional complex array

Package-specific datatypes are classes which can be divided into two distinct groups:

"struct-like" classes with public fields which you can access directly.
"object-like" classes which have no public fields. You should use ALGLIB functions to work with them.

5.4 Constants

The most important constants (defined in the ap.h header) from ALGLIB namespace are:

alglib::machineepsilon - small number which is close to the double precision &eps;, but is slightly larger
alglib::maxrealnumber - very large number which is close to the maximum real number, but is slightly smaller
alglib::minrealnumber - very small number which is close to the minimum nonzero real number, but is slightly larger
alglib::fp_nan - NAN (non-signalling under most platforms except for PA-RISC, where it is signalling; but when PA-RISC CPU is in its default state, it is silently converted to the quiet NAN)
alglib::fp_posinf - positive infinity
alglib::fp_neginf - negative infinity

5.5 Functions

The most important "basic" functions from ALGLIB namespace (ap.h header) are:

alglib::randomreal() - returns random real number from [0,1)
alglib::randominteger(mx) - returns random integer number from [0,nx); mx must be less than RAND_MAX
alglib::fp_eq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers. If numbers are represented with greater precision than specified by IEEE 754 (as with Intel 80-bit FPU), this functions converts them to 64 bits before comparing.
alglib::fp_neq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_less(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_less_eq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_greater(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_greater_eq(v1,v2) - makes IEEE-compliant comparison of two double precision numbers.
alglib::fp_isnan - checks whether number is NAN
alglib::fp_isposinf - checks whether number is +INF
alglib::fp_isneginf - checks whether number is -INF
alglib::fp_isinf - checks whether number is +INF or -INF
alglib::fp_isfinite - checks whether number is finite value (possibly subnormalized)

5.6 Working with vectors and matrices

ALGLIB (ap.h header) supports matrixes and vectors (one-dimensional and two-dimensional arrays) of variable size, with numeration starting from zero.

Everything starts from array creation. You should distinguish the creation of array class instance and the memory allocation for the array. When creating the class instance, you can use constructor without any parameters, that creates an empty array without any elements. An attempt to address them may cause the program failure.

You can use copy and assignment constructors that copy one array into another. If, during the copy operation, the source array has no memory allocated for the array elements, destination array will contain no elements either. If the source array has memory allocated for its elements, destination array will allocate the same amount of memory and copy the elements there. That is, the copy operation yields into two independent arrays with indentical contents.

You can also create array from formatted string like "[]", "[true,FALSE,tRUe]", "[[]]]" or "[[1,2],[3.2,4],[5.2]]" (note: '.' is used as decimal point independently from locale settings).

alglib::boolean_1d_array b1;
b1 = "[true]";

alglib::real_2d_array r2("[[2,3],[3,4]]");
alglib::real_2d_array r2_1("[[]]");
alglib::real_2d_array r2_2(r2);
r2_1 = r2;

alglib::complex_1d_array c2;
c2 = "[]";
c2 = "[0]";
c2 = "[1,2i]";
c2 = "[+1-2i,-1+5i]";
c2 = "[ 4i-2,  8i+2]";
c2 = "[+4i-2, +8i+2]";
c2 = "[-4i-2, -8i+2]";

After an empty array has been created, you can allocate memory for its elements, using the setlength() method. The content of the created array elements is not defined. If the setlength method is called for the array with already allocated memory, then, after changing its parameters, the newly allocated elements also become undefined and the old content is destroyed.

alglib::boolean_1d_array b1;
b1.setlength(2);

alglib::integer_2d_array r2;
r2.setlength(4,3);

Another way to initialize array is to call setcontent() method. This method accepts pointer to data which are copied into newly allocated array. Vectors are stored in contiguous order, matrices are stored row by row.

alglib::real_1d_array r1;
double _r1[] = {2, 3};
r1.setcontent(2,_r1);

alglib::real_2d_array r2;
double _r2[] = {11, 12, 13, 21, 22, 23};
r2.setcontent(2,3,_r2);

You can also attach real vector/matrix object to already allocated double precision array (attaching to boolean/integer/complex arrays is not supported). In this case, no actual data is copied, and attached vector/matrix object becomes a read/write proxy for external array.

alglib::real_1d_array r1;
double a1[] = {2, 3};
r1.attach_to_ptr(2,a1);

alglib::real_2d_array r2;
double a2[] = {11, 12, 13, 21, 22, 23};
r2.attach_to_ptr(2,3,_r2);

To access the array elements, an overloaded operator() or operator[] can used. That is, the code addressing the element of array a with indexes [i,j] can look like a(i,j) or a[i][j].

alglib::integer_1d_array a("[1,2,3]");
alglib::integer_1d_array b("[3,9,27]");
a[0] = b(0);

alglib::integer_2d_array c("[[1,2,3],[9,9,9]]");
alglib::integer_2d_array d("[[3,9,27],[8,8,8]]");
d[1][1] = c(0,0);

You can access contents of 1-dimensional array by calling getcontent() method which returns pointer to the array memory. For historical reasons 2-dimensional arrays do not provide getcontent() method, but you can use create reference to any element of array. 2-dimensional arrays store data in row-major order with aligned rows (i.e. generally distance between rows is not equal to number of columns). You can get stride (distance between consequtive elements in different rows) with getstride() call.

alglib::integer_1d_array a("[1,2]");
alglib::real_2d_array b("[[0,1],[10,11]]");

alglib::ae_int_t *a_row = a.getcontent();

// all three pointers point to the same location
double *b_row0 = &b[0][0];
double *b_row0_2 = &b(0,0);
double *b_row0_3 = b[0];

// advancing to the next row of 2-dimensional array
double *b_row1 = b_row0 + b.getstride();

Finally, you can get array size with length(), rows() or cols() methods:

alglib::integer_1d_array a("[1,2]");
alglib::real_2d_array b("[[0,1],[10,11]]");

printf("%ld\n", (long)a.length());
printf("%ld\n", (long)b.rows());
printf("%ld\n", (long)b.cols());

5.7 Using functions: 'expert' and 'friendly' interfaces

Most ALGLIB functions provide two interfaces: 'expert' and 'friendly'. What is the difference between two? When you use 'friendly' interface, ALGLIB:

tries to automatically determine size of input arguments
throws an exception when arguments have inconsistent size (for example, square matrix is expected, but non-square is passed; another example - two parameters must have same size, but have different size)

When you use 'expert' interface, ALGLIB requires caller to explicitly specify size of input arguments. If vector/matrix is larger than size being specified (say, N), only N leading elements are used.

Here are several examples of 'friendly' and 'expert' interfaces:

#include "interpolation.h"

...

alglib::real_1d_array    x("[0,1,2,3]");
alglib::real_1d_array    y("[1,5,3,9]");
alglib::real_1d_array   y2("[1,5,3,9,0]");
alglib::spline1dinterpolant s;

alglib::spline1dbuildlinear(x, y, 4, s);  // 'expert' interface is used
alglib::spline1dbuildlinear(x, y, s);     // 'friendly' interface - input size is
                                          // automatically determined

alglib::spline1dbuildlinear(x, y2, 4, s); // y2.length() is 5, but it will work

alglib::spline1dbuildlinear(x, y2, s);    // it won't work because sizes of x and y2
                                          // are inconsistent

5.8 Handling errors

ALGLIB uses two error handling strategies:

returning error code
throwing exception

What is actually done depends on function being used and error being reported:

if function returns some error code and has corresponding value for this kind of error, ALLGIB returns error code
if function does not return error code (or returns error code, but there is no code for error being reported), ALGLIB throws alglib::ap_error exception. Exception object has msg parameter which contains short description of error.

To make things clear we consider several examples of error handling.

Example 1. mincgreate function creates nonlinear CG optimizer. It accepts problem size N and initial point X. Several things can go wrong - you may pass array which is too short, filled by NAN's, or otherwise pass incorrect data. However, this function returns no error code - so it throws an exception in case something goes wrong. There is no other way to tell caller that something went wrong.

Example 2. rmatrixinverse function calculates inverse matrix. It returns error code, which is set to +1 when problem is solved and is set to -3 if singular matrix was passed to the function. However, there is no error code for matrix which is non-square or contains infinities. Well, we could have created corresponding error codes - but we didn't. So if you pass singular matrix to rmatrixinverse, you will get completion code -3. But if you pass matrix which contains INF in one of its elements, alglib::ap_error will be thrown.

First error handling strategy (error codes) is used to report "frequent" errors which can occur during normal execution of user program. Second error handling strategy (exceptions) is used to report "rare" errors which are result of serious flaws in your program (or ALGLIB) - infinities/NAN's in the inputs, inconsistent inputs, etc.

5.9 Working with Level 1 BLAS functions

ALGLIB (ap.h header) includes following Level 1 BLAS functions:

alglib::vdotproduct() family, which allows to calculate dot product of two real or complex vectors
alglib::vmove() family, which allows to copy real/complex vector to another location with optimal multiplication by real/complex value
alglib::vmoveneg() family, which allows to copy real/complex vector to another location with multiplication by -1
alglib::vadd() and alglib::vsub() families, which allows to add or subtract two real/complex vectors with optimal multiplication by real/complex value
alglib::vmul() family, which implements in-place multiplication of real/complex vector by real/complex value

Each Level 1 BLAS function accepts input stride and output stride, which are expected to be positive. Input and output vectors should not overlap. Functions operating with complex vectors accept additional parameter conj_src, which specifies whether input vector is conjugated or not.

For each real/complex function there exists "simple" companion which accepts no stride or conjugation modifier. "Simple" function assumes that input/output stride is +1, and no input conjugation is required.

alglib::real_1d_array    rvec("[0,1,2,3]");
alglib::real_2d_array    rmat("[[1,2],[3,4]]");
alglib::complex_1d_array cvec("[0+1i,1+2i,2-1i,3-2i]");
alglib::complex_2d_array cmat("[[3i,1],[9,2i]]");

alglib::vmove(&rvec[0],  1, &rmat[0][0], rmat.getstride(), 2); // now rvec is [1,3,2,3]

alglib::vmove(&cvec[0],  1, &cmat[0][0], rmat.getstride(), "No conj", 2); // now cvec is [3i, 9, 2-1i, 3-2i]
alglib::vmove(&cvec[2],  1, &cmat[0][0], 1,                "Conj", 2);    // now cvec is [3i, 9, -3i,  1]

Here is full list of Level 1 BLAS functions implemented in ALGLIB:

double vdotproduct(
    const double *v0,
     ae_int_t stride0,
     const double *v1,
     ae_int_t stride1,
     ae_int_t n);
double vdotproduct(
    const double *v1,
     const double *v2,
     ae_int_t N);

alglib::complex vdotproduct(
    const alglib::complex *v0,
     ae_int_t stride0,
     const char *conj0,
     const alglib::complex *v1,
     ae_int_t stride1,
     const char *conj1,
     ae_int_t n);
alglib::complex vdotproduct(
    const alglib::complex *v1,
     const alglib::complex *v2,
     ae_int_t N);

void vmove(
    double *vdst,
      ae_int_t stride_dst,
     const double* vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vmove(
    double *vdst,
     const double* vsrc,
     ae_int_t N);

void vmove(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vmove(
    alglib::complex *vdst,
     const alglib::complex* vsrc,
     ae_int_t N);

void vmoveneg(
    double *vdst,
      ae_int_t stride_dst,
     const double* vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vmoveneg(
    double *vdst,
     const double *vsrc,
     ae_int_t N);

void vmoveneg(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vmoveneg(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N);

void vmove(
    double *vdst,
      ae_int_t stride_dst,
     const double* vsrc,
      ae_int_t stride_src,
     ae_int_t n,
     double alpha);
void vmove(
    double *vdst,
     const double *vsrc,
     ae_int_t N,
     double alpha);

void vmove(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     double alpha);
void vmove(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     double alpha);

void vmove(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex* vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     alglib::complex alpha);
void vmove(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     alglib::complex alpha);

void vadd(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vadd(
    double *vdst,
     const double *vsrc,
     ae_int_t N);

void vadd(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vadd(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N);

void vadd(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n,
     double alpha);
void vadd(
    double *vdst,
     const double *vsrc,
     ae_int_t N,
     double alpha);

void vadd(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     double alpha);
void vadd(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     double alpha);

void vadd(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     alglib::complex alpha);
void vadd(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     alglib::complex alpha);

void vsub(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n);
void vsub(
    double *vdst,
     const double *vsrc,
     ae_int_t N);

void vsub(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n);
void vsub(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N);

void vsub(
    double *vdst,
      ae_int_t stride_dst,
     const double *vsrc,
      ae_int_t stride_src,
     ae_int_t n,
     double alpha);
void vsub(
    double *vdst,
     const double *vsrc,
     ae_int_t N,
     double alpha);

void vsub(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     double alpha);
void vsub(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     double alpha);

void vsub(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     const alglib::complex *vsrc,
     ae_int_t stride_src,
     const char *conj_src,
     ae_int_t n,
     alglib::complex alpha);
void vsub(
    alglib::complex *vdst,
     const alglib::complex *vsrc,
     ae_int_t N,
     alglib::complex alpha);

void vmul(
    double *vdst,
      ae_int_t stride_dst,
     ae_int_t n,
     double alpha);
void vmul(
    double *vdst,
     ae_int_t N,
     double alpha);

void vmul(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     ae_int_t n,
     double alpha);
void vmul(
    alglib::complex *vdst,
     ae_int_t N,
     double alpha);

void vmul(
    alglib::complex *vdst,
     ae_int_t stride_dst,
     ae_int_t n,
     alglib::complex alpha);
void vmul(
    alglib::complex *vdst,
     ae_int_t N,
     alglib::complex alpha);

5.10 Reading data from CSV files

ALGLIB (ap.h header) has alglib::read_csv() function which allows to read data from CSV file. Entire file is loaded into memory as double precision 2D array (alglib::real_2d_array object). This function provides following features:

support for ASCI encoding and UTF-8 without BOM (in header names)
ability to use any character (comma/tab/space) as field separator (as long as it is distinct from one used for decimal point)
ability to use both comma and full stop as decimal point (as long is decimal point is distinct from field separator).
support for Unix and Windows text files (CR vs CRLF)

See comments on alglib::read_csv() function for more information about its functionality.

6 Working with commercial version

6.1 Benefits of commercial version

Commercial version of ALGLIB for C++ features four important improvements over open source one:

License. Commercial license used by ALGLIB is friendly to closed source applications. Unlike GPL, it does not require you to open source your application. Thus, almost any commercial software developer is interested in obtaining commercial license.
Low-level optimizations. Commercial version of ALGLIB includes SIMD-optimized versions of many computationally intensive functions. It allows to increase performance on Intel/AMD platforms while still being able to use software under non-x86 CPU's.
Multithreading. Commercial version of ALGLIB can utilize multicore capabilities of modern CPU's. Large computational problems can be automatically split between different cores. ALGLIB uses its own multithreading framework which does not depend on vendor/compiler support for technologies like OpenMP/MPI/... It gives ALGLIB unprecedented portability across operating systems and compilers.
Integrated Intel MKL. Commercial version of ALGLIB includes special MKL extensions - special lightweight distribution of Intel MKL, high-performance numerical analysis library, accompanied by ALGLIB-MKL interface. We obtained license from Intel which allows us to integrate MKL into ALGLIB distribution. Linking with MKL accelerates many ALGLIB functions, however due to license restrictions you can not use MKL directly (i.e. bypass ALGLIB interface between your program and MKL).

6.2 Working with SIMD support (Intel/AMD users)

ALGLIB for C++ can utilize SIMD instructions supported by Intel and AMD processors. This feature is optional and must be explicitly turned on during compile-time. If you do not activate it, ALGLIB will use generic C code, without any processor-specific assembly/intrinsics.

Thus, if you turn on this feature, your code will run faster on x86_32 and x86_64 processors, but will be unportable to non-x86 platforms (and Intel MIC platform, which is not exactly x86!). From the other side, if you do not activate this feature, your code will be portable to almost any modern CPU (SPARC, ARM, ...).

In order to turn on x86-specific optimizations, you should define AE_CPU=AE_INTEL preprocessor definition at global level. It will tell ALGLIB to use SIMD intrinsics supported by GCC, MSVC and Intel compilers. Additionally you should tell compiler to generate SIMD-capable code. It can be done in the project settings of your IDE or in the command line:


GCC example:
> g++ -msse2 -I. -DAE_CPU=AE_INTEL *.cpp -lm

MSVC example:
> cl /I. /EHsc /DAE_CPU=AE_INTEL *.cpp

6.3 Using multithreading

6.3.1 General information

Commercial version of ALGLIB includes out-of-the-box support for multithreading. Many (not all) computationally intensive problems can be solved in multithreaded mode. You should read comments on specific ALGLIB functions to determine what can be multithreaded and what can not.

ALGLIB does not depend on vendor/compiler support for technologies like OpenMP/MPI/... Under Windows ALGLIB uses OS threads and custom synchronization framework. Under POSIX-compatible OS (Solaris, Linux, FreeBSD, NetBSD, OpenBSD, ...) ALGLIB uses POSIX Threads (standard *nix library which is shipped with any POSIX system) with its threading and synchronization primitives. It gives ALGLIB unprecedented portability across operating systems and compilers. ALGLIB does not depend on presence of any custom multithreading library or compiler support for any multithreading technology.

6.3.2 Compiling ALGLIB in the multithreaded mode

You should compile ALGLIB in OS-specific mode by #defining either AE_OS=AE_WINDOWS or AE_OS=AE_POSIX (or AE_OS=AE_LINUX, which means "POSIX with Linux extensions") at the compile time, depending on the OS being used. The former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family, while the latter two mean almost any POSIX-compatible OS or any OS from the Linux family. When compiling on POSIX/Linux, do not forget to link ALGLIB with libpthread library.

When compiled without OS-specific switches, ALGLIB runs in OS-agnostic mode and ignores all kinds of parallelism settings.

6.3.3 Two kinds of parallelism

ALGLIB supports two types of parallelism:

Internal parallelism
Callback parallelism

Internal parallelism refers to ALGLIB's capability to speed up operations that are embedded within its own functions. When this feature is active, its presence is mostly noticeable through the reduced execution time of computation-heavy functions. To use internal parallelism, you only need to enable it in ALGLIB, without any modifications to your existing code. This parallelism is available in a range of ALGLIB functions, covering areas from linear algebra to data analysis.

Callback parallelism, on the other hand, allows certain ALGLIB optimizers to send multiple parallel requests to user-defined callbacks. For example, numerical differentiation is inherently parallel as each component of a gradient can be computed independently. Similarly, optimizers like differential evolution can issue several requests simultaneously, which can be processed in parallel.

While it is ALGLIB who manages the creation and termination of computing threads to parallelize these batch requests, callback parallelism still requires you to provide a thread-safe callback. This callback must be capable of handling multiple simultaneous calls from multiple threads. Unlike internal parallelism, callback parallelism is not automatically managed by ALGLIB; it requires active cooperation from your side to ensure that your code is re-entrant and thread-safe.

Both types of parallelism can be used in various combinations. You might choose to employ only internal parallelism, only callback parallelism, both, or neither.

6.3.4 Activating parallelism

ALGLIB provides the flexibility to define parallelism settings at both global and function-specific levels. Global settings can be configured using the alglib::setglobalthreading function. For function-local settings, you specify them by passing an instance of the alglib::xparams type to the particular computational function you are using.

It's important to note that these function-local settings are actually call-local. This means that making simultaneous calls to the same function with different settings will result in the use of different parallelism types for each call.

Another important point is that function-local settings do not propagate to other (even connected) functions. E.g., if you create an instance of MINNLC optimizer with minnlccreate and then start optimization with minnlcoptimize, then it is the latter function which actually does the job. Any parallelism-related settings should be passed directly to minnlcoptimize and not to minnlccreate, because minnlccreate does nothing except for allocating data structures.

The alglib::xparams structure, which stores parallelism settings, can be set to the following values:

alglib::serial - indicates no internal parallelism.
alglib::parallel - enables internal parallelism.
alglib::serial_callbacks - disables callback parallelism.
alglib::parallel_callbacks - enables parallel callbacks.

You can combine values pertaining to different types of parallelism (internal and callback) using the OR operator, such as alglib::serial|alglib::parallel_callbacks. However, attempting to combine conflicting values like alglib::parallel_callbacks|alglib::serial_callbacks will result in an invalid setting.

If a value for a specific type of parallelism is not provided at the function-local level — such as alglib::serial, which defines settings for internal parallelism but omits callback parallelism — then that specific setting defaults to the global settings as determined by alglib::setglobalthreading. In cases where no global settings are specified for a particular parallelism type, serial execution is the default choice.

IMPORTANT: Activating internal parallelism in ALGLIB does not automatically mean that computations will be parallelized. ALGLIB assesses whether the problem at hand is sufficiently large to warrant efficient parallelization. This decision is made automatically by the library. In contrast, enabling callback parallelism always leads to ALGLIB creating worker threads for handling parallel requests.

This occurs even if the requests are too lightweight to benefit from multithreading. Be aware that this could potentially slow down your application due to the overhead of parallel synchronization, rather than speeding it up. The reason is that ALGLIB sees user-defined callbacks as black boxes. Therefore, when using callback parallelism, the decision on its effectiveness rests with you.

Another important point is that only some ALGLIB optimizers support callback parallelism. This feature was introduced only in ALGLIB 4.01.0, and some optimizers still have to be refactored.

6.3.5 Controlling cores count

ALGLIB automatically determines number of cores on application startup. On Windows it is done using GetSystemInfo() call. On POSIX systems ALGLIB performs sysconf(_SC_NPROCESSORS_ONLN) system call. This system call is supported by all modern POSIX-compatible systems: Solaris, Linux, FreeBSD, NetBSD, OpenBSD.

By default, ALGLIB uses all available cores (when told to parallelize calculations). Such behavior may be changed with setnworkers() call:

alglib::setnworkers(0) = use all cores
alglib::setnworkers(-1) = leave one core unused
alglib::setnworkers(-2) = leave two cores unused
alglib::setnworkers(+2) = use 2 cores (even if you have more)

You may want to specify maximum number of worker threads during compile time by means of preprocessor definition AE_NWORKERS=N. You can add this definition to compiler command line or change corresponding project settings in your IDE. Here N can be any positive number. ALGLIB will use exactly N worker threads, unless being told to use less by setnworkers() call.

Some old POSIX-compatible operating systems do not support sysconf(_SC_NPROCESSORS_ONLN) system call which is required in order to automatically determine number of active cores. On these systems you should specify number of cores manually at compile time. Without it ALGLIB will run in single-threaded mode.

6.3.6 More on parallel callbacks

As mentioned earlier, callback parallelism is a powerful feature that enables faster numerical differentiation and faster highly parallel optimization methods like differential evolution. To effectively use callback parallelism, it is essential that:

your callback is thread-safe and can be simultaneously called from several threads
your callback uses as little synchronization as possible

Achieving the latter can be challenging if your callback uses temporary data structures (one instance per invocation). Generally, it is inefficient to allocate temporary arrays on heap every time the callback is invoked. Memory allocation and deallocation requires synchronization which hampers parallelism. The only exception is when evaluating your objective function is so costly that memory allocation becomes a negligible factor.

Similarly, storing preallocated temporaries in a lock-protected pool can also be inefficient. Lock-free data structures relying on interlocked operations offer a somewhat better option, but even lock-free code still involves some synchronization, which can become a bottleneck.

The ideal strategy is to design your callback code to avoid synchronization entirely. This can be achieved by preallocating an array of alglib::getmaxnworkers() temporaries at the outset and using alglib::getcallbackworkeridx() to identify the worker thread that invoked the callback. This index allows retrieval of a temporary structure from the array. Since different worker threads have distinct indexes, this approach achieves thread safety with no synchronization required.

This option needs C++11 or later because it relies on thread-local storage being properly supported by the language. When compiled under earlier versions of the standard, alglib::getcallbackworkeridx() is not available.

6.3.7 SMT (CMT/hyper-threading) issues

Simultaneous multithreading (SMT) also known as Hyper-threading (Intel) and Cluster-based Multithreading (AMD) is a CPU design where several (usually two) logical cores share resources of one physical core. Say, on dual-core system with 2x HT scale factor you will see 4 logical cores. Each pair of these 4 cores, however, share same hardware resources. Thus, you may get only marginal speedup when running highly optimized software which fully utilizes CPU resources.

Say, if one thread occupies floating-point unit, another thread on the same physical core may work with integer numbers at the same time without any performance penalties. In this case you may get some speedup due to having additional cores. But if both threads keep FPU unit 100% busy, they won't get any multithreaded speedup.

So, if 2 math-intensive threads are dispatched by OS scheduler to different physical cores, you will get 2x speedup due to use of multithreading. But if these threads are dispatched to different logical cores - but same physical core - you won't get any speedup at all! One physical core will be 100% busy, and another one will be 100% idle. From the other side, if you start four threads instead of two, your system will be 100% utilized independently of thread scheduling details.

Let we stress it one more time - multithreading speedup on SMT systems is highly dependent on number of threads you are running and decisions made by OS scheduler. It is not 100% deterministic! With "true SMP" when you run 2 threads, you get 2x speedup (or 1.95, or 1.80 - it depends on algorithm, but this factor is always same). With SMT when you run 2 threads you may get your 2x speedup - or no speedup at all. Modern OS schedulers do a good job on single-socket hardware, but even in this "simple" case they give no guarantees of fair distribution of hardware resources. And things become a bit tricky when you work with multi-socket hardware. On SMT systems the only guaranteed way to 100% utilize your CPU is to create as many worker threads as there are logical cores. In this case OS scheduler has no chance to make its work in a wrong way.

6.4 Linking with Intel MKL

6.4.1 Using lightweight Intel MKL supplied by ALGLIB Project

Commercial edition of ALGLIB includes MKL extensions - special lightweight distribution of Intel MKL, highly optimized numerical library from Intel - and precompiled ALGLIB-MKL interface libraries. Linking your programs with MKL extensions allows you to run ALGLIB with maximum performance. MKL binaries are delivered for x86/x64 Windows and x64 Linux platforms.

Unlike the rest of the library, MKL extensions are distributed in binary-only form. ALGLIB itself is still distributed in source code form, but Intel MKL and ALGLIB-MKL interface are distributed as precompiled dynamic/static libraries. We can not distribute them in source because of license restrictions associated with Intel MKL. Also due to license restrictions we can not give you direct access to MKL functionality. You may use MKL to accelerate ALGLIB - without paying for MKL license - but you may not call its functions directly. It is technically possible, but strictly prohibited by both MKL's EULA and ALGLIB License Agreement. If you want to work with MKL, you should obtain separate license from Intel (as of 2018, free licenses are available).

MKL extensions are located in the /alglib-cpp/addons-mkl subdirectory of the ALGLIB distribution. This directory includes following files:

alglib???_32mkl.lib - x86 Windows import library (MSVC) for Intel MKL extensions
alglib???_64mkl.lib - x64 Windows import library (MSVC) for Intel MKL extensions
alglib???_32mkl.dll - x86 Windows binary with Intel MKL inside
alglib???_64mkl.dll - x64 Windows binary with Intel MKL inside
libalglib???_64mkl.so - x64 Linux binary with Intel MKL inside

Here ??? stands for specific ALGLIB version: 313 for ALGLIB 3.13, and so on. Files above are just MKL extensions - ALGLIB itself is not included in these binaries, and you still have to compile primary ALGLIB distribution.

In order to activate MKL extensions you should:

#define globally AE_MKL to activate ALGLIB-MKL connection.
additionally, #define globally AE_OS=AE_WINDOWS or AE_OS=AE_POSIX to activate multithreading capabilities
additionally, #define globally AE_CPU=AE_INTEL to use SIMD instructions provided by x86/x64 CPU's in the rest of ALGLIB
Depending on your OS, perform one of the following:
- Windows: choose 32-bit or 64-bit import library alglib???_32/64mkl.lib and link it with your application
- Linux: choose 64-bit SO file alglib???_64mkl.so and link it with your application
place DLL/SO binary into directory where it can be found during application startup (Windows: application dir; Linux: one of the system directories)

Several examples of ALGLIB+MKL usage are given in the 'compiling ALGLIB: examples' section.

6.4.2 Using your own installation of Intel MKL

If you bought separate license for Intel MKL, and want to use your own installation of MKL - and not our lightweight distribution - then you should compile ALGLIB as it was told in the previous section, with all necessary preprocessor definitions (AE_OS=AE_WINDOWS or AE_OS=AE_POSIX, AE_CPU=AE_INTEL and AE_MKL defined). But instead of linking with MKL Extensions binary, you should add to your project alglib2mkl.c file from addons-mkl directory and compile it (as C file) along with the rest of ALGLIB.

This C file implements interface between MKL and ALGLIB. Having this file in your project and defining AE_MKL preprocessor definition results in ALGLIB using MKL functions.

However, this C file is just interface! It is your responsibility to make sure that C/C++ compiler can find MKL headers, and appropriate MKL static/dynamic libraries are linked to your application.

7 Advanced topics

7.1 Using Red Zones to find memory access violations

Red Zone is a fixed size area added before and after each dynamically allocated block. Red Zone is filled by the special red zone control value during its allocation. When the dynamically allocated block is freed, its control value is checked. Any change means that someone (either ALGLIB or user code that works with ALGLIB-allocated arrays) performed an out-of-bounds write. Red Zones are essential for finding memory access errors that silently corrupt your data and/or crash your program.

ALGLIB for C++ comes with Red Zones support disabled by default. It can be enabled by #defining ALGLIB_REDZONE=512 (or some other multiple of 64 bytes) at the global level. Larger values mean that a larger buffer is added, and a better opportunity to detect large misses. E.g., if your write is just 8 bytes away from the array, a 64-byte red zone is enough to detect it. On the other hand, a 1024-byte miss needs a red zone at least that large.

ALGLIB checks the red zone of the dynamic array when it is deallocated by its destructor (or reallocated). In case the red zone is damaged, ALGLIB prints a message to stderr and terminates the program. Two kinds of errors are distinguished: writes past the end, and writes prior to the beginning. You can find out the exact variable being damaged by looking to the trace stack. It is one which is currently deallocated or reallocated.

Red zones add some memory overhead (insignificant for large arrays) and some modest performance overhead. Your program may become a few percent slower (in the very worst case - a few tens of percent).

7.2 Replacing stdlib rand() as an entropy source with OpenSSL implementation

ALGLIB uses and provides to its users its own random numbers generator. Whilst it is not cryptographically secure, it is good enough for scientific purposes. However, this generator has to be primed with an external entropy source - one which provides random or pseudorandom seeds.

In its default configuration ALGLIB obtains these seeds with stdlib rand(), whish was chosen as the default entropy source for portability reasons. In most cases it is acceptable - all we need is just two integers which will be used to prime a high-quality generator implemented in ALGLIB. However, some users may prefer to use other sources of entropy. Entropy source used by ALGLIB can be configured by #defining ALGLIB_ENTROPY_SRC to be one of the following:

ALGLIB_ENTROPY_SRC_STDRAND to use the default source - rand() function provided by stdlib.h
ALGLIB_ENTROPY_SRC_OPENSSL to use OpenSSL facilities as the entropy source. One has to have OpenSSL installed on its system to use this option.

7.3 Exception-free mode

ALGLIB for C++ can be compiled in exception-free mode, with exceptions (throw/try/catch constructs) being disabled at compiler level. Such feature is sometimes used by developers of embedded software.

ALGLIB uses two-level model of errors: "expected" errors (like degeneracy of linear system or inconsistency of linear constraints) are reported with dedicated completion codes, and "critical" errors (like malloc failures, unexpected NANs/INFs in the input data and so on) are reported with exceptions. The idea is that it is hard to put (and handle) completion codes in every ALGLIB function, so we use exceptions to signal errors which should never happen under normal circumstances.

Internally ALGLIB for C++ is implemented as C++ wrapper around computational core written in pure C. Thus, internals of ALGLIB core use C-specific methods of error handling - completion codes and setjmp/longjmp functions. These error handling strategies are combined with sophisticated machinery of C memory management which makes sure that not even a byte of dynamic memory is lost when we make longjmp to the error handler. So, the only point where C++ exceptions are actually used is a boundary between C core and C++ interface.

If you choose to use exceptions (default mode), ALGLIB will throw an exception with short textual description of the situation. And if you choose to work without exceptions, ALGLIB will set global error flag and silently return from the current function/constructor/... instead of throwing an exception. Due to portability issues this error flag is made to be a non-TLS variable, i.e. it is shared between different threads. So, you can use exception-free error handling only in single-threaded programs - although multithreaded programs won't break, there is no way to determine which thread caused an "exception without exceptions".

Exception-free method of reporting critical errors can be activated by #defining two preprocessor symbols at global level:

AE_NO_EXCEPTIONS - to switch from exception-based to exception-free code
AE_THREADING=AE_SERIAL_UNSAFE - to confirm that you are aware of limitations associated with exception-free mode (it does not support multithreading)

We must also note that exception-free mode is incompatible with OS-aware compiling: you can not have AE_OS=??? defined together with AE_NO_EXCEPTIONS.

After you #define all the necessary preprocessor symbols, two functions will appear in alglib namespace:

bool alglib::get_error_flag(const char **p_msg = NULL), which returns current error status (true is returned on error), with optional char** parameter used to get human-readable error message.
void alglib::clear_error_flag(), which clears error flag (ALGLIB functions set flag on failure, but do not clear it on successful calls)

You must check error flag after EVERY operation with ALGLIB objects and functions. In addition to calling computational ALGLIB functions, following kinds of operations may result in "exception":

calling default constructor for ALGLIB object (simply instantiating object may result in "exception"). Due to large size ALGLIB objects are allocated dynamically (they look like value types, but internally everything is stored in the heap memory). Thus every constructor, even default one, makes at least one malloc() call which may fail.
calling copy/assignment constructors for ALGLIB objects (same reason - malloc)
resizing arrays with setlength()/setcontents() and attaching to external memory with attach_to_ptr()

7.4 Partial compilation

Due to ALGLIB modular structure it is possible to selectively enable/disable some of its subpackages along with their dependencies. Deactivation of ALGLIB source code is performed at preprocessor level - compiler does not even see disabled code. Partial compilation can be used for two purposes:

to reduce code size in cases when linker is unable to remove unused code (happens with some compilers when ALGLIB is compiled as part of the shared library)
to reduce build time (disabled code is silently removed by preprocessor, thus no time is spent in the compiler)

You can activate partial compilation by #defining at global level following symbols:

AE_PARTIAL_BUILD - to disable everything not enabled by default
AE_COMPILE_SUBPACKAGE - to selectively enable subpackage SUBPACKAGE, with its name given in upper case (case sensitive). You may combine several definitions like this in order to enable several subpackages. Subpackage names can be found in the list of ALGLIB packages and subpackages.

7.5 Testing ALGLIB

There are three test suites in ALGLIB: computational tests, interface tests, extended tests. Computational tests are located in /tests/test_c.cpp. They are focused on numerical properties of algorithms, stress testing and "deep" tests (large automatically generated problems). They require significant amount of time to finish (tens of minutes).

Interface tests are located in /tests/test_i.cpp. These tests are focused on ability to correctly pass data between computational core and caller, ability to detect simple problems in inputs, and on ability to at least compile ALGLIB with your compiler. They are very fast (about a minute to finish including compilation time).

Extended tests are located in /tests/test_x.cpp. These tests are focused on testing some special properties (say, testing that cloning object indeed results in 100% independent copy being created) and performance of several chosen algorithms.

Running test suite is easy - just

compile one of these files (test_c.cpp, test_i.cpp or test_x.cpp) along with the rest of the library
launch executable you will get. It may take from several seconds (interface tests) to several minutes (computational tests) to get final results

If you want to be sure that ALGLIB will work with some sophisticated optimization settings, set corresponding flags during compile time. If your compiler/system are not in the list of supported ones, we recommend you to run both test suites. But if you are running out of time, run at least test_i.cpp.

8 ALGLIB packages and subpackages

8.1 `AlglibMisc` package
hqrnd		High quality random numbers generator
nearestneighbor		Nearest neighbor search: approximate and exact
xdebug		Debug functions to test ALGLIB interface generator

8.2 `DataAnalysis` package
bdss		Basic dataset functions
clustering		Clustering functions (hierarchical, k-means, k-means++)
datacomp		Backward compatibility functions
dforest		Decision forest classifier (regression model)
filters		Different filters used in data analysis
knn		K Nearest Neighbors classification/regression
lda		Linear discriminant analysis
linreg		Linear models
logit		Logit models
mcpd		Markov Chains for Population/proportional Data
mlpbase		Basic functions for neural networks
mlpe		Basic functions for neural ensemble models
mlptrain		Neural network training
pca		Principal component analysis
ssa		Singular Spectrum Analysis

8.3 `DiffEquations` package
odesolver		Ordinary differential equation solver

8.4 `FastTransforms` package
conv		Fast real/complex convolution
corr		Fast real/complex cross-correlation
fft		Real/complex FFT
fht		Real Fast Hartley Transform

8.5 `Integration` package
autogk		Adaptive 1-dimensional integration
gkq		Gauss-Kronrod quadrature generator
gq		Gaussian quadrature generator

8.6 `Interpolation` package
fitsphere		Fitting circle/sphere to data (least squares, minimum circumscribed, maximum inscribed, minimum zone)
idw		Inverse distance weighting: interpolation/fitting with improved Shepard-like algorithm
intcomp		Backward compatibility functions
lsfit		Fitting with least squates target function (linear and nonlinear least-squares)
parametric		Parametric curves
polint		Polynomial interpolation/fitting
ratint		Rational interpolation/fitting
rbf		Scattered N-dimensional interpolation with RBF models
spline1d		1D spline interpolation/fitting
spline2d		2D spline interpolation and fitting
spline3d		3D spline interpolation

8.7 `LinAlg` package
ablas		Level 2 and Level 3 BLAS operations
bdsvd		Bidiagonal SVD
evd		Direct and iterative eigensolvers
inverseupdate		Sherman-Morrison update of the inverse matrix
matdet		Determinant calculation
matgen		Random matrix generation
matinv		Matrix inverse
normestimator		Estimates norm of the sparse matrix (from below)
ortfac		Real/complex QR/LQ, bi(tri)diagonal, Hessenberg decompositions
rcond		Condition number estimates
schur		Schur decomposition
sparse		Sparse matrices
spdgevd		Generalized symmetric eigensolver
svd		Singular value decomposition
trfac		LU and Cholesky decompositions (dense and sparse)

8.8 `Optimization` package
minbc		Box constrained optimizer with fast activation of multiple constraints per step
minbleic		Bound constrained optimizer with additional linear equality/inequality constraints
mincg		Conjugate gradient optimizer
mincomp		Backward compatibility functions
mindf		Derivative-free and global optimization
minlbfgs		Limited memory BFGS optimizer
minlm		Improved Levenberg-Marquardt optimizer
minlp		Linear programming suite
minmo		Multi-objective optimizer
minnlc		Nonlinear programming solver (analytic gradient, numdiff, model-based DFO)
minns		Nonsmooth constrained optimizer
minqp		Quadratic optimization with linear, quadratic and conic constraints
nls		Nonlinear least squares (derivative-free)
optguardapi		OptGuard integrity checking for nonlinear models
opts		Internal service functions

8.9 `Solvers` package
directdensesolvers		Direct dense linear solvers
directsparsesolvers		Direct sparse linear solvers
iterativesparse		Sparse linear iterative solvers (GMRES)
lincg		Sparse linear CG solver
linlsqr		Sparse linear LSQR solver
nleq		Solvers for nonlinear equations
polynomialsolver		Polynomial solver

8.10 `SpecialFunctions` package
airyf		Airy functions
bessel		Bessel functions
betaf		Beta function
binomialdistr		Binomial distribution
chebyshev		Chebyshev polynomials
chisquaredistr		Chi-Square distribution
dawson		Dawson integral
elliptic		Elliptic integrals
expintegrals		Exponential integrals
fdistr		F-distribution
fresnel		Fresnel integrals
gammafunc		Gamma function
hermite		Hermite polynomials
ibetaf		Incomplete beta function
igammaf		Incomplete gamma function
jacobianelliptic		Jacobian elliptic functions
laguerre		Laguerre polynomials
legendre		Legendre polynomials
normaldistr		Univarite and bivariate normal distribution PDF and CDF
poissondistr		Poisson distribution
psif		Psi function
studenttdistr		Student's t-distribution
trigintegrals		Trigonometric integrals

8.11 `Statistics` package
basestat		Mean, variance, covariance, correlation, etc.
correlationtests		Hypothesis testing: correlation tests
jarquebera		Hypothesis testing: Jarque-Bera test
mannwhitneyu		Hypothesis testing: Mann-Whitney-U test
stest		Hypothesis testing: sign test
studentttests		Hypothesis testing: Student's t-test
variancetests		Hypothesis testing: F-test and one-sample variance test
wsr		Hypothesis testing: Wilcoxon signed rank test

`ablas` subpackage

Functions

cmatrixcopy
cmatrixgemm
cmatrixherk
cmatrixlefttrsm
cmatrixmv
cmatrixrank1
cmatrixrighttrsm
cmatrixsyrk
cmatrixtranspose
rmatrixcopy
rmatrixenforcesymmetricity
rmatrixgemm
rmatrixgemv
rmatrixgencopy
rmatrixger
rmatrixlefttrsm
rmatrixmv
rmatrixrank1
rmatrixrighttrsm
rmatrixsymv
rmatrixsyrk
rmatrixsyvmv
rmatrixtranspose
rmatrixtrsv
rvectorcopy

Examples

ablas_d_gemm		Matrix multiplication (single-threaded)
ablas_d_syrk		Symmetric rank-K update (single-threaded)

`cmatrixcopy` function

/*************************************************************************
Copy

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void cmatrixcopy(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);

`cmatrixgemm` function

/*************************************************************************
This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
* C is MxN general matrix
* op1(A) is MxK matrix
* op2(B) is KxN matrix
* "op" may be identity transformation, transposition, conjugate transposition

Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

IMPORTANT:

This function does NOT preallocate output matrix C, it MUST be preallocated
by caller prior to calling this function. In case C does not have  enough
space to store result, exception will be generated.

INPUT PARAMETERS
    M       -   matrix size, M>0
    N       -   matrix size, N>0
    K       -   matrix size, K>0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset
    JA      -   submatrix offset
    OpTypeA -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    B       -   matrix
    IB      -   submatrix offset
    JB      -   submatrix offset
    OpTypeB -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    Beta    -   coefficient
    C       -   matrix (PREALLOCATED, large enough to store result)
    IC      -   submatrix offset
    JC      -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     2009-2019
     Bochkanov Sergey
*************************************************************************/
void cmatrixgemm(const ae_int_t m, const ae_int_t n, const ae_int_t k, const alglib::complex alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const ae_int_t optypeb, const alglib::complex beta, complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixherk` function

/*************************************************************************
This subroutine calculates  C=alpha*A*A^H+beta*C  or  C=alpha*A^H*A+beta*C
where:
* C is NxN Hermitian matrix given by its upper/lower triangle
* A is NxK matrix when A*A^H is calculated, KxN matrix otherwise

Additional info:
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

INPUT PARAMETERS
    N       -   matrix size, N>=0
    K       -   matrix size, K>=0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset (row index)
    JA      -   submatrix offset (column index)
    OpTypeA -   multiplication type:
                * 0 - A*A^H is calculated
                * 2 - A^H*A is calculated
    Beta    -   coefficient
    C       -   preallocated input/output matrix
    IC      -   submatrix offset (row index)
    JC      -   submatrix offset (column index)
    IsUpper -   whether upper or lower triangle of C is updated;
                this function updates only one half of C, leaving
                other half unchanged (not referenced at all).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     16.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void cmatrixherk(const ae_int_t n, const ae_int_t k, const double alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixlefttrsm` function

/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
Multiplication result replaces X.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void cmatrixlefttrsm(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, complex_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);

`cmatrixmv` function

/*************************************************************************
Matrix-vector product: y := op(A)*x

INPUT PARAMETERS:
    M   -   number of rows of op(A)
            M>=0
    N   -   number of columns of op(A)
            N>=0
    A   -   target matrix
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    OpA -   operation type:
            * OpA=0     =>  op(A) = A
            * OpA=1     =>  op(A) = A^T
            * OpA=2     =>  op(A) = A^H
    X   -   input vector
    IX  -   subvector offset
    IY  -   subvector offset
    Y   -   preallocated matrix, must be large enough to store result

OUTPUT PARAMETERS:
    Y   -   vector which stores result

if M=0, then subroutine does nothing.
if N=0, Y is filled by zeros.


  -- ALGLIB routine --

     28.01.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixmv(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const complex_1d_array &x, const ae_int_t ix, complex_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);

`cmatrixrank1` function

/*************************************************************************
Rank-1 correction: A := A + u*v'

INPUT PARAMETERS:
    M   -   number of rows
    N   -   number of columns
    A   -   target matrix, MxN submatrix is updated
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset
*************************************************************************/
void cmatrixrank1(const ae_int_t m, const ae_int_t n, complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const complex_1d_array &u, const ae_int_t iu, const complex_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);

`cmatrixrighttrsm` function

/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition, conjugate transposition
Multiplication result replaces X.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
                * 2 - conjugate transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     20.01.2018
     Bochkanov Sergey
*************************************************************************/
void cmatrixrighttrsm(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, complex_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);

`cmatrixsyrk` function

/*************************************************************************
This subroutine is an older version of CMatrixHERK(), one with wrong  name
(it is HErmitian update, not SYmmetric). It  is  left  here  for  backward
compatibility.

  -- ALGLIB routine --
     16.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixsyrk(const ae_int_t n, const ae_int_t k, const double alpha, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, complex_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);

`cmatrixtranspose` function

/*************************************************************************
Cache-oblivous complex "copy-and-transpose"

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void cmatrixtranspose(const ae_int_t m, const ae_int_t n, const complex_2d_array &a, const ae_int_t ia, const ae_int_t ja, complex_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);

`rmatrixcopy` function

/*************************************************************************
Copy

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void rmatrixcopy(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);

`rmatrixenforcesymmetricity` function

/*************************************************************************
This code enforces symmetricy of the matrix by copying Upper part to lower
one (or vice versa).

INPUT PARAMETERS:
    A   -   matrix
    N   -   number of rows/columns
    IsUpper - whether we want to copy upper triangle to lower one (True)
            or vice versa (False).
*************************************************************************/
void rmatrixenforcesymmetricity(real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`rmatrixgemm` function

/*************************************************************************
This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where:
* C is MxN general matrix
* op1(A) is MxK matrix
* op2(B) is KxN matrix
* "op" may be identity transformation, transposition

Additional info:
* cache-oblivious algorithm is used.
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

IMPORTANT:

This function does NOT preallocate output matrix C, it MUST be preallocated
by caller prior to calling this function. In case C does not have  enough
space to store result, exception will be generated.

INPUT PARAMETERS
    M       -   matrix size, M>0
    N       -   matrix size, N>0
    K       -   matrix size, K>0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset
    JA      -   submatrix offset
    OpTypeA -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    B       -   matrix
    IB      -   submatrix offset
    JB      -   submatrix offset
    OpTypeB -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    Beta    -   coefficient
    C       -   PREALLOCATED output matrix, large enough to store result
    IC      -   submatrix offset
    JC      -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     2009-2019
     Bochkanov Sergey
*************************************************************************/
void rmatrixgemm(const ae_int_t m, const ae_int_t n, const ae_int_t k, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const ae_int_t optypeb, const double beta, real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixgemv` function

/*************************************************************************

*************************************************************************/
void rmatrixgemv(const ae_int_t m, const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const real_1d_array &x, const ae_int_t ix, const double beta, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);

`rmatrixgencopy` function

/*************************************************************************
Performs generalized copy: B := Beta*B + Alpha*A.

If Beta=0, then previous contents of B is simply ignored. If Alpha=0, then
A is ignored and not referenced. If both Alpha and Beta  are  zero,  B  is
filled by zeros.

Input parameters:
    M   -   number of rows
    N   -   number of columns
    Alpha-  coefficient
    A   -   source matrix, MxN submatrix is copied
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    Beta-   coefficient
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void rmatrixgencopy(const ae_int_t m, const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const double beta, real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);

`rmatrixger` function

/*************************************************************************
Rank-1 correction: A := A + alpha*u*v'

NOTE: this  function  expects  A  to  be  large enough to store result. No
      automatic preallocation happens for  smaller  arrays.  No  integrity
      checks is performed for sizes of A, u, v.

INPUT PARAMETERS:
    M   -   number of rows
    N   -   number of columns
    A   -   target matrix, MxN submatrix is updated
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    Alpha-  coefficient
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset


  -- ALGLIB routine --

     16.10.2017
     Bochkanov Sergey
*************************************************************************/
void rmatrixger(const ae_int_t m, const ae_int_t n, real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const double alpha, const real_1d_array &u, const ae_int_t iu, const real_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);

`rmatrixlefttrsm` function

/*************************************************************************
This subroutine calculates op(A^-1)*X where:
* X is MxN general matrix
* A is MxM upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition
Multiplication result replaces X.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void rmatrixlefttrsm(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, real_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);

`rmatrixmv` function

/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGEMV()
           which is more generic version of this function.

Matrix-vector product: y := op(A)*x

INPUT PARAMETERS:
    M   -   number of rows of op(A)
    N   -   number of columns of op(A)
    A   -   target matrix
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    OpA -   operation type:
            * OpA=0     =>  op(A) = A
            * OpA=1     =>  op(A) = A^T
    X   -   input vector
    IX  -   subvector offset
    IY  -   subvector offset
    Y   -   preallocated matrix, must be large enough to store result

OUTPUT PARAMETERS:
    Y   -   vector which stores result

if M=0, then subroutine does nothing.
if N=0, Y is filled by zeros.


  -- ALGLIB routine --

     28.01.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixmv(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t opa, const real_1d_array &x, const ae_int_t ix, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);

`rmatrixrank1` function

/*************************************************************************
IMPORTANT: this function is deprecated since ALGLIB 3.13. Use RMatrixGER()
           which is more generic version of this function.

Rank-1 correction: A := A + u*v'

INPUT PARAMETERS:
    M   -   number of rows
    N   -   number of columns
    A   -   target matrix, MxN submatrix is updated
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    U   -   vector #1
    IU  -   subvector offset
    V   -   vector #2
    IV  -   subvector offset
*************************************************************************/
void rmatrixrank1(const ae_int_t m, const ae_int_t n, real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const real_1d_array &u, const ae_int_t iu, const real_1d_array &v, const ae_int_t iv, const xparams _xparams = alglib::xdefault);

`rmatrixrighttrsm` function

/*************************************************************************
This subroutine calculates X*op(A^-1) where:
* X is MxN general matrix
* A is NxN upper/lower triangular/unitriangular matrix
* "op" may be identity transformation, transposition
Multiplication result replaces X.

INPUT PARAMETERS
    N   -   matrix size, N>=0
    M   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1]
    I1      -   submatrix offset
    J1      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X   -   matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1]
    I2  -   submatrix offset
    J2  -   submatrix offset

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void rmatrixrighttrsm(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t i1, const ae_int_t j1, const bool isupper, const bool isunit, const ae_int_t optype, real_2d_array &x, const ae_int_t i2, const ae_int_t j2, const xparams _xparams = alglib::xdefault);

`rmatrixsymv` function

/*************************************************************************

*************************************************************************/
void rmatrixsymv(const ae_int_t n, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const real_1d_array &x, const ae_int_t ix, const double beta, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);

`rmatrixsyrk` function

/*************************************************************************
This subroutine calculates  C=alpha*A*A^T+beta*C  or  C=alpha*A^T*A+beta*C
where:
* C is NxN symmetric matrix given by its upper/lower triangle
* A is NxK matrix when A*A^T is calculated, KxN matrix otherwise

Additional info:
* multiplication result replaces C. If Beta=0, C elements are not used in
  calculations (not multiplied by zero - just not referenced)
* if Alpha=0, A is not used (not multiplied by zero - just not referenced)
* if both Beta and Alpha are zero, C is filled by zeros.

INPUT PARAMETERS
    N       -   matrix size, N>=0
    K       -   matrix size, K>=0
    Alpha   -   coefficient
    A       -   matrix
    IA      -   submatrix offset (row index)
    JA      -   submatrix offset (column index)
    OpTypeA -   multiplication type:
                * 0 - A*A^T is calculated
                * 2 - A^T*A is calculated
    Beta    -   coefficient
    C       -   preallocated input/output matrix
    IC      -   submatrix offset (row index)
    JC      -   submatrix offset (column index)
    IsUpper -   whether C is upper triangular or lower triangular

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     16.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
void rmatrixsyrk(const ae_int_t n, const ae_int_t k, const double alpha, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const ae_int_t optypea, const double beta, real_2d_array &c, const ae_int_t ic, const ae_int_t jc, const bool isupper, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixsyvmv` function

/*************************************************************************

*************************************************************************/
double rmatrixsyvmv(const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const real_1d_array &x, const ae_int_t ix, real_1d_array &tmp, const xparams _xparams = alglib::xdefault);

`rmatrixtranspose` function

/*************************************************************************
Cache-oblivous real "copy-and-transpose"

Input parameters:
    M   -   number of rows
    N   -   number of columns
    A   -   source matrix, MxN submatrix is copied and transposed
    IA  -   submatrix offset (row index)
    JA  -   submatrix offset (column index)
    B   -   destination matrix, must be large enough to store result
    IB  -   submatrix offset (row index)
    JB  -   submatrix offset (column index)
*************************************************************************/
void rmatrixtranspose(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, real_2d_array &b, const ae_int_t ib, const ae_int_t jb, const xparams _xparams = alglib::xdefault);

`rmatrixtrsv` function

/*************************************************************************
This subroutine solves linear system op(A)*x=b where:
* A is NxN upper/lower triangular/unitriangular matrix
* X and B are Nx1 vectors
* "op" may be identity transformation or transposition

Solution replaces X.

IMPORTANT: * no overflow/underflow/denegeracy tests is performed.
           * no integrity checks for operand sizes, out-of-bounds accesses
             and so on is performed

INPUT PARAMETERS
    N   -   matrix size, N>=0
    A       -   matrix, actial matrix is stored in A[IA:IA+N-1,JA:JA+N-1]
    IA      -   submatrix offset
    JA      -   submatrix offset
    IsUpper -   whether matrix is upper triangular
    IsUnit  -   whether matrix is unitriangular
    OpType  -   transformation type:
                * 0 - no transformation
                * 1 - transposition
    X       -   right part, actual vector is stored in X[IX:IX+N-1]
    IX      -   offset

OUTPUT PARAMETERS
    X       -   solution replaces elements X[IX:IX+N-1]

  -- ALGLIB routine / remastering of LAPACK's DTRSV --
     (c) 2017 Bochkanov Sergey - converted to ALGLIB
     (c) 2016 Reference BLAS level1 routine (LAPACK version 3.7.0)
     Reference BLAS is a software package provided by Univ. of Tennessee,
     Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd.
*************************************************************************/
void rmatrixtrsv(const ae_int_t n, const real_2d_array &a, const ae_int_t ia, const ae_int_t ja, const bool isupper, const bool isunit, const ae_int_t optype, real_1d_array &x, const ae_int_t ix, const xparams _xparams = alglib::xdefault);

`rvectorcopy` function

/*************************************************************************
Copy

Input parameters:
    N   -   subvector size
    A   -   source vector, N elements are copied
    IA  -   source offset (first element index)
    B   -   destination vector, must be large enough to store result
    IB  -   destination offset (first element index)
*************************************************************************/
void rvectorcopy(const ae_int_t n, const real_1d_array &a, const ae_int_t ia, real_1d_array &b, const ae_int_t ib, const xparams _xparams = alglib::xdefault);

ablas_d_gemm example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array a = "[[2,1],[1,3]]";
        real_2d_array b = "[[2,1],[0,1]]";
        real_2d_array c = "[[0,0],[0,0]]";

        //
        // rmatrixgemm() function allows us to calculate matrix product C:=A*B or
        // to perform more general operation, C:=alpha*op1(A)*op2(B)+beta*C,
        // where A, B, C are rectangular matrices, op(X) can be X or X^T,
        // alpha and beta are scalars.
        //
        // This function:
        // * can apply transposition and/or multiplication by scalar to operands
        // * can use arbitrary part of matrices A/B (given by submatrix offset)
        // * can store result into arbitrary part of C
        // * for performance reasons requires C to be preallocated
        //
        // Parameters of this function are:
        // * M, N, K            -   sizes of op1(A) (which is MxK), op2(B) (which
        //                          is KxN) and C (which is MxN)
        // * Alpha              -   coefficient before A*B
        // * A, IA, JA          -   matrix A and offset of the submatrix
        // * OpTypeA            -   transformation type:
        //                          0 - no transformation
        //                          1 - transposition
        // * B, IB, JB          -   matrix B and offset of the submatrix
        // * OpTypeB            -   transformation type:
        //                          0 - no transformation
        //                          1 - transposition
        // * Beta               -   coefficient before C
        // * C, IC, JC          -   preallocated matrix C and offset of the submatrix
        //
        // Below we perform simple product C:=A*B (alpha=1, beta=0)
        //
        // IMPORTANT: this function works with preallocated C, which must be large
        //            enough to store multiplication result.
        //
        ae_int_t m = 2;
        ae_int_t n = 2;
        ae_int_t k = 2;
        double alpha = 1.0;
        ae_int_t ia = 0;
        ae_int_t ja = 0;
        ae_int_t optypea = 0;
        ae_int_t ib = 0;
        ae_int_t jb = 0;
        ae_int_t optypeb = 0;
        double beta = 0.0;
        ae_int_t ic = 0;
        ae_int_t jc = 0;
        rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
        printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[4,3],[2,4]]

        //
        // Now we try to apply some simple transformation to operands: C:=A*B^T
        //
        optypeb = 1;
        rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
        printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[5,1],[5,3]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

ablas_d_syrk example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // rmatrixsyrk() function allows us to calculate symmetric rank-K update
        // C := beta*C + alpha*A'*A, where C is square N*N matrix, A is square K*N
        // matrix, alpha and beta are scalars. It is also possible to update by
        // adding A*A' instead of A'*A.
        //
        // Parameters of this function are:
        // * N, K       -   matrix size
        // * Alpha      -   coefficient before A
        // * A, IA, JA  -   matrix and submatrix offsets
        // * OpTypeA    -   multiplication type:
        //                  * 0 - A*A^T is calculated
        //                  * 2 - A^T*A is calculated
        // * Beta       -   coefficient before C
        // * C, IC, JC  -   preallocated input/output matrix and submatrix offsets
        // * IsUpper    -   whether upper or lower triangle of C is updated;
        //                  this function updates only one half of C, leaving
        //                  other half unchanged (not referenced at all).
        //
        // Below we will show how to calculate simple product C:=A'*A
        //
        // NOTE: beta=0 and we do not use previous value of C, but still it
        //       MUST be preallocated.
        //
        ae_int_t n = 2;
        ae_int_t k = 1;
        double alpha = 1.0;
        ae_int_t ia = 0;
        ae_int_t ja = 0;
        ae_int_t optypea = 2;
        double beta = 0.0;
        ae_int_t ic = 0;
        ae_int_t jc = 0;
        bool isupper = true;
        real_2d_array a = "[[1,2]]";

        // preallocate space to store result
        real_2d_array c = "[[0,0],[0,0]]";

        // calculate product, store result into upper part of c
        rmatrixsyrk(n, k, alpha, a, ia, ja, optypea, beta, c, ic, jc, isupper);

        // output result.
        // IMPORTANT: lower triangle of C was NOT updated!
        printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[1,2],[0,4]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`airyf` subpackage

Functions

airy

Examples

`airy` function

/*************************************************************************
Airy function

Solution of the differential equation

y"(x) = xy.

The function returns the two independent solutions Ai, Bi
and their first derivatives Ai'(x), Bi'(x).

Evaluation is by power series summation for small x,
by rational minimax approximations for large x.



ACCURACY:
Error criterion is absolute when function <= 1, relative
when function > 1, except * denotes relative error criterion.
For large negative x, the absolute error increases as x^1.5.
For large positive x, the relative error increases as x^1.5.

Arithmetic  domain   function  # trials      peak         rms
IEEE        -10, 0     Ai        10000       1.6e-15     2.7e-16
IEEE          0, 10    Ai        10000       2.3e-14*    1.8e-15*
IEEE        -10, 0     Ai'       10000       4.6e-15     7.6e-16
IEEE          0, 10    Ai'       10000       1.8e-14*    1.5e-15*
IEEE        -10, 10    Bi        30000       4.2e-15     5.3e-16
IEEE        -10, 10    Bi'       30000       4.9e-15     7.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void airy(const double x, double &ai, double &aip, double &bi, double &bip, const xparams _xparams = alglib::xdefault);

`autogk` subpackage

Integrating f=exp(x) by adaptive integrator

`autogkreport` class

/*************************************************************************
Integration report:
* TerminationType = completetion code:
    * -5    non-convergence of Gauss-Kronrod nodes
            calculation subroutine.
    * -1    incorrect parameters were specified
    *  1    OK
* Rep.NFEV countains number of function calculations
* Rep.NIntervals contains number of intervals [a,b]
  was partitioned into.
*************************************************************************/
class autogkreport
{
public:
    autogkreport();
    autogkreport(const autogkreport &rhs);
    autogkreport& operator=(const autogkreport &rhs);
    virtual ~autogkreport();
    ae_int_t terminationtype;
    ae_int_t nfev;
    ae_int_t nintervals;
};

`autogkstate` class

/*************************************************************************
This structure stores state of the integration algorithm.

Although this class has public fields,  they are not intended for external
use. You should use ALGLIB functions to work with this class:
* autogksmooth()/AutoGKSmoothW()/... to create objects
* autogkintegrate() to begin integration
* autogkresults() to get results
*************************************************************************/
class autogkstate
{
public:
    autogkstate();
    autogkstate(const autogkstate &rhs);
    autogkstate& operator=(const autogkstate &rhs);
    virtual ~autogkstate();
};

`autogkintegrate` function

/*************************************************************************
This function is used to start iterations of the 1-dimensional integrator

It accepts following parameters:
    func    -   callback which calculates f(x) for given x
    ptr     -   optional pointer which is passed to func; can be NULL


  -- ALGLIB --
     Copyright 07.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogkintegrate(autogkstate &state,
    void (*func)(double x, double xminusa, double bminusx, double &y, void *ptr),
    void *ptr = NULL, const xparams _xparams = alglib::xdefault);

Examples: [1]

`autogkiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool autogkiteration(autogkstate &state, const xparams _xparams = alglib::xdefault);

`autogkresults` function

/*************************************************************************
Adaptive integration results

Called after AutoGKIteration returned False.

Input parameters:
    State   -   algorithm state (used by AutoGKIteration).

Output parameters:
    V       -   integral(f(x)dx,a,b)
    Rep     -   optimization report (see AutoGKReport description)

  -- ALGLIB --
     Copyright 14.11.2007 by Bochkanov Sergey
*************************************************************************/
void autogkresults(const autogkstate &state, double &v, autogkreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`autogksingular` function

/*************************************************************************
Integration on a finite interval [A,B].
Integrand have integrable singularities at A/B.

F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B,  with known
alpha/beta (alpha>-1, beta>-1).  If alpha/beta  are  not known,  estimates
from below can be used (but these estimates should be greater than -1 too).

One  of  alpha/beta variables (or even both alpha/beta) may be equal to 0,
which means than function F(x) is non-singular at A/B. Anyway (singular at
bounds or not), function F(x) is supposed to be continuous on (A,B).

Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)
    Alpha   -   power-law coefficient of the F(x) at A,
                Alpha>-1
    Beta    -   power-law coefficient of the F(x) at B,
                Beta>-1

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmooth, AutoGKSmoothW, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogksingular(const double a, const double b, const double alpha, const double beta, autogkstate &state, const xparams _xparams = alglib::xdefault);

`autogksmooth` function

/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].

Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.

Algorithm works well only with smooth integrands.  It  may  be  used  with
continuous non-smooth integrands, but with  less  performance.

It should never be used with integrands which have integrable singularities
at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
cases.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmoothW, AutoGKSingular, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogksmooth(const double a, const double b, autogkstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`autogksmoothw` function

/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].

This subroutine is same as AutoGKSmooth(), but it guarantees that interval
[a,b] is partitioned into subintervals which have width at most XWidth.

Subroutine  can  be  used  when  integrating nearly-constant function with
narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
subroutine can overlook them.

INPUT PARAMETERS:
    A, B    -   interval boundaries (A<B, A=B or A>B)

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state

SEE ALSO
    AutoGKSmooth, AutoGKSingular, AutoGKResults.


  -- ALGLIB --
     Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
void autogksmoothw(const double a, const double b, const double xwidth, autogkstate &state, const xparams _xparams = alglib::xdefault);

autogk_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "integration.h"

using namespace alglib;
void int_function_1_func(double x, double xminusa, double bminusx, double &y, void *ptr) 
{
    // this callback calculates f(x)=exp(x)
    y = exp(x);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates integration of f=exp(x) on [0,1]:
        // * first, autogkstate is initialized
        // * then we call integration function
        // * and finally we obtain results with autogkresults() call
        //
        double a = 0;
        double b = 1;
        autogkstate s;
        double v;
        autogkreport rep;

        autogksmooth(a, b, s);
        alglib::autogkintegrate(s, int_function_1_func);
        autogkresults(s, v, rep);

        printf("%.2f\n", double(v)); // EXPECTED: 1.7182
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`basestat` subpackage

Functions

cov2
covm
covm2
pearsoncorr2
pearsoncorrelation
pearsoncorrm
pearsoncorrm2
rankdata
rankdatacentered
sampleadev
samplekurtosis
samplemean
samplemedian
samplemoments
samplepercentile
sampleskewness
samplevariance
spearmancorr2
spearmancorrm
spearmancorrm2
spearmanrankcorrelation

Examples

basestat_d_base		Basic functionality (moments, adev, median, percentile)
basestat_d_c2		Correlation (covariance) between two random variables
basestat_d_cm		Correlation (covariance) between components of random vector
basestat_d_cm2		Correlation (covariance) between two random vectors

`cov2` function

/*************************************************************************
2-sample covariance

Input parameters:
    X       -   sample 1 (array indexes: [0..N-1])
    Y       -   sample 2 (array indexes: [0..N-1])
    N       -   N>=0, sample size:
                * if given, only N leading elements of X/Y are processed
                * if not given, automatically determined from input sizes

Result:
    covariance (zero for N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
double cov2(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double cov2(const real_1d_array &x, const real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1]

`covm` function

/*************************************************************************
Covariance matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X are used
            * if not given, automatically determined from input size
    M   -   M>0, number of variables:
            * if given, only leading M columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M,M], covariance matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void covm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c, const xparams _xparams = alglib::xdefault);
void covm(const real_2d_array &x, real_2d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`covm2` function

/*************************************************************************
Cross-covariance matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M1], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    Y   -   array[N,M2], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X/Y are used
            * if not given, automatically determined from input sizes
    M1  -   M1>0, number of variables in X:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size
    M2  -   M2>0, number of variables in Y:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M1,M2], cross-covariance matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void covm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c, const xparams _xparams = alglib::xdefault);
void covm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`pearsoncorr2` function

/*************************************************************************
Pearson product-moment correlation coefficient

Input parameters:
    X       -   sample 1 (array indexes: [0..N-1])
    Y       -   sample 2 (array indexes: [0..N-1])
    N       -   N>=0, sample size:
                * if given, only N leading elements of X/Y are processed
                * if not given, automatically determined from input sizes

Result:
    Pearson product-moment correlation coefficient
    (zero for N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
double pearsoncorr2(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double pearsoncorr2(const real_1d_array &x, const real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1]

`pearsoncorrelation` function

/*************************************************************************
Obsolete function, we recommend to use PearsonCorr2().

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double pearsoncorrelation(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`pearsoncorrm` function

/*************************************************************************
Pearson product-moment correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X are used
            * if not given, automatically determined from input size
    M   -   M>0, number of variables:
            * if given, only leading M columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M,M], correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void pearsoncorrm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c, const xparams _xparams = alglib::xdefault);
void pearsoncorrm(const real_2d_array &x, real_2d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`pearsoncorrm2` function

/*************************************************************************
Pearson product-moment cross-correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M1], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    Y   -   array[N,M2], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X/Y are used
            * if not given, automatically determined from input sizes
    M1  -   M1>0, number of variables in X:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size
    M2  -   M2>0, number of variables in Y:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void pearsoncorrm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c, const xparams _xparams = alglib::xdefault);
void pearsoncorrm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rankdata` function

/*************************************************************************
This function replaces data in XY by their ranks:
* XY is processed row-by-row
* rows are processed separately
* tied data are correctly handled (tied ranks are calculated)
* ranking starts from 0, ends at NFeatures-1
* sum of within-row values is equal to (NFeatures-1)*NFeatures/2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points
    NFeatures-  number of features

OUTPUT PARAMETERS:
    XY      -   data are replaced by their within-row ranks;
                ranking starts from 0, ends at NFeatures-1

  -- ALGLIB --
     Copyright 18.04.2013 by Bochkanov Sergey
*************************************************************************/
void rankdata(real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures, const xparams _xparams = alglib::xdefault);
void rankdata(real_2d_array &xy, const xparams _xparams = alglib::xdefault);

`rankdatacentered` function

/*************************************************************************
This function replaces data in XY by their CENTERED ranks:
* XY is processed row-by-row
* rows are processed separately
* tied data are correctly handled (tied ranks are calculated)
* centered ranks are just usual ranks, but centered in such way  that  sum
  of within-row values is equal to 0.0.
* centering is performed by subtracting mean from each row, i.e it changes
  mean value, but does NOT change higher moments

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points
    NFeatures-  number of features

OUTPUT PARAMETERS:
    XY      -   data are replaced by their within-row ranks;
                ranking starts from 0, ends at NFeatures-1

  -- ALGLIB --
     Copyright 18.04.2013 by Bochkanov Sergey
*************************************************************************/
void rankdatacentered(real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures, const xparams _xparams = alglib::xdefault);
void rankdatacentered(real_2d_array &xy, const xparams _xparams = alglib::xdefault);

`sampleadev` function

/*************************************************************************
ADev

Input parameters:
    X   -   sample
    N   -   N>=0, sample size:
            * if given, only leading N elements of X are processed
            * if not given, automatically determined from size of X

Output parameters:
    ADev-   ADev

  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void sampleadev(const real_1d_array &x, const ae_int_t n, double &adev, const xparams _xparams = alglib::xdefault);
void sampleadev(const real_1d_array &x, double &adev, const xparams _xparams = alglib::xdefault);

Examples: [1]

`samplekurtosis` function

/*************************************************************************
Calculation of the kurtosis.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Kurtosis' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double samplekurtosis(const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double samplekurtosis(const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`samplemean` function

/*************************************************************************
Calculation of the mean.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Mean' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double samplemean(const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double samplemean(const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`samplemedian` function

/*************************************************************************
Median calculation.

Input parameters:
    X   -   sample (array indexes: [0..N-1])
    N   -   N>=0, sample size:
            * if given, only leading N elements of X are processed
            * if not given, automatically determined from size of X

Output parameters:
    Median

  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void samplemedian(const real_1d_array &x, const ae_int_t n, double &median, const xparams _xparams = alglib::xdefault);
void samplemedian(const real_1d_array &x, double &median, const xparams _xparams = alglib::xdefault);

Examples: [1]

`samplemoments` function

/*************************************************************************
Calculation of the distribution moments: mean, variance, skewness, kurtosis.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

OUTPUT PARAMETERS
    Mean    -   mean.
    Variance-   variance.
    Skewness-   skewness (if variance<>0; zero otherwise).
    Kurtosis-   kurtosis (if variance<>0; zero otherwise).

NOTE: variance is calculated by dividing sum of squares by N-1, not N.

  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
void samplemoments(const real_1d_array &x, const ae_int_t n, double &mean, double &variance, double &skewness, double &kurtosis, const xparams _xparams = alglib::xdefault);
void samplemoments(const real_1d_array &x, double &mean, double &variance, double &skewness, double &kurtosis, const xparams _xparams = alglib::xdefault);

Examples: [1]

`samplepercentile` function

/*************************************************************************
Percentile calculation.

Input parameters:
    X   -   sample (array indexes: [0..N-1])
    N   -   N>=0, sample size:
            * if given, only leading N elements of X are processed
            * if not given, automatically determined from size of X
    P   -   percentile (0<=P<=1)

Output parameters:
    V   -   percentile

  -- ALGLIB --
     Copyright 01.03.2008 by Bochkanov Sergey
*************************************************************************/
void samplepercentile(const real_1d_array &x, const ae_int_t n, const double p, double &v, const xparams _xparams = alglib::xdefault);
void samplepercentile(const real_1d_array &x, const double p, double &v, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sampleskewness` function

/*************************************************************************
Calculation of the skewness.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Skewness' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double sampleskewness(const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double sampleskewness(const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`samplevariance` function

/*************************************************************************
Calculation of the variance.

INPUT PARAMETERS:
    X       -   sample
    N       -   N>=0, sample size:
                * if given, only leading N elements of X are processed
                * if not given, automatically determined from size of X

NOTE:

This function return result  which calculated by 'SampleMoments' function
and stored at 'Variance' variable.


  -- ALGLIB --
     Copyright 06.09.2006 by Bochkanov Sergey
*************************************************************************/
double samplevariance(const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double samplevariance(const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`spearmancorr2` function

/*************************************************************************
Spearman's rank correlation coefficient

Input parameters:
    X       -   sample 1 (array indexes: [0..N-1])
    Y       -   sample 2 (array indexes: [0..N-1])
    N       -   N>=0, sample size:
                * if given, only N leading elements of X/Y are processed
                * if not given, automatically determined from input sizes

Result:
    Spearman's rank correlation coefficient
    (zero for N=0 or N=1)

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double spearmancorr2(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double spearmancorr2(const real_1d_array &x, const real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spearmancorrm` function

/*************************************************************************
Spearman's rank correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X are used
            * if not given, automatically determined from input size
    M   -   M>0, number of variables:
            * if given, only leading M columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M,M], correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void spearmancorrm(const real_2d_array &x, const ae_int_t n, const ae_int_t m, real_2d_array &c, const xparams _xparams = alglib::xdefault);
void spearmancorrm(const real_2d_array &x, real_2d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spearmancorrm2` function

/*************************************************************************
Spearman's rank cross-correlation matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    X   -   array[N,M1], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    Y   -   array[N,M2], sample matrix:
            * J-th column corresponds to J-th variable
            * I-th row corresponds to I-th observation
    N   -   N>=0, number of observations:
            * if given, only leading N rows of X/Y are used
            * if not given, automatically determined from input sizes
    M1  -   M1>0, number of variables in X:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size
    M2  -   M2>0, number of variables in Y:
            * if given, only leading M1 columns of X are used
            * if not given, automatically determined from input size

OUTPUT PARAMETERS:
    C   -   array[M1,M2], cross-correlation matrix (zero if N=0 or N=1)

  -- ALGLIB --
     Copyright 28.10.2010 by Bochkanov Sergey
*************************************************************************/
void spearmancorrm2(const real_2d_array &x, const real_2d_array &y, const ae_int_t n, const ae_int_t m1, const ae_int_t m2, real_2d_array &c, const xparams _xparams = alglib::xdefault);
void spearmancorrm2(const real_2d_array &x, const real_2d_array &y, real_2d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spearmanrankcorrelation` function

/*************************************************************************
Obsolete function, we recommend to use SpearmanCorr2().

    -- ALGLIB --
    Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
double spearmanrankcorrelation(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const xparams _xparams = alglib::xdefault);

basestat_d_base example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_1d_array x = "[0,1,4,9,16,25,36,49,64,81]";
        double mean;
        double variance;
        double skewness;
        double kurtosis;
        double adev;
        double p;
        double v;

        //
        // Here we demonstrate calculation of sample moments
        // (mean, variance, skewness, kurtosis)
        //
        samplemoments(x, mean, variance, skewness, kurtosis);
        printf("%.1f\n", double(mean)); // EXPECTED: 28.5
        printf("%.1f\n", double(variance)); // EXPECTED: 801.1667
        printf("%.1f\n", double(skewness)); // EXPECTED: 0.5751
        printf("%.1f\n", double(kurtosis)); // EXPECTED: -1.2666

        //
        // Average deviation
        //
        sampleadev(x, adev);
        printf("%.1f\n", double(adev)); // EXPECTED: 23.2

        //
        // Median and percentile
        //
        samplemedian(x, v);
        printf("%.1f\n", double(v)); // EXPECTED: 20.5
        p = 0.5;
        samplepercentile(x, p, v);
        printf("%.1f\n", double(v)); // EXPECTED: 20.5
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

basestat_d_c2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We have two samples - x and y, and want to measure dependency between them
        //
        real_1d_array x = "[0,1,4,9,16,25,36,49,64,81]";
        real_1d_array y = "[0,1,2,3,4,5,6,7,8,9]";
        double v;

        //
        // Three dependency measures are calculated:
        // * covariation
        // * Pearson correlation
        // * Spearman rank correlation
        //
        v = cov2(x, y);
        printf("%.2f\n", double(v)); // EXPECTED: 82.5
        v = pearsoncorr2(x, y);
        printf("%.2f\n", double(v)); // EXPECTED: 0.9627
        v = spearmancorr2(x, y);
        printf("%.2f\n", double(v)); // EXPECTED: 1.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

basestat_d_cm example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // X is a sample matrix:
        // * I-th row corresponds to I-th observation
        // * J-th column corresponds to J-th variable
        //
        real_2d_array x = "[[1,0,1],[1,1,0],[-1,1,0],[-2,-1,1],[-1,0,9]]";
        real_2d_array c;

        //
        // Three dependency measures are calculated:
        // * covariation
        // * Pearson correlation
        // * Spearman rank correlation
        //
        // Result is stored into C, with C[i,j] equal to correlation
        // (covariance) between I-th and J-th variables of X.
        //
        covm(x, c);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.80,0.60,-1.40],[0.60,0.70,-0.80],[-1.40,-0.80,14.70]]
        pearsoncorrm(x, c);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.000,0.535,-0.272],[0.535,1.000,-0.249],[-0.272,-0.249,1.000]]
        spearmancorrm(x, c);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.000,0.556,-0.306],[0.556,1.000,-0.750],[-0.306,-0.750,1.000]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

basestat_d_cm2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // X and Y are sample matrices:
        // * I-th row corresponds to I-th observation
        // * J-th column corresponds to J-th variable
        //
        real_2d_array x = "[[1,0,1],[1,1,0],[-1,1,0],[-2,-1,1],[-1,0,9]]";
        real_2d_array y = "[[2,3],[2,1],[-1,6],[-9,9],[7,1]]";
        real_2d_array c;

        //
        // Three dependency measures are calculated:
        // * covariation
        // * Pearson correlation
        // * Spearman rank correlation
        //
        // Result is stored into C, with C[i,j] equal to correlation
        // (covariance) between I-th variable of X and J-th variable of Y.
        //
        covm2(x, y, c);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[4.100,-3.250],[2.450,-1.500],[13.450,-5.750]]
        pearsoncorrm2(x, y, c);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[0.519,-0.699],[0.497,-0.518],[0.596,-0.433]]
        spearmancorrm2(x, y, c);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[0.541,-0.649],[0.216,-0.433],[0.433,-0.135]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`bdss` subpackage

Functions

dsoptimalsplit2
dsoptimalsplit2fast

Examples

`dsoptimalsplit2` function

/*************************************************************************
Optimal binary classification

Algorithms finds optimal (=with minimal cross-entropy) binary partition.
Internal subroutine.

INPUT PARAMETERS:
    A       -   array[0..N-1], variable
    C       -   array[0..N-1], class numbers (0 or 1).
    N       -   array size

OUTPUT PARAMETERS:
    Info    -   completetion code:
                * -3, all values of A[] are same (partition is impossible)
                * -2, one of C[] is incorrect (<0, >1)
                * -1, incorrect pararemets were passed (N<=0).
                *  1, OK
    Threshold-  partiton boundary. Left part contains values which are
                strictly less than Threshold. Right part contains values
                which are greater than or equal to Threshold.
    PAL, PBL-   probabilities P(0|v<Threshold) and P(1|v<Threshold)
    PAR, PBR-   probabilities P(0|v>=Threshold) and P(1|v>=Threshold)
    CVE     -   cross-validation estimate of cross-entropy

  -- ALGLIB --
     Copyright 22.05.2008 by Bochkanov Sergey
*************************************************************************/
void dsoptimalsplit2(const real_1d_array &a, const integer_1d_array &c, const ae_int_t n, ae_int_t &info, double &threshold, double &pal, double &pbl, double &par, double &pbr, double &cve, const xparams _xparams = alglib::xdefault);

`dsoptimalsplit2fast` function

/*************************************************************************
Optimal partition, internal subroutine. Fast version.

Accepts:
    A       array[0..N-1]       array of attributes     array[0..N-1]
    C       array[0..N-1]       array of class labels
    TiesBuf array[0..N]         temporaries (ties)
    CntBuf  array[0..2*NC-1]    temporaries (counts)
    Alpha                       centering factor (0<=alpha<=1, recommended value - 0.05)
    BufR    array[0..N-1]       temporaries
    BufI    array[0..N-1]       temporaries

Output:
    Info    error code (">0"=OK, "<0"=bad)
    RMS     training set RMS error
    CVRMS   leave-one-out RMS error

Note:
    content of all arrays is changed by subroutine;
    it doesn't allocate temporaries.

  -- ALGLIB --
     Copyright 11.12.2008 by Bochkanov Sergey
*************************************************************************/
void dsoptimalsplit2fast(real_1d_array &a, integer_1d_array &c, integer_1d_array &tiesbuf, integer_1d_array &cntbuf, real_1d_array &bufr, integer_1d_array &bufi, const ae_int_t n, const ae_int_t nc, const double alpha, ae_int_t &info, double &threshold, double &rms, double &cvrms, const xparams _xparams = alglib::xdefault);

`bdsvd` subpackage

Functions

rmatrixbdsvd

Examples

`rmatrixbdsvd` function

/*************************************************************************
Singular value decomposition of a bidiagonal matrix (extended algorithm)

COMMERCIAL EDITION OF ALGLIB:

  ! Commercial version of ALGLIB includes one  important  improvement   of
  ! this function, which can be used from C++ and C#:
  ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
  !
  ! Intel MKL gives approximately constant  (with  respect  to  number  of
  ! worker threads) acceleration factor which depends on CPU  being  used,
  ! problem  size  and  "baseline"  ALGLIB  edition  which  is  used   for
  ! comparison.
  !
  ! Generally, commercial ALGLIB is several times faster than  open-source
  ! generic C edition, and many times faster than open-source C# edition.
  !
  ! Multithreaded acceleration is NOT supported for this function.
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The algorithm performs the singular value decomposition  of  a  bidiagonal
matrix B (upper or lower) representing it as B = Q*S*P^T, where Q and  P -
orthogonal matrices, S - diagonal matrix with non-negative elements on the
main diagonal, in descending order.

The  algorithm  finds  singular  values.  In  addition,  the algorithm can
calculate  matrices  Q  and P (more precisely, not the matrices, but their
product  with  given  matrices U and VT - U*Q and (P^T)*VT)).  Of  course,
matrices U and VT can be of any type, including identity. Furthermore, the
algorithm can calculate Q'*C (this product is calculated more  effectively
than U*Q,  because  this calculation operates with rows instead  of matrix
columns).

The feature of the algorithm is its ability to find  all  singular  values
including those which are arbitrarily close to 0  with  relative  accuracy
close to  machine precision. If the parameter IsFractionalAccuracyRequired
is set to True, all singular values will have high relative accuracy close
to machine precision. If the parameter is set to False, only  the  biggest
singular value will have relative accuracy  close  to  machine  precision.
The absolute error of other singular values is equal to the absolute error
of the biggest singular value.

Input parameters:
    D       -   main diagonal of matrix B.
                Array whose index ranges within [0..N-1].
    E       -   superdiagonal (or subdiagonal) of matrix B.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix B.
    IsUpper -   True, if the matrix is upper bidiagonal.
    IsFractionalAccuracyRequired -
                THIS PARAMETER IS IGNORED SINCE ALGLIB 3.5.0
                SINGULAR VALUES ARE ALWAYS SEARCHED WITH HIGH ACCURACY.
    U       -   matrix to be multiplied by Q.
                Array whose indexes range within [0..NRU-1, 0..N-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..NRU-1, 0..N-1] will be multiplied by Q.
    NRU     -   number of rows in matrix U.
    C       -   matrix to be multiplied by Q'.
                Array whose indexes range within [0..N-1, 0..NCC-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..N-1, 0..NCC-1] will be multiplied by Q'.
    NCC     -   number of columns in matrix C.
    VT      -   matrix to be multiplied by P^T.
                Array whose indexes range within [0..N-1, 0..NCVT-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..N-1, 0..NCVT-1] will be multiplied by P^T.
    NCVT    -   number of columns in matrix VT.

Output parameters:
    D       -   singular values of matrix B in descending order.
    U       -   if NRU>0, contains matrix U*Q.
    VT      -   if NCVT>0, contains matrix (P^T)*VT.
    C       -   if NCC>0, contains matrix Q'*C.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

NOTE: multiplication U*Q is performed by means of transposition to internal
      buffer, multiplication and backward transposition. It helps to avoid
      costly columnwise operations and speed-up algorithm.

Additional information:
    The type of convergence is controlled by the internal  parameter  TOL.
    If the parameter is greater than 0, the singular values will have
    relative accuracy TOL. If TOL<0, the singular values will have
    absolute accuracy ABS(TOL)*norm(B).
    By default, |TOL| falls within the range of 10*Epsilon and 100*Epsilon,
    where Epsilon is the machine precision. It is not  recommended  to  use
    TOL less than 10*Epsilon since this will  considerably  slow  down  the
    algorithm and may not lead to error decreasing.

History:
    * 31 March, 2007.
        changed MAXITR from 6 to 12.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1999.
*************************************************************************/
bool rmatrixbdsvd(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const bool isupper, const bool isfractionalaccuracyrequired, real_2d_array &u, const ae_int_t nru, real_2d_array &c, const ae_int_t ncc, real_2d_array &vt, const ae_int_t ncvt, const xparams _xparams = alglib::xdefault);

`bessel` subpackage

Functions

besseli0
besseli1
besselj0
besselj1
besseljn
besselk0
besselk1
besselkn
bessely0
bessely1
besselyn

Examples

`besseli0` function

/*************************************************************************
Modified Bessel function of order zero

Returns modified Bessel function of order zero of the
argument.

The function is defined as i0(x) = j0( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       5.8e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli0(const double x, const xparams _xparams = alglib::xdefault);

`besseli1` function

/*************************************************************************
Modified Bessel function of order one

Returns modified Bessel function of order one of the
argument.

The function is defined as i1(x) = -i j1( ix ).

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.9e-15     2.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseli1(const double x, const xparams _xparams = alglib::xdefault);

`besselj0` function

/*************************************************************************
Bessel function of order zero

Returns Bessel function of order zero of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval the following rational
approximation is used:


       2         2
(w - r  ) (w - r  ) P (w) / Q (w)
      1         2    3       8

           2
where w = x  and the two r's are zeros of the function.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.

ACCURACY:

                     Absolute error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       60000       4.2e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj0(const double x, const xparams _xparams = alglib::xdefault);

`besselj1` function

/*************************************************************************
Bessel function of order one

Returns Bessel function of order one of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 24 term Chebyshev
expansion is used. In the second, the asymptotic
trigonometric representation is employed using two
rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       2.6e-16     1.1e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double besselj1(const double x, const xparams _xparams = alglib::xdefault);

`besseljn` function

/*************************************************************************
Bessel function of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The ratio of jn(x) to j0(x) is computed by backward
recurrence.  First the ratio jn/jn-1 is found by a
continued fraction expansion.  Then the recurrence
relating successive orders is applied until j0 or j1 is
reached.

If n = 0 or 1 the routine for j0 or j1 is called
directly.

ACCURACY:

                     Absolute error:
arithmetic   range      # trials      peak         rms
   IEEE      0, 30        5000       4.4e-16     7.9e-17


Not suitable for large n or x. Use jv() (fractional order) instead.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besseljn(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`besselk0` function

/*************************************************************************
Modified Bessel function, second kind, order zero

Returns modified Bessel function of the second kind
of order zero of the argument.

The range is partitioned into the two intervals [0,8] and
(8, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

Tested at 2000 random points between 0 and 8.  Peak absolute
error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk0(const double x, const xparams _xparams = alglib::xdefault);

`besselk1` function

/*************************************************************************
Modified Bessel function, second kind, order one

Computes the modified Bessel function of the second kind
of order one of the argument.

The range is partitioned into the two intervals [0,2] and
(2, infinity).  Chebyshev polynomial expansions are employed
in each interval.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.2e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselk1(const double x, const xparams _xparams = alglib::xdefault);

`besselkn` function

/*************************************************************************
Modified Bessel function, second kind, integer order

Returns modified Bessel function of the second kind
of order n of the argument.

The range is partitioned into the two intervals [0,9.55] and
(9.55, infinity).  An ascending power series is used in the
low range, and an asymptotic expansion in the high range.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        90000       1.8e-8      3.0e-10

Error is high only near the crossover point x = 9.55
between the two expansions used.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier
*************************************************************************/
double besselkn(const ae_int_t nn, const double x, const xparams _xparams = alglib::xdefault);

`bessely0` function

/*************************************************************************
Bessel function of the second kind, order zero

Returns Bessel function of the second kind, of order
zero, of the argument.

The domain is divided into the intervals [0, 5] and
(5, infinity). In the first interval a rational approximation
R(x) is employed to compute
  y0(x)  = R(x)  +   2 * log(x) * j0(x) / PI.
Thus a call to j0() is required.

In the second interval, the Hankel asymptotic expansion
is employed with two rational functions of degree 6/6
and 7/7.



ACCURACY:

 Absolute error, when y0(x) < 1; else relative error:

arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       1.3e-15     1.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely0(const double x, const xparams _xparams = alglib::xdefault);

`bessely1` function

/*************************************************************************
Bessel function of second kind of order one

Returns Bessel function of the second kind of order one
of the argument.

The domain is divided into the intervals [0, 8] and
(8, infinity). In the first interval a 25 term Chebyshev
expansion is used, and a call to j1() is required.
In the second, the asymptotic trigonometric representation
is employed using two rational functions of degree 5/5.

ACCURACY:

                     Absolute error:
arithmetic   domain      # trials      peak         rms
   IEEE      0, 30       30000       1.0e-15     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double bessely1(const double x, const xparams _xparams = alglib::xdefault);

`besselyn` function

/*************************************************************************
Bessel function of second kind of integer order

Returns Bessel function of order n, where n is a
(possibly negative) integer.

The function is evaluated by forward recurrence on
n, starting with values computed by the routines
y0() and y1().

If n = 0 or 1 the routine for y0 or y1 is called
directly.

ACCURACY:
                     Absolute error, except relative
                     when y > 1:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       30000       3.4e-15     4.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double besselyn(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`betaf` subpackage

Functions

beta

Examples

`beta` function

/*************************************************************************
Beta function


                  -     -
                 | (a) | (b)
beta( a, b )  =  -----------.
                    -
                   | (a+b)

For large arguments the logarithm of the function is
evaluated using lgam(), then exponentiated.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,30       30000       8.1e-14     1.1e-14

Cephes Math Library Release 2.0:  April, 1987
Copyright 1984, 1987 by Stephen L. Moshier
*************************************************************************/
double beta(const double a, const double b, const xparams _xparams = alglib::xdefault);

`binomialdistr` subpackage

Functions

binomialcdistribution
binomialdistribution
invbinomialdistribution

Examples

`binomialcdistribution` function

/*************************************************************************
Complemented binomial distribution

Returns the sum of the terms k+1 through n of the Binomial
probability density:

  n
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=k+1

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      6.7e-15     8.2e-16
 For p between 0 and .001:
   IEEE     0,100       100000      1.5e-13     2.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialcdistribution(const ae_int_t k, const ae_int_t n, const double p, const xparams _xparams = alglib::xdefault);

`binomialdistribution` function

/*************************************************************************
Binomial distribution

Returns the sum of the terms 0 through k of the Binomial
probability density:

  k
  --  ( n )   j      n-j
  >   (   )  p  (1-p)
  --  ( j )
 j=0

The terms are not summed directly; instead the incomplete
beta integral is employed, according to the formula

y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ).

The arguments must be positive, with p ranging from 0 to 1.

ACCURACY:

Tested at random points (a,b,p), with p between 0 and 1.

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      4.3e-15     2.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double binomialdistribution(const ae_int_t k, const ae_int_t n, const double p, const xparams _xparams = alglib::xdefault);

`invbinomialdistribution` function

/*************************************************************************
Inverse binomial distribution

Finds the event probability p such that the sum of the
terms 0 through k of the Binomial probability density
is equal to the given cumulative probability y.

This is accomplished using the inverse beta integral
function and the relation

1 - p = incbi( n-k, k+1, y ).

ACCURACY:

Tested at random points (a,b,p).

              a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between 0.001 and 1:
   IEEE     0,100       100000      2.3e-14     6.4e-16
   IEEE     0,10000     100000      6.6e-12     1.2e-13
 For p between 10^-6 and 0.001:
   IEEE     0,100       100000      2.0e-12     1.3e-14
   IEEE     0,10000     100000      1.5e-12     3.2e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invbinomialdistribution(const ae_int_t k, const ae_int_t n, const double y, const xparams _xparams = alglib::xdefault);

`chebyshev` subpackage

Functions

chebyshevcalculate
chebyshevcoefficients
chebyshevsum
fromchebyshev

Examples

`chebyshevcalculate` function

/*************************************************************************
Calculation of the value of the Chebyshev polynomials of the
first and second kinds.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument, -1 <= x <= 1

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevcalculate(const ae_int_t r, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`chebyshevcoefficients` function

/*************************************************************************
Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void chebyshevcoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`chebyshevsum` function

/*************************************************************************
Summation of Chebyshev polynomials using Clenshaw's recurrence formula.

This routine calculates
    c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x)
or
    c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x)
depending on the R.

Parameters:
    r   -   polynomial kind, either 1 or 2.
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Chebyshev polynomial at x
*************************************************************************/
double chebyshevsum(const real_1d_array &c, const ae_int_t r, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`fromchebyshev` function

/*************************************************************************
Conversion of a series of Chebyshev polynomials to a power series.

Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as
B[0] + B[1]*X + ... + B[N]*X^N.

Input parameters:
    A   -   Chebyshev series coefficients
    N   -   degree, N>=0

Output parameters
    B   -   power series coefficients
*************************************************************************/
void fromchebyshev(const real_1d_array &a, const ae_int_t n, real_1d_array &b, const xparams _xparams = alglib::xdefault);

`chisquaredistr` subpackage

Functions

chisquarecdistribution
chisquaredistribution
invchisquaredistribution

Examples

`chisquarecdistribution` function

/*************************************************************************
Complemented Chi-square distribution

Returns the area under the right hand tail (from x to
infinity) of the Chi square probability density function
with v degrees of freedom:

                                 inf.
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  x

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquarecdistribution(const double v, const double x, const xparams _xparams = alglib::xdefault);

`chisquaredistribution` function

/*************************************************************************
Chi-square distribution

Returns the area under the left hand tail (from 0 to x)
of the Chi square probability density function with
v degrees of freedom.


                                  x
                                   -
                       1          | |  v/2-1  -t/2
 P( x | v )   =   -----------     |   t      e     dt
                   v/2  -       | |
                  2    | (v/2)   -
                                  0

where x is the Chi-square variable.

The incomplete gamma integral is used, according to the
formula

y = chdtr( v, x ) = igam( v/2.0, x/2.0 ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double chisquaredistribution(const double v, const double x, const xparams _xparams = alglib::xdefault);

`invchisquaredistribution` function

/*************************************************************************
Inverse of complemented Chi-square distribution

Finds the Chi-square argument x such that the integral
from x to infinity of the Chi-square density is equal
to the given cumulative probability y.

This is accomplished using the inverse gamma integral
function and the relation

   x/2 = igami( df/2, y );

ACCURACY:

See inverse incomplete gamma function


Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double invchisquaredistribution(const double v, const double y, const xparams _xparams = alglib::xdefault);

`clustering` subpackage

Classes

ahcreport
clusterizerstate
kmeansreport

Functions

clusterizercreate
clusterizergetdistances
clusterizergetkclusters
clusterizerrunahc
clusterizerrunkmeans
clusterizerseparatedbycorr
clusterizerseparatedbydist
clusterizersetahcalgo
clusterizersetdistances
clusterizersetkmeansinit
clusterizersetkmeanslimits
clusterizersetpoints
clusterizersetseed

Examples

clst_ahc		Simple hierarchical clusterization with Euclidean distance function
clst_distance		Clusterization with different metric types
clst_kclusters		Obtaining K top clusters from clusterization tree
clst_kmeans		Simple k-means clusterization
clst_linkage		Clusterization with different linkage types

`ahcreport` class

/*************************************************************************
This structure  is used to store results of the agglomerative hierarchical
clustering (AHC).

Following information is returned:

* TerminationType - completion code:
  * 1   for successful completion of algorithm
  * -5  inappropriate combination of  clustering  algorithm  and  distance
        function was used. As for now, it  is  possible  only when  Ward's
        method is called for dataset with non-Euclidean distance function.
  In case negative completion code is returned,  other  fields  of  report
  structure are invalid and should not be used.

* NPoints contains number of points in the original dataset

* Z contains information about merges performed  (see below).  Z  contains
  indexes from the original (unsorted) dataset and it can be used when you
  need to know what points were merged. However, it is not convenient when
  you want to build a dendrograd (see below).

* if  you  want  to  build  dendrogram, you  can use Z, but it is not good
  option, because Z contains  indexes from  unsorted  dataset.  Dendrogram
  built from such dataset is likely to have intersections. So, you have to
  reorder you points before building dendrogram.
  Permutation which reorders point is returned in P. Another representation
  of  merges,  which  is  more  convenient for dendorgram construction, is
  returned in PM.

* more information on format of Z, P and PM can be found below and in the
  examples from ALGLIB Reference Manual.

FORMAL DESCRIPTION OF FIELDS:
    NPoints         number of points
    Z               array[NPoints-1,2],  contains   indexes   of  clusters
                    linked in pairs to  form  clustering  tree.  I-th  row
                    corresponds to I-th merge:
                    * Z[I,0] - index of the first cluster to merge
                    * Z[I,1] - index of the second cluster to merge
                    * Z[I,0]<Z[I,1]
                    * clusters are  numbered  from 0 to 2*NPoints-2,  with
                      indexes from 0 to NPoints-1 corresponding to  points
                      of the original dataset, and indexes from NPoints to
                      2*NPoints-2  correspond  to  clusters  generated  by
                      subsequent  merges  (I-th  row  of Z creates cluster
                      with index NPoints+I).

                    IMPORTANT: indexes in Z[] are indexes in the ORIGINAL,
                    unsorted dataset. In addition to  Z algorithm  outputs
                    permutation which rearranges points in such  way  that
                    subsequent merges are  performed  on  adjacent  points
                    (such order is needed if you want to build dendrogram).
                    However,  indexes  in  Z  are  related  to   original,
                    unrearranged sequence of points.

    P               array[NPoints], permutation which reorders points  for
                    dendrogram  construction.  P[i] contains  index of the
                    position  where  we  should  move  I-th  point  of the
                    original dataset in order to apply merges PZ/PM.

    PZ              same as Z, but for permutation of points given  by  P.
                    The  only  thing  which  changed  are  indexes  of the
                    original points; indexes of clusters remained same.

    MergeDist       array[NPoints-1], contains distances between  clusters
                    being merged (MergeDist[i] correspond to merge  stored
                    in Z[i,...]):
                    * CLINK, SLINK and  average  linkage algorithms report
                      "raw", unmodified distance metric.
                    * Ward's   method   reports   weighted   intra-cluster
                      variance, which is equal to ||Ca-Cb||^2 * Sa*Sb/(Sa+Sb).
                      Here  A  and  B  are  clusters being merged, Ca is a
                      center of A, Cb is a center of B, Sa is a size of A,
                      Sb is a size of B.

    PM              array[NPoints-1,6], another representation of  merges,
                    which is suited for dendrogram construction. It  deals
                    with rearranged points (permutation P is applied)  and
                    represents merges in a form which different  from  one
                    used by Z.
                    For each I from 0 to NPoints-2, I-th row of PM represents
                    merge performed on two clusters C0 and C1. Here:
                    * C0 contains points with indexes PM[I,0]...PM[I,1]
                    * C1 contains points with indexes PM[I,2]...PM[I,3]
                    * indexes stored in PM are given for dataset sorted
                      according to permutation P
                    * PM[I,1]=PM[I,2]-1 (only adjacent clusters are merged)
                    * PM[I,0]<=PM[I,1], PM[I,2]<=PM[I,3], i.e. both
                      clusters contain at least one point
                    * heights of "subdendrograms" corresponding  to  C0/C1
                      are stored in PM[I,4]  and  PM[I,5].  Subdendrograms
                      corresponding   to   single-point   clusters    have
                      height=0. Dendrogram of the merge result has  height
                      H=max(H0,H1)+1.

NOTE: there is one-to-one correspondence between merges described by Z and
      PM. I-th row of Z describes same merge of clusters as I-th row of PM,
      with "left" cluster from Z corresponding to the "left" one from PM.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
class ahcreport
{
public:
    ahcreport();
    ahcreport(const ahcreport &rhs);
    ahcreport& operator=(const ahcreport &rhs);
    virtual ~ahcreport();
    ae_int_t terminationtype;
    ae_int_t npoints;
    integer_1d_array p;
    integer_2d_array z;
    integer_2d_array pz;
    integer_2d_array pm;
    real_1d_array mergedist;
};

`clusterizerstate` class

/*************************************************************************
This structure is a clusterization engine.

You should not try to access its fields directly.
Use ALGLIB functions in order to work with this object.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
class clusterizerstate
{
public:
    clusterizerstate();
    clusterizerstate(const clusterizerstate &rhs);
    clusterizerstate& operator=(const clusterizerstate &rhs);
    virtual ~clusterizerstate();
};

`kmeansreport` class

/*************************************************************************
This  structure   is  used  to  store  results of the  k-means  clustering
algorithm.

Following information is always returned:
* NPoints contains number of points in the original dataset
* TerminationType contains completion code, negative on failure, positive
  on success
* K contains number of clusters

For positive TerminationType we return:
* NFeatures contains number of variables in the original dataset
* C, which contains centers found by algorithm
* CIdx, which maps points of the original dataset to clusters

FORMAL DESCRIPTION OF FIELDS:
    NPoints         number of points, >=0
    NFeatures       number of variables, >=1
    TerminationType completion code:
                    * -5 if  distance  type  is  anything  different  from
                         Euclidean metric
                    * -3 for degenerate dataset: a) less  than  K  distinct
                         points, b) K=0 for non-empty dataset.
                    * +1 for successful completion
    K               number of clusters
    C               array[K,NFeatures], rows of the array store centers
    CIdx            array[NPoints], which contains cluster indexes
    IterationsCount actual number of iterations performed by clusterizer.
                    If algorithm performed more than one random restart,
                    total number of iterations is returned.
    Energy          merit function, "energy", sum  of  squared  deviations
                    from cluster centers

  -- ALGLIB --
     Copyright 27.11.2012 by Bochkanov Sergey
*************************************************************************/
class kmeansreport
{
public:
    kmeansreport();
    kmeansreport(const kmeansreport &rhs);
    kmeansreport& operator=(const kmeansreport &rhs);
    virtual ~kmeansreport();
    ae_int_t npoints;
    ae_int_t nfeatures;
    ae_int_t terminationtype;
    ae_int_t iterationscount;
    double energy;
    ae_int_t k;
    real_2d_array c;
    integer_1d_array cidx;
};

`clusterizercreate` function

/*************************************************************************
This function initializes clusterizer object. Newly initialized object  is
empty, i.e. it does not contain dataset. You should use it as follows:
1. creation
2. dataset is added with ClusterizerSetPoints()
3. additional parameters are set
3. clusterization is performed with one of the clustering functions

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizercreate(clusterizerstate &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizergetdistances` function

/*************************************************************************
This function returns distance matrix for dataset

INPUT PARAMETERS:
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points, >=0
    NFeatures-  number of features, >=1
    DistType-   distance function:
                *  0    Chebyshev distance  (L-inf norm)
                *  1    city block distance (L1 norm)
                *  2    Euclidean distance  (L2 norm, non-squared)
                * 10    Pearson correlation:
                        dist(a,b) = 1-corr(a,b)
                * 11    Absolute Pearson correlation:
                        dist(a,b) = 1-|corr(a,b)|
                * 12    Uncentered Pearson correlation (cosine of the angle):
                        dist(a,b) = a'*b/(|a|*|b|)
                * 13    Absolute uncentered Pearson correlation
                        dist(a,b) = |a'*b|/(|a|*|b|)
                * 20    Spearman rank correlation:
                        dist(a,b) = 1-rankcorr(a,b)
                * 21    Absolute Spearman rank correlation
                        dist(a,b) = 1-|rankcorr(a,b)|

OUTPUT PARAMETERS:
    D       -   array[NPoints,NPoints], distance matrix
                (full matrix is returned, with lower and upper triangles)

NOTE:  different distance functions have different performance penalty:
       * Euclidean or Pearson correlation distances are the fastest ones
       * Spearman correlation distance function is a bit slower
       * city block and Chebyshev distances are order of magnitude slower

       The reason behing difference in performance is that correlation-based
       distance functions are computed using optimized linear algebra kernels,
       while Chebyshev and city block distance functions are computed using
       simple nested loops with two branches at each iteration.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizergetdistances(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures, const ae_int_t disttype, real_2d_array &d, const xparams _xparams = alglib::xdefault);

`clusterizergetkclusters` function

/*************************************************************************
This function takes as input clusterization report Rep,  desired  clusters
count K, and builds top K clusters from hierarchical clusterization  tree.
It returns assignment of points to clusters (array of cluster indexes).

INPUT PARAMETERS:
    Rep     -   report from ClusterizerRunAHC() performed on XY
    K       -   desired number of clusters, 1<=K<=NPoints.
                K can be zero only when NPoints=0.

OUTPUT PARAMETERS:
    CIdx    -   array[NPoints], I-th element contains cluster index  (from
                0 to K-1) for I-th point of the dataset.
    CZ      -   array[K]. This array allows  to  convert  cluster  indexes
                returned by this function to indexes used by  Rep.Z.  J-th
                cluster returned by this function corresponds to  CZ[J]-th
                cluster stored in Rep.Z/PZ/PM.
                It is guaranteed that CZ[I]<CZ[I+1].

NOTE: K clusters built by this subroutine are assumed to have no hierarchy.
      Although  they  were  obtained  by  manipulation with top K nodes of
      dendrogram  (i.e.  hierarchical  decomposition  of  dataset),   this
      function does not return information about hierarchy.  Each  of  the
      clusters stand on its own.

NOTE: Cluster indexes returned by this function  does  not  correspond  to
      indexes returned in Rep.Z/PZ/PM. Either you work  with  hierarchical
      representation of the dataset (dendrogram), or you work with  "flat"
      representation returned by this function.  Each  of  representations
      has its own clusters indexing system (former uses [0, 2*NPoints-2]),
      while latter uses [0..K-1]), although  it  is  possible  to  perform
      conversion from one system to another by means of CZ array, returned
      by this function, which allows you to convert indexes stored in CIdx
      to the numeration system used by Rep.Z.

NOTE: this subroutine is optimized for moderate values of K. Say, for  K=5
      it will perform many times faster than  for  K=100.  Its  worst-case
      performance is O(N*K), although in average case  it  perform  better
      (up to O(N*log(K))).

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizergetkclusters(const ahcreport &rep, const ae_int_t k, integer_1d_array &cidx, integer_1d_array &cz, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`clusterizerrunahc` function

/*************************************************************************
This function performs agglomerative hierarchical clustering

NOTE: Agglomerative  hierarchical  clustering  algorithm  has two  phases:
      distance matrix calculation and clustering  itself. Only first phase
      (distance matrix calculation) is accelerated by SIMD and SMP.  Thus,
      acceleration is significant  only  for  medium  or  high-dimensional
      problems.

      Although activating multithreading gives some speedup  over  single-
      threaded execution, you  should  not  expect  nearly-linear  scaling
      with respect to cores count.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()

OUTPUT PARAMETERS:
    Rep     -   clustering results; see description of AHCReport
                structure for more information.

NOTE 1: hierarchical clustering algorithms require large amounts of memory.
        In particular, this implementation needs  sizeof(double)*NPoints^2
        bytes, which are used to store distance matrix. In  case  we  work
        with user-supplied matrix, this amount is multiplied by 2 (we have
        to store original matrix and to work with its copy).

        For example, problem with 10000 points  would require 800M of RAM,
        even when working in a 1-dimensional space.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizerrunahc(clusterizerstate &s, ahcreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizerrunkmeans` function

/*************************************************************************
This function performs clustering by k-means++ algorithm.

You may change algorithm properties by calling:
* ClusterizerSetKMeansLimits() to change number of restarts or iterations
* ClusterizerSetKMeansInit() to change initialization algorithm

By  default,  one  restart  and  unlimited number of iterations are  used.
Initialization algorithm is chosen automatically.

NOTE: k-means clustering  algorithm has two  phases:  selection of initial
      centers and clustering  itself.  ALGLIB  parallelizes  both  phases.
      Parallel version is optimized for the following  scenario: medium or
      high-dimensional problem (8 or more dimensions) with large number of
      points and clusters. However, some speed-up  can  be  obtained  even
      when assumptions above are violated.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    K       -   number of clusters, K>=0.
                K  can  be  zero only when algorithm is called  for  empty
                dataset,  in   this   case   completion  code  is  set  to
                success (+1).
                If  K=0  and  dataset  size  is  non-zero,  we   can   not
                meaningfully assign points to some center  (there  are  no
                centers because K=0) and  return  -3  as  completion  code
                (failure).

OUTPUT PARAMETERS:
    Rep     -   clustering results; see description of KMeansReport
                structure for more information.

NOTE 1: k-means  clustering  can  be  performed  only  for  datasets  with
        Euclidean  distance  function.  Algorithm  will  return   negative
        completion code in Rep.TerminationType in case dataset  was  added
        to clusterizer with DistType other than Euclidean (or dataset  was
        specified by distance matrix instead of explicitly given points).

NOTE 2: by default, k-means uses non-deterministic seed to initialize  RNG
        which is used to select initial centers. As  result,  each  run of
        algorithm may return different values. If you  need  deterministic
        behavior, use ClusterizerSetSeed() function.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizerrunkmeans(clusterizerstate &s, const ae_int_t k, kmeansreport &rep, const xparams _xparams = alglib::xdefault);

`clusterizerseparatedbycorr` function

/*************************************************************************
This  function  accepts  AHC  report  Rep,  desired  maximum  intercluster
correlation and returns top clusters from hierarchical clusterization tree
which are separated by correlation R or LOWER.

It returns assignment of points to clusters (array of cluster indexes).

There is one more function with similar name - ClusterizerSeparatedByDist,
which returns clusters with intercluster distance equal  to  R  or  HIGHER
(note: higher for distance, lower for correlation).

INPUT PARAMETERS:
    Rep     -   report from ClusterizerRunAHC() performed on XY
    R       -   desired maximum intercluster correlation, -1<=R<=+1

OUTPUT PARAMETERS:
    K       -   number of clusters, 1<=K<=NPoints
    CIdx    -   array[NPoints], I-th element contains cluster index  (from
                0 to K-1) for I-th point of the dataset.
    CZ      -   array[K]. This array allows  to  convert  cluster  indexes
                returned by this function to indexes used by  Rep.Z.  J-th
                cluster returned by this function corresponds to  CZ[J]-th
                cluster stored in Rep.Z/PZ/PM.
                It is guaranteed that CZ[I]<CZ[I+1].

NOTE: K clusters built by this subroutine are assumed to have no hierarchy.
      Although  they  were  obtained  by  manipulation with top K nodes of
      dendrogram  (i.e.  hierarchical  decomposition  of  dataset),   this
      function does not return information about hierarchy.  Each  of  the
      clusters stand on its own.

NOTE: Cluster indexes returned by this function  does  not  correspond  to
      indexes returned in Rep.Z/PZ/PM. Either you work  with  hierarchical
      representation of the dataset (dendrogram), or you work with  "flat"
      representation returned by this function.  Each  of  representations
      has its own clusters indexing system (former uses [0, 2*NPoints-2]),
      while latter uses [0..K-1]), although  it  is  possible  to  perform
      conversion from one system to another by means of CZ array, returned
      by this function, which allows you to convert indexes stored in CIdx
      to the numeration system used by Rep.Z.

NOTE: this subroutine is optimized for moderate values of K. Say, for  K=5
      it will perform many times faster than  for  K=100.  Its  worst-case
      performance is O(N*K), although in average case  it  perform  better
      (up to O(N*log(K))).

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizerseparatedbycorr(const ahcreport &rep, const double r, ae_int_t &k, integer_1d_array &cidx, integer_1d_array &cz, const xparams _xparams = alglib::xdefault);

`clusterizerseparatedbydist` function

/*************************************************************************
This  function  accepts  AHC  report  Rep,  desired  minimum  intercluster
distance and returns top clusters from  hierarchical  clusterization  tree
which are separated by distance R or HIGHER.

It returns assignment of points to clusters (array of cluster indexes).

There is one more function with similar name - ClusterizerSeparatedByCorr,
which returns clusters with intercluster correlation equal to R  or  LOWER
(note: higher for distance, lower for correlation).

INPUT PARAMETERS:
    Rep     -   report from ClusterizerRunAHC() performed on XY
    R       -   desired minimum intercluster distance, R>=0

OUTPUT PARAMETERS:
    K       -   number of clusters, 1<=K<=NPoints
    CIdx    -   array[NPoints], I-th element contains cluster index  (from
                0 to K-1) for I-th point of the dataset.
    CZ      -   array[K]. This array allows  to  convert  cluster  indexes
                returned by this function to indexes used by  Rep.Z.  J-th
                cluster returned by this function corresponds to  CZ[J]-th
                cluster stored in Rep.Z/PZ/PM.
                It is guaranteed that CZ[I]<CZ[I+1].

NOTE: K clusters built by this subroutine are assumed to have no hierarchy.
      Although  they  were  obtained  by  manipulation with top K nodes of
      dendrogram  (i.e.  hierarchical  decomposition  of  dataset),   this
      function does not return information about hierarchy.  Each  of  the
      clusters stand on its own.

NOTE: Cluster indexes returned by this function  does  not  correspond  to
      indexes returned in Rep.Z/PZ/PM. Either you work  with  hierarchical
      representation of the dataset (dendrogram), or you work with  "flat"
      representation returned by this function.  Each  of  representations
      has its own clusters indexing system (former uses [0, 2*NPoints-2]),
      while latter uses [0..K-1]), although  it  is  possible  to  perform
      conversion from one system to another by means of CZ array, returned
      by this function, which allows you to convert indexes stored in CIdx
      to the numeration system used by Rep.Z.

NOTE: this subroutine is optimized for moderate values of K. Say, for  K=5
      it will perform many times faster than  for  K=100.  Its  worst-case
      performance is O(N*K), although in average case  it  perform  better
      (up to O(N*log(K))).

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizerseparatedbydist(const ahcreport &rep, const double r, ae_int_t &k, integer_1d_array &cidx, integer_1d_array &cz, const xparams _xparams = alglib::xdefault);

`clusterizersetahcalgo` function

/*************************************************************************
This function sets agglomerative hierarchical clustering algorithm

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    Algo    -   algorithm type:
                * 0     complete linkage (default algorithm)
                * 1     single linkage
                * 2     unweighted average linkage
                * 3     weighted average linkage
                * 4     Ward's method

NOTE: Ward's method works correctly only with Euclidean  distance,  that's
      why algorithm will return negative termination  code  (failure)  for
      any other distance type.

      It is possible, however,  to  use  this  method  with  user-supplied
      distance matrix. It  is  your  responsibility  to pass one which was
      calculated with Euclidean distance function.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizersetahcalgo(clusterizerstate &s, const ae_int_t algo, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizersetdistances` function

/*************************************************************************
This function adds dataset given by distance  matrix  to  the  clusterizer
structure. It is important that dataset is not  given  explicitly  -  only
distance matrix is given.

This function overrides all previous calls  of  ClusterizerSetPoints()  or
ClusterizerSetDistances().

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    D       -   array[NPoints,NPoints], distance matrix given by its upper
                or lower triangle (main diagonal is  ignored  because  its
                entries are expected to be zero).
    NPoints -   number of points
    IsUpper -   whether upper or lower triangle of D is given.

NOTE 1: different clustering algorithms have different limitations:
        * agglomerative hierarchical clustering algorithms may be used with
          any kind of distance metric, including one  which  is  given  by
          distance matrix
        * k-means++ clustering algorithm may be used only  with  Euclidean
          distance function and explicitly given points - it  can  not  be
          used with dataset given by distance matrix
        Thus, if you call this function, you will be unable to use k-means
        clustering algorithm to process your problem.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizersetdistances(clusterizerstate &s, const real_2d_array &d, const ae_int_t npoints, const bool isupper, const xparams _xparams = alglib::xdefault);
void clusterizersetdistances(clusterizerstate &s, const real_2d_array &d, const bool isupper, const xparams _xparams = alglib::xdefault);

Examples: [1]

`clusterizersetkmeansinit` function

/*************************************************************************
This function sets k-means  initialization  algorithm.  Several  different
algorithms can be chosen, including k-means++.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    InitAlgo-   initialization algorithm:
                * 0  automatic selection ( different  versions  of  ALGLIB
                     may select different algorithms)
                * 1  random initialization
                * 2  k-means++ initialization  (best  quality  of  initial
                     centers, but long  non-parallelizable  initialization
                     phase with bad cache locality)
                * 3  "fast-greedy"  algorithm  with  efficient,  easy   to
                     parallelize initialization. Quality of initial centers
                     is  somewhat  worse  than  that  of  k-means++.  This
                     algorithm is a default one in the current version  of
                     ALGLIB.
                *-1  "debug" algorithm which always selects first  K  rows
                     of dataset; this algorithm is used for debug purposes
                     only. Do not use it in the industrial code!

  -- ALGLIB --
     Copyright 21.01.2015 by Bochkanov Sergey
*************************************************************************/
void clusterizersetkmeansinit(clusterizerstate &s, const ae_int_t initalgo, const xparams _xparams = alglib::xdefault);

`clusterizersetkmeanslimits` function

/*************************************************************************
This  function  sets k-means properties:  number  of  restarts and maximum
number of iterations per one run.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    Restarts-   restarts count, >=1.
                k-means++ algorithm performs several restarts and  chooses
                best set of centers (one with minimum squared distance).
    MaxIts  -   maximum number of k-means iterations performed during  one
                run. >=0, zero value means that algorithm performs unlimited
                number of iterations.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizersetkmeanslimits(clusterizerstate &s, const ae_int_t restarts, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`clusterizersetpoints` function

/*************************************************************************
This function adds dataset to the clusterizer structure.

This function overrides all previous calls  of  ClusterizerSetPoints()  or
ClusterizerSetDistances().

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    XY      -   array[NPoints,NFeatures], dataset
    NPoints -   number of points, >=0
    NFeatures-  number of features, >=1
    DistType-   distance function:
                *  0    Chebyshev distance  (L-inf norm)
                *  1    city block distance (L1 norm)
                *  2    Euclidean distance  (L2 norm), non-squared
                * 10    Pearson correlation:
                        dist(a,b) = 1-corr(a,b)
                * 11    Absolute Pearson correlation:
                        dist(a,b) = 1-|corr(a,b)|
                * 12    Uncentered Pearson correlation (cosine of the angle):
                        dist(a,b) = a'*b/(|a|*|b|)
                * 13    Absolute uncentered Pearson correlation
                        dist(a,b) = |a'*b|/(|a|*|b|)
                * 20    Spearman rank correlation:
                        dist(a,b) = 1-rankcorr(a,b)
                * 21    Absolute Spearman rank correlation
                        dist(a,b) = 1-|rankcorr(a,b)|

NOTE 1: different distance functions have different performance penalty:
        * Euclidean or Pearson correlation distances are the fastest ones
        * Spearman correlation distance function is a bit slower
        * city block and Chebyshev distances are order of magnitude slower

        The reason behing difference in performance is that correlation-based
        distance functions are computed using optimized linear algebra kernels,
        while Chebyshev and city block distance functions are computed using
        simple nested loops with two branches at each iteration.

NOTE 2: different clustering algorithms have different limitations:
        * agglomerative hierarchical clustering algorithms may be used with
          any kind of distance metric
        * k-means++ clustering algorithm may be used only  with  Euclidean
          distance function
        Thus, list of specific clustering algorithms you may  use  depends
        on distance function you specify when you set your dataset.

  -- ALGLIB --
     Copyright 10.07.2012 by Bochkanov Sergey
*************************************************************************/
void clusterizersetpoints(clusterizerstate &s, const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nfeatures, const ae_int_t disttype, const xparams _xparams = alglib::xdefault);
void clusterizersetpoints(clusterizerstate &s, const real_2d_array &xy, const ae_int_t disttype, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`clusterizersetseed` function

/*************************************************************************
This  function  sets  seed  which  is  used to initialize internal RNG. By
default, deterministic seed is used - same for each run of clusterizer. If
you specify non-deterministic  seed  value,  then  some  algorithms  which
depend on random initialization (in current version: k-means)  may  return
slightly different results after each run.

INPUT PARAMETERS:
    S       -   clusterizer state, initialized by ClusterizerCreate()
    Seed    -   seed:
                * positive values = use deterministic seed for each run of
                  algorithms which depend on random initialization
                * zero or negative values = use non-deterministic seed

  -- ALGLIB --
     Copyright 08.06.2017 by Bochkanov Sergey
*************************************************************************/
void clusterizersetseed(clusterizerstate &s, const ae_int_t seed, const xparams _xparams = alglib::xdefault);

clst_ahc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple clusterization example
        //
        // We have a set of points in 2D space:
        //     (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
        //
        //  |
        //  |     P3
        //  |
        //  | P1          
        //  |             P4
        //  | P0          P2
        //  |-------------------------
        //
        // We want to perform Agglomerative Hierarchic Clusterization (AHC),
        // using complete linkage (default algorithm) and Euclidean distance
        // (default metric).
        //
        // In order to do that, we:
        // * create clusterizer with clusterizercreate()
        // * set points XY and metric (2=Euclidean) with clusterizersetpoints()
        // * run AHC algorithm with clusterizerrunahc
        //
        // You may see that clusterization itself is a minor part of the example,
        // most of which is dominated by comments :)
        //
        clusterizerstate s;
        ahcreport rep;
        real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";

        clusterizercreate(s);
        clusterizersetpoints(s, xy, 2);
        clusterizerrunahc(s, rep);

        //
        // Now we've built our clusterization tree. Rep.z contains information which
        // is required to build dendrogram. I-th row of rep.z represents one merge
        // operation, with first cluster to merge having index rep.z[I,0] and second
        // one having index rep.z[I,1]. Merge result has index NPoints+I.
        //
        // Clusters with indexes less than NPoints are single-point initial clusters,
        // while ones with indexes from NPoints to 2*NPoints-2 are multi-point
        // clusters created during merges.
        //
        // In our example, Z=[[2,4], [0,1], [3,6], [5,7]]
        //
        // It means that:
        // * first, we merge C2=(P2) and C4=(P4),    and create C5=(P2,P4)
        // * then, we merge  C2=(P0) and C1=(P1),    and create C6=(P0,P1)
        // * then, we merge  C3=(P3) and C6=(P0,P1), and create C7=(P0,P1,P3)
        // * finally, we merge C5 and C7 and create C8=(P0,P1,P2,P3,P4)
        //
        // Thus, we have following dendrogram:
        //  
        //      ------8-----
        //      |          |
        //      |      ----7----
        //      |      |       |
        //   ---5---   |    ---6---
        //   |     |   |    |     |
        //   P2   P4   P3   P0   P1
        //
        printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[2,4],[0,1],[3,6],[5,7]]

        //
        // We've built dendrogram above by reordering our dataset.
        //
        // Without such reordering it would be impossible to build dendrogram without
        // intersections. Luckily, ahcreport structure contains two additional fields
        // which help to build dendrogram from your data:
        // * rep.p, which contains permutation applied to dataset
        // * rep.pm, which contains another representation of merges 
        //
        // In our example we have:
        // * P=[3,4,0,2,1]
        // * PZ=[[0,0,1,1,0,0],[3,3,4,4,0,0],[2,2,3,4,0,1],[0,1,2,4,1,2]]
        //
        // Permutation array P tells us that P0 should be moved to position 3,
        // P1 moved to position 4, P2 moved to position 0 and so on:
        //
        //   (P0 P1 P2 P3 P4) => (P2 P4 P3 P0 P1)
        //
        // Merges array PZ tells us how to perform merges on the sorted dataset.
        // One row of PZ corresponds to one merge operations, with first pair of
        // elements denoting first of the clusters to merge (start index, end
        // index) and next pair of elements denoting second of the clusters to
        // merge. Clusters being merged are always adjacent, with first one on
        // the left and second one on the right.
        //
        // For example, first row of PZ tells us that clusters [0,0] and [1,1] are
        // merged (single-point clusters, with first one containing P2 and second
        // one containing P4). Third row of PZ tells us that we merge one single-
        // point cluster [2,2] with one two-point cluster [3,4].
        //
        // There are two more elements in each row of PZ. These are the helper
        // elements, which denote HEIGHT (not size) of left and right subdendrograms.
        // For example, according to PZ, first two merges are performed on clusterization
        // trees of height 0, while next two merges are performed on 0-1 and 1-2
        // pairs of trees correspondingly.
        //
        printf("%s\n", rep.p.tostring().c_str()); // EXPECTED: [3,4,0,2,1]
        printf("%s\n", rep.pm.tostring().c_str()); // EXPECTED: [[0,0,1,1,0,0],[3,3,4,4,0,0],[2,2,3,4,0,1],[0,1,2,4,1,2]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

clst_distance example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We have three points in 4D space:
        //     (P0,P1,P2) = ((1, 2, 1, 2), (6, 7, 6, 7), (7, 6, 7, 6))
        //
        // We want to try clustering them with different distance functions.
        // Distance function is chosen when we add dataset to the clusterizer.
        // We can choose several distance types - Euclidean, city block, Chebyshev,
        // several correlation measures or user-supplied distance matrix.
        //
        // Here we'll try three distances: Euclidean, Pearson correlation,
        // user-supplied distance matrix. Different distance functions lead
        // to different choices being made by algorithm during clustering.
        //
        clusterizerstate s;
        ahcreport rep;
        ae_int_t disttype;
        real_2d_array xy = "[[1, 2, 1, 2], [6, 7, 6, 7], [7, 6, 7, 6]]";
        clusterizercreate(s);

        // With Euclidean distance function (disttype=2) two closest points
        // are P1 and P2, thus:
        // * first, we merge P1 and P2 to form C3=[P1,P2]
        // * second, we merge P0 and C3 to form C4=[P0,P1,P2]
        disttype = 2;
        clusterizersetpoints(s, xy, disttype);
        clusterizerrunahc(s, rep);
        printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[1,2],[0,3]]

        // With Pearson correlation distance function (disttype=10) situation
        // is different - distance between P0 and P1 is zero, thus:
        // * first, we merge P0 and P1 to form C3=[P0,P1]
        // * second, we merge P2 and C3 to form C4=[P0,P1,P2]
        disttype = 10;
        clusterizersetpoints(s, xy, disttype);
        clusterizerrunahc(s, rep);
        printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[0,1],[2,3]]

        // Finally, we try clustering with user-supplied distance matrix:
        //     [ 0 3 1 ]
        // P = [ 3 0 3 ], where P[i,j] = dist(Pi,Pj)
        //     [ 1 3 0 ]
        //
        // * first, we merge P0 and P2 to form C3=[P0,P2]
        // * second, we merge P1 and C3 to form C4=[P0,P1,P2]
        real_2d_array d = "[[0,3,1],[3,0,3],[1,3,0]]";
        clusterizersetdistances(s, d, true);
        clusterizerrunahc(s, rep);
        printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[0,2],[1,3]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

clst_kclusters example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We have a set of points in 2D space:
        //     (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
        //
        //  |
        //  |     P3
        //  |
        //  | P1          
        //  |             P4
        //  | P0          P2
        //  |-------------------------
        //
        // We perform Agglomerative Hierarchic Clusterization (AHC) and we want
        // to get top K clusters from clusterization tree for different K.
        //
        clusterizerstate s;
        ahcreport rep;
        real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";
        integer_1d_array cidx;
        integer_1d_array cz;

        clusterizercreate(s);
        clusterizersetpoints(s, xy, 2);
        clusterizerrunahc(s, rep);

        // with K=5, every points is assigned to its own cluster:
        // C0=P0, C1=P1 and so on...
        clusterizergetkclusters(rep, 5, cidx, cz);
        printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,1,2,3,4]

        // with K=1 we have one large cluster C0=[P0,P1,P2,P3,P4,P5]
        clusterizergetkclusters(rep, 1, cidx, cz);
        printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,0,0,0,0]

        // with K=3 we have three clusters C0=[P3], C1=[P2,P4], C2=[P0,P1]
        clusterizergetkclusters(rep, 3, cidx, cz);
        printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [2,2,1,0,1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

clst_kmeans example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple clusterization example
        //
        // We have a set of points in 2D space:
        //     (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
        //
        //  |
        //  |     P3
        //  |
        //  | P1          
        //  |             P4
        //  | P0          P2
        //  |-------------------------
        //
        // We want to perform k-means++ clustering with K=2.
        //
        // In order to do that, we:
        // * create clusterizer with clusterizercreate()
        // * set points XY and metric (must be Euclidean, distype=2) with clusterizersetpoints()
        // * (optional) set number of restarts from random positions to 5
        // * run k-means algorithm with clusterizerrunkmeans()
        //
        // You may see that clusterization itself is a minor part of the example,
        // most of which is dominated by comments :)
        //
        clusterizerstate s;
        kmeansreport rep;
        real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";

        clusterizercreate(s);
        clusterizersetpoints(s, xy, 2);
        clusterizersetkmeanslimits(s, 5, 0);
        clusterizerrunkmeans(s, 2, rep);

        //
        // We've performed clusterization, and it succeeded (completion code is +1).
        //
        // Now first center is stored in the first row of rep.c, second one is stored
        // in the second row. rep.cidx can be used to determine which center is
        // closest to some specific point of the dataset.
        //
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

        // We called clusterizersetpoints() with disttype=2 because k-means++
        // algorithm does NOT support metrics other than Euclidean. But what if we
        // try to use some other metric?
        //
        // We change metric type by calling clusterizersetpoints() one more time,
        // and try to run k-means algo again. It fails.
        //
        clusterizersetpoints(s, xy, 0);
        clusterizerrunkmeans(s, 2, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: -5
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

clst_linkage example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We have a set of points in 1D space:
        //     (P0,P1,P2,P3,P4) = (1, 3, 10, 16, 20)
        //
        // We want to perform Agglomerative Hierarchic Clusterization (AHC),
        // using either complete or single linkage and Euclidean distance
        // (default metric).
        //
        // First two steps merge P0/P1 and P3/P4 independently of the linkage type.
        // However, third step depends on linkage type being used:
        // * in case of complete linkage P2=10 is merged with [P0,P1]
        // * in case of single linkage P2=10 is merged with [P3,P4]
        //
        clusterizerstate s;
        ahcreport rep;
        real_2d_array xy = "[[1],[3],[10],[16],[20]]";
        integer_1d_array cidx;
        integer_1d_array cz;

        clusterizercreate(s);
        clusterizersetpoints(s, xy, 2);

        // use complete linkage, reduce set down to 2 clusters.
        // print clusterization with clusterizergetkclusters(2).
        // P2 must belong to [P0,P1]
        clusterizersetahcalgo(s, 0);
        clusterizerrunahc(s, rep);
        clusterizergetkclusters(rep, 2, cidx, cz);
        printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [1,1,1,0,0]

        // use single linkage, reduce set down to 2 clusters.
        // print clusterization with clusterizergetkclusters(2).
        // P2 must belong to [P2,P3]
        clusterizersetahcalgo(s, 1);
        clusterizerrunahc(s, rep);
        clusterizergetkclusters(rep, 2, cidx, cz);
        printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,0,1,1,1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`conv` subpackage

Functions

convc1d
convc1dbuf
convc1dcircular
convc1dcircularbuf
convc1dcircularinv
convc1dcircularinvbuf
convc1dinv
convc1dinvbuf
convr1d
convr1dbuf
convr1dcircular
convr1dcircularbuf
convr1dcircularinv
convr1dcircularinvbuf
convr1dinv
convr1dinvbuf

Examples

`convc1d` function

/*************************************************************************
1-dimensional complex convolution.

For given A/B returns conv(A,B) (non-circular). Subroutine can automatically
choose between three implementations: straightforward O(M*N)  formula  for
very small N (or M), overlap-add algorithm for  cases  where  max(M,N)  is
significantly larger than min(M,N), but O(M*N) algorithm is too slow,  and
general FFT-based formula for cases where two previous algorithms are  too
slow.

Algorithm has max(M,N)*log(max(M,N)) complexity for any M/N.

INPUT PARAMETERS
    A   -   array[M] - complex function to be transformed
    M   -   problem size
    B   -   array[N] - complex function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[N+M-1]

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
    functions have non-zero values at negative T's, you can still use this
    subroutine - just shift its result correspondingly.

NOTE: there is a buffered version of this  function,  ConvC1DBuf(),  which
      can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convc1d(const complex_1d_array &a, const ae_int_t m, const complex_1d_array &b, const ae_int_t n, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`convc1dbuf` function

/*************************************************************************
1-dimensional complex convolution, buffered version of ConvC1DBuf(), which
does not reallocate R[] if its length is enough to store the result  (i.e.
it reuses previously allocated memory as much as possible).

  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convc1dbuf(const complex_1d_array &a, const ae_int_t m, const complex_1d_array &b, const ae_int_t n, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`convc1dcircular` function

/*************************************************************************
1-dimensional circular complex convolution.

For given S/R returns conv(S,R) (circular). Algorithm has linearithmic
complexity for any M/N.

IMPORTANT:  normal convolution is commutative,  i.e.   it  is symmetric  -
conv(A,B)=conv(B,A).  Cyclic convolution IS NOT.  One function - S - is  a
signal,  periodic function, and another - R - is a response,  non-periodic
function with limited length.

INPUT PARAMETERS
    S   -   array[M] - complex periodic signal
    M   -   problem size
    B   -   array[N] - complex non-periodic response
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[M].

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

NOTE: there is a buffered version of this  function,  ConvC1DCircularBuf(),
      which can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convc1dcircular(const complex_1d_array &s, const ae_int_t m, const complex_1d_array &r, const ae_int_t n, complex_1d_array &c, const xparams _xparams = alglib::xdefault);

`convc1dcircularbuf` function

/*************************************************************************
1-dimensional circular complex convolution.

Buffered version of ConvC1DCircular(), which does not  reallocate  C[]  if
its length is enough to  store  the  result  (i.e.  it  reuses  previously
allocated memory as much as possible).

  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convc1dcircularbuf(const complex_1d_array &s, const ae_int_t m, const complex_1d_array &r, const ae_int_t n, complex_1d_array &c, const xparams _xparams = alglib::xdefault);

`convc1dcircularinv` function

/*************************************************************************
1-dimensional circular complex deconvolution (inverse of ConvC1DCircular()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved periodic signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - non-periodic response
    N   -   response length

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-1].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

NOTE: there is a buffered version of this  function,  ConvC1DCircularInvBuf(),
      which can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convc1dcircularinv(const complex_1d_array &a, const ae_int_t m, const complex_1d_array &b, const ae_int_t n, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`convc1dcircularinvbuf` function

/*************************************************************************
1-dimensional circular complex deconvolution (inverse of ConvC1DCircular()).

Buffered version of ConvC1DCircularInv(), which does not reallocate R[] if
its length is enough to  store  the  result  (i.e.  it  reuses  previously
allocated memory as much as possible).

  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convc1dcircularinvbuf(const complex_1d_array &a, const ae_int_t m, const complex_1d_array &b, const ae_int_t n, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`convc1dinv` function

/*************************************************************************
1-dimensional complex non-circular deconvolution (inverse of ConvC1D()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - response
    N   -   response length, N<=M

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-N].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

NOTE: there is a buffered version of this  function,  ConvC1DInvBuf(),
      which can reuse space previously allocated in its output parameter R

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convc1dinv(const complex_1d_array &a, const ae_int_t m, const complex_1d_array &b, const ae_int_t n, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`convc1dinvbuf` function

/*************************************************************************
1-dimensional complex non-circular deconvolution (inverse of ConvC1D()).

A buffered version, which does not reallocate R[] if its length is  enough
to store the result (i.e. it reuses previously allocated memory as much as
possible).

  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convc1dinvbuf(const complex_1d_array &a, const ae_int_t m, const complex_1d_array &b, const ae_int_t n, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`convr1d` function

/*************************************************************************
1-dimensional real convolution.

Analogous to ConvC1D(), see ConvC1D() comments for more details.

INPUT PARAMETERS
    A   -   array[0..M-1] - real function to be transformed
    M   -   problem size
    B   -   array[0..N-1] - real function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..N+M-2].

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

NOTE: there is a buffered version of this  function,  ConvR1DBuf(),
      which can reuse space previously allocated in its output parameter R.


  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convr1d(const real_1d_array &a, const ae_int_t m, const real_1d_array &b, const ae_int_t n, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`convr1dbuf` function

/*************************************************************************
1-dimensional real convolution.

Buffered version of ConvR1D(), which does not reallocate R[] if its length
is enough to store the result (i.e. it reuses previously allocated  memory
as much as possible).

  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convr1dbuf(const real_1d_array &a, const ae_int_t m, const real_1d_array &b, const ae_int_t n, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`convr1dcircular` function

/*************************************************************************
1-dimensional circular real convolution.

Analogous to ConvC1DCircular(), see ConvC1DCircular() comments for more details.

INPUT PARAMETERS
    S   -   array[0..M-1] - real signal
    M   -   problem size
    B   -   array[0..N-1] - real response
    N   -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

NOTE: there is a buffered version of this  function,  ConvR1DCurcularBuf(),
      which can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convr1dcircular(const real_1d_array &s, const ae_int_t m, const real_1d_array &r, const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`convr1dcircularbuf` function

/*************************************************************************
1-dimensional circular real convolution, buffered version, which  does not
reallocate C[] if its length is enough to store the result (i.e. it reuses
previously allocated memory as much as possible).

  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convr1dcircularbuf(const real_1d_array &s, const ae_int_t m, const real_1d_array &r, const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`convr1dcircularinv` function

/*************************************************************************
1-dimensional complex deconvolution (inverse of ConvC1D()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - response
    N   -   response length

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-N].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that B is zero at T<0. If  it  has  non-zero  values  at
negative T's, you can still use this subroutine - just  shift  its  result
correspondingly.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convr1dcircularinv(const real_1d_array &a, const ae_int_t m, const real_1d_array &b, const ae_int_t n, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`convr1dcircularinvbuf` function

/*************************************************************************
1-dimensional complex deconvolution, inverse of ConvR1DCircular().

Buffered version, which does not reallocate R[] if its length is enough to
store the result (i.e. it reuses previously allocated memory  as  much  as
possible).

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convr1dcircularinvbuf(const real_1d_array &a, const ae_int_t m, const real_1d_array &b, const ae_int_t n, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`convr1dinv` function

/*************************************************************************
1-dimensional real deconvolution (inverse of ConvC1D()).

Algorithm has M*log(M)) complexity for any M (composite or prime).

INPUT PARAMETERS
    A   -   array[0..M-1] - convolved signal, A = conv(R, B)
    M   -   convolved signal length
    B   -   array[0..N-1] - response
    N   -   response length, N<=M

OUTPUT PARAMETERS
    R   -   deconvolved signal. array[0..M-N].

NOTE:
    deconvolution is unstable process and may result in division  by  zero
(if your response function is degenerate, i.e. has zero Fourier coefficient).

NOTE:
    It is assumed that A is zero at T<0, B is zero too.  If  one  or  both
functions have non-zero values at negative T's, you  can  still  use  this
subroutine - just shift its result correspondingly.

NOTE: there is a buffered version of this  function,  ConvR1DInvBuf(),
      which can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void convr1dinv(const real_1d_array &a, const ae_int_t m, const real_1d_array &b, const ae_int_t n, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`convr1dinvbuf` function

/*************************************************************************
1-dimensional real deconvolution (inverse of ConvR1D()), buffered version,
which does not reallocate R[] if its length is enough to store the  result
(i.e. it reuses previously allocated memory as much as possible).


  -- ALGLIB --
     Copyright 30.11.2023 by Bochkanov Sergey
*************************************************************************/
void convr1dinvbuf(const real_1d_array &a, const ae_int_t m, const real_1d_array &b, const ae_int_t n, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`corr` subpackage

Functions

corrc1d
corrc1dbuf
corrc1dcircular
corrc1dcircularbuf
corrr1d
corrr1dbuf
corrr1dcircular
corrr1dcircularbuf

Examples

`corrc1d` function

/*************************************************************************
1-dimensional complex cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (non-circular).

Correlation is calculated using reduction to  convolution.  Algorithm with
max(N,N)*log(max(N,N)) complexity is used (see  ConvC1D()  for  more  info
about performance).

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order: CorrC1D(Signal, Pattern) = Pattern x Signal (using  traditional
    definition of cross-correlation, denoting cross-correlation as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - complex function to be transformed,
                signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - complex function to be transformed,
                pattern to 'search' within a signal
    M       -   problem size

OUTPUT PARAMETERS
    R       -   cross-correlation, array[0..N+M-2]:
                * positive lags are stored in R[0..N-1],
                  R[i] = sum(conj(pattern[j])*signal[i+j]
                * negative lags are stored in R[N..N+M-2],
                  R[N+M-1-i] = sum(conj(pattern[j])*signal[-i+j]

NOTE:
    It is assumed that pattern domain is [0..M-1].  If Pattern is non-zero
on [-K..M-1],  you can still use this subroutine, just shift result by K.

NOTE: there is a buffered version of this  function,  CorrC1DBuf(), which
     can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrc1d(const complex_1d_array &signal, const ae_int_t n, const complex_1d_array &pattern, const ae_int_t m, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`corrc1dbuf` function

/*************************************************************************
1-dimensional complex cross-correlation, a buffered version  of  CorrC1D()
which does not reallocate R[] if its length is enough to store the  result
(i.e. it reuses previously allocated memory as much as possible).

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrc1dbuf(const complex_1d_array &signal, const ae_int_t n, const complex_1d_array &pattern, const ae_int_t m, complex_1d_array &r, const xparams _xparams = alglib::xdefault);

`corrc1dcircular` function

/*************************************************************************
1-dimensional circular complex cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (circular).
Algorithm has linearithmic complexity for any M/N.

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order:   CorrC1DCircular(Signal, Pattern) = Pattern x Signal    (using
    traditional definition of cross-correlation, denoting cross-correlation
    as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - complex function to be transformed,
                periodic signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - complex function to be transformed,
                non-periodic pattern to 'search' within a signal
    M       -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].

NOTE: there is a buffered version of this  function,  CorrC1DCircular(),
      which can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrc1dcircular(const complex_1d_array &signal, const ae_int_t m, const complex_1d_array &pattern, const ae_int_t n, complex_1d_array &c, const xparams _xparams = alglib::xdefault);

`corrc1dcircularbuf` function

/*************************************************************************
1-dimensional circular complex cross-correlation.

A buffered function which does not reallocate C[] if its length is  enough
to store the result (i.e. it reuses previously allocated memory as much as
possible).

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrc1dcircularbuf(const complex_1d_array &signal, const ae_int_t m, const complex_1d_array &pattern, const ae_int_t n, complex_1d_array &c, const xparams _xparams = alglib::xdefault);

`corrr1d` function

/*************************************************************************
1-dimensional real cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (non-circular).

Correlation is calculated using reduction to  convolution.  Algorithm with
max(N,N)*log(max(N,N)) complexity is used (see  ConvC1D()  for  more  info
about performance).

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order: CorrR1D(Signal, Pattern) = Pattern x Signal (using  traditional
    definition of cross-correlation, denoting cross-correlation as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - real function to be transformed,
                signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - real function to be transformed,
                pattern to 'search' withing signal
    M       -   problem size

OUTPUT PARAMETERS
    R       -   cross-correlation, array[0..N+M-2]:
                * positive lags are stored in R[0..N-1],
                  R[i] = sum(pattern[j]*signal[i+j]
                * negative lags are stored in R[N..N+M-2],
                  R[N+M-1-i] = sum(pattern[j]*signal[-i+j]

NOTE:
    It is assumed that pattern domain is [0..M-1].  If Pattern is non-zero
on [-K..M-1],  you can still use this subroutine, just shift result by K.

NOTE: there is a buffered version of this  function,  CorrR1DBuf(),  which
      can reuse space previously allocated in its output parameter R.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrr1d(const real_1d_array &signal, const ae_int_t n, const real_1d_array &pattern, const ae_int_t m, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`corrr1dbuf` function

/*************************************************************************
1-dimensional real cross-correlation, buffered function,  which  does  not
reallocate R[] if its length is enough to store the result (i.e. it reuses
previously allocated memory as much as possible).

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrr1dbuf(const real_1d_array &signal, const ae_int_t n, const real_1d_array &pattern, const ae_int_t m, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`corrr1dcircular` function

/*************************************************************************
1-dimensional circular real cross-correlation.

For given Pattern/Signal returns corr(Pattern,Signal) (circular).
Algorithm has linearithmic complexity for any M/N.

IMPORTANT:
    for  historical reasons subroutine accepts its parameters in  reversed
    order:   CorrR1DCircular(Signal, Pattern) = Pattern x Signal    (using
    traditional definition of cross-correlation, denoting cross-correlation
    as "x").

INPUT PARAMETERS
    Signal  -   array[0..N-1] - real function to be transformed,
                periodic signal containing pattern
    N       -   problem size
    Pattern -   array[0..M-1] - real function to be transformed,
                non-periodic pattern to search withing signal
    M       -   problem size

OUTPUT PARAMETERS
    R   -   convolution: A*B. array[0..M-1].

NOTE: there is a buffered version of this  function,  CorrR1DCircularBuf(),
      which can reuse space previously allocated in its output parameter C.

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrr1dcircular(const real_1d_array &signal, const ae_int_t m, const real_1d_array &pattern, const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`corrr1dcircularbuf` function

/*************************************************************************
1-dimensional circular real cross-correlation,  buffered  version ,  which
does not reallocate C[] if its length is enough to store the result  (i.e.
it reuses previously allocated memory as much as possible).

  -- ALGLIB --
     Copyright 21.07.2009 by Bochkanov Sergey
*************************************************************************/
void corrr1dcircularbuf(const real_1d_array &signal, const ae_int_t m, const real_1d_array &pattern, const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`correlationtests` subpackage

Functions

pearsoncorrelationsignificance
spearmanrankcorrelationsignificance

Examples

`pearsoncorrelationsignificance` function

/*************************************************************************
Pearson's correlation coefficient significance test

This test checks hypotheses about whether X  and  Y  are  samples  of  two
continuous  distributions  having  zero  correlation  or   whether   their
correlation is non-zero.

The following tests are performed:
    * two-tailed test (null hypothesis - X and Y have zero correlation)
    * left-tailed test (null hypothesis - the correlation  coefficient  is
      greater than or equal to 0)
    * right-tailed test (null hypothesis - the correlation coefficient  is
      less than or equal to 0).

Requirements:
    * the number of elements in each sample is not less than 5
    * normality of distributions of X and Y.

Input parameters:
    R   -   Pearson's correlation coefficient for X and Y
    N   -   number of elements in samples, N>=5.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void pearsoncorrelationsignificance(const double r, const ae_int_t n, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`spearmanrankcorrelationsignificance` function

/*************************************************************************
Spearman's rank correlation coefficient significance test

This test checks hypotheses about whether X  and  Y  are  samples  of  two
continuous  distributions  having  zero  correlation  or   whether   their
correlation is non-zero.

The following tests are performed:
    * two-tailed test (null hypothesis - X and Y have zero correlation)
    * left-tailed test (null hypothesis - the correlation  coefficient  is
      greater than or equal to 0)
    * right-tailed test (null hypothesis - the correlation coefficient  is
      less than or equal to 0).

Requirements:
    * the number of elements in each sample is not less than 5.

The test is non-parametric and doesn't require distributions X and Y to be
normal.

Input parameters:
    R   -   Spearman's rank correlation coefficient for X and Y
    N   -   number of elements in samples, N>=5.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void spearmanrankcorrelationsignificance(const double r, const ae_int_t n, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`datacomp` subpackage

Functions

kmeansgenerate

Examples

`kmeansgenerate` function

/*************************************************************************
k-means++ clusterization.
Backward compatibility function, we recommend to use CLUSTERING subpackage
as better replacement.

  -- ALGLIB --
     Copyright 21.03.2009 by Bochkanov Sergey
*************************************************************************/
void kmeansgenerate(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t k, const ae_int_t restarts, ae_int_t &info, real_2d_array &c, integer_1d_array &xyc, const xparams _xparams = alglib::xdefault);

`dawson` subpackage

Functions

dawsonintegral

Examples

`dawsonintegral` function

/*************************************************************************
Dawson's Integral

Approximates the integral

                            x
                            -
                     2     | |        2
 dawsn(x)  =  exp( -x  )   |    exp( t  ) dt
                         | |
                          -
                          0

Three different rational approximations are employed, for
the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,10        10000       6.9e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double dawsonintegral(const double x, const xparams _xparams = alglib::xdefault);

`dforest` subpackage

Classes

decisionforest
decisionforestbuffer
decisionforestbuilder
dfreport

Functions

dfavgce
dfavgerror
dfavgrelerror
dfbinarycompression
dfbuilderbuildrandomforest
dfbuildercreate
dfbuildergetprogress
dfbuilderpeekprogress
dfbuildersetdataset
dfbuildersetimportancenone
dfbuildersetimportanceoobgini
dfbuildersetimportancepermutation
dfbuildersetimportancetrngini
dfbuildersetrdfalgo
dfbuildersetrdfsplitstrength
dfbuildersetrndvars
dfbuildersetrndvarsauto
dfbuildersetrndvarsratio
dfbuildersetseed
dfbuildersetsubsampleratio
dfbuildrandomdecisionforest
dfbuildrandomdecisionforestx1
dfclassify
dfcreatebuffer
dfprocess
dfprocess0
dfprocessi
dfrelclserror
dfrmserror
dfserialize
dftsprocess
dfunserialize

Examples

randomforest_cls		Simple classification with random forests
randomforest_reg		Simple regression with decision forest

`decisionforest` class

/*************************************************************************
Decision forest (random forest) model.
*************************************************************************/
class decisionforest
{
public:
    decisionforest();
    decisionforest(const decisionforest &rhs);
    decisionforest& operator=(const decisionforest &rhs);
    virtual ~decisionforest();
};

`decisionforestbuffer` class

/*************************************************************************
Buffer object which is used to perform  various  requests  (usually  model
inference) in the multithreaded mode (multiple threads working  with  same
DF object).

This object should be created with DFCreateBuffer().
*************************************************************************/
class decisionforestbuffer
{
public:
    decisionforestbuffer();
    decisionforestbuffer(const decisionforestbuffer &rhs);
    decisionforestbuffer& operator=(const decisionforestbuffer &rhs);
    virtual ~decisionforestbuffer();
};

`decisionforestbuilder` class

/*************************************************************************
A random forest (decision forest) builder object.

Used to store dataset and specify decision forest training algorithm settings.
*************************************************************************/
class decisionforestbuilder
{
public:
    decisionforestbuilder();
    decisionforestbuilder(const decisionforestbuilder &rhs);
    decisionforestbuilder& operator=(const decisionforestbuilder &rhs);
    virtual ~decisionforestbuilder();
};

`dfreport` class

/*************************************************************************
Decision forest training report.

=== training/oob errors ==================================================

Following fields store training set errors:
* relclserror           -   fraction of misclassified cases, [0,1]
* avgce                 -   average cross-entropy in bits per symbol
* rmserror              -   root-mean-square error
* avgerror              -   average error
* avgrelerror           -   average relative error

Out-of-bag estimates are stored in fields with same names, but "oob" prefix.

For classification problems:
* RMS, AVG and AVGREL errors are calculated for posterior probabilities

For regression problems:
* RELCLS and AVGCE errors are zero

=== variable importance ==================================================

Following fields are used to store variable importance information:

* topvars               -   variables ordered from the most  important  to
                            less  important  ones  (according  to  current
                            choice of importance raiting).
                            For example, topvars[0] contains index of  the
                            most important variable, and topvars[0:2]  are
                            indexes of 3 most important ones and so on.

* varimportances        -   array[nvars], ratings (the  larger,  the  more
                            important the variable  is,  always  in  [0,1]
                            range).
                            By default, filled  by  zeros  (no  importance
                            ratings are  provided  unless  you  explicitly
                            request them).
                            Zero rating means that variable is not important,
                            however you will rarely encounter such a thing,
                            in many cases  unimportant  variables  produce
                            nearly-zero (but nonzero) ratings.

Variable importance report must be EXPLICITLY requested by calling:
* dfbuildersetimportancegini() function, if you need out-of-bag Gini-based
  importance rating also known as MDI  (fast to  calculate,  resistant  to
  overfitting  issues,   but   has   some   bias  towards  continuous  and
  high-cardinality categorical variables)
* dfbuildersetimportancetrngini() function, if you need training set Gini-
  -based importance rating (what other packages typically report).
* dfbuildersetimportancepermutation() function, if you  need  permutation-
  based importance rating also known as MDA (slower to calculate, but less
  biased)
* dfbuildersetimportancenone() function,  if  you  do  not  need  importance
  ratings - ratings will be zero, topvars[] will be [0,1,2,...]

Different importance ratings (Gini or permutation) produce  non-comparable
values. Although in all cases rating values lie in [0,1] range, there  are
exist differences:
* informally speaking, Gini importance rating tends to divide "unit amount
  of importance"  between  several  important  variables, i.e. it produces
  estimates which roughly sum to 1.0 (or less than 1.0, if your  task  can
  not be solved exactly). If all variables  are  equally  important,  they
  will have same rating,  roughly  1/NVars,  even  if  every  variable  is
  critically important.
* from the other side, permutation importance tells us what percentage  of
  the model predictive power will be ruined  by  permuting  this  specific
  variable. It does not produce estimates which  sum  to  one.  Critically
  important variable will have rating close  to  1.0,  and  you  may  have
  multiple variables with such a rating.

More information on variable importance ratings can be found  in  comments
on the dfbuildersetimportancegini() and dfbuildersetimportancepermutation()
functions.
*************************************************************************/
class dfreport
{
public:
    dfreport();
    dfreport(const dfreport &rhs);
    dfreport& operator=(const dfreport &rhs);
    virtual ~dfreport();
    double relclserror;
    double avgce;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double oobrelclserror;
    double oobavgce;
    double oobrmserror;
    double oobavgerror;
    double oobavgrelerror;
    integer_1d_array topvars;
    real_1d_array varimportances;
};

`dfavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/(NPoints*LN(2)).
    Zero if model solves regression task.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double dfavgce(const decisionforest &df, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`dfavgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for
    classification task, it means average error when estimating posterior
    probabilities.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double dfavgerror(const decisionforest &df, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`dfavgrelerror` function

/*************************************************************************
Average relative error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for
    classification task, it means average relative error when estimating
    posterior probability of belonging to the correct class.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double dfavgrelerror(const decisionforest &df, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`dfbinarycompression` function

/*************************************************************************
This function performs binary compression of the decision forest.

Original decision forest produced by the  forest  builder  is stored using
64-bit representation for all numbers - offsets, variable  indexes,  split
points.

It is possible to significantly reduce model size by means of:
* using compressed  dynamic encoding for integers  (offsets  and  variable
  indexes), which uses just 1 byte to store small ints  (less  than  128),
  just 2 bytes for larger values (less than 128^2) and so on
* storing floating point numbers using 8-bit exponent and 16-bit mantissa

As  result,  model  needs  significantly  less  memory (compression factor
depends on  variable and class counts). In particular:
* NVars<128   and NClasses<128 result in 4.4x-5.7x model size reduction
* NVars<16384 and NClasses<128 result in 3.7x-4.5x model size reduction

Such storage format performs lossless compression  of  all  integers,  but
compression of floating point values (split values) is lossy, with roughly
0.01% relative error introduced during rounding. Thus, we recommend you to
re-evaluate model accuracy after compression.

Another downside  of  compression  is  ~1.5x reduction  in  the  inference
speed due to necessity of dynamic decompression of the compressed model.

INPUT PARAMETERS:
    DF      -   decision forest built by forest builder

OUTPUT PARAMETERS:
    DF      -   replaced by compressed forest

RESULT:
    compression factor (in-RAM size of the compressed model vs than of the
    uncompressed one), positive number larger than 1.0

  -- ALGLIB --
     Copyright 22.07.2019 by Bochkanov Sergey
*************************************************************************/
double dfbinarycompression(decisionforest &df, const xparams _xparams = alglib::xdefault);

`dfbuilderbuildrandomforest` function

/*************************************************************************
This subroutine builds decision forest according to current settings using
dataset internally stored in the builder object. Dense algorithm is used.

NOTE: this   function   uses   dense  algorithm  for  forest  construction
      independently from the dataset format (dense or sparse).

NOTE: forest built with this function is  stored  in-memory  using  64-bit
      data structures for offsets/indexes/split values. It is possible  to
      convert  forest  into  more  memory-efficient   compressed    binary
      representation.  Depending  on  the  problem  properties,  3.7x-5.7x
      compression factors are possible.

      The downsides of compression are (a) slight reduction in  the  model
      accuracy and (b) ~1.5x reduction in  the  inference  speed  (due  to
      increased complexity of the storage format).

      See comments on dfbinarycompression() for more info.

Default settings are used by the algorithm; you can tweak  them  with  the
help of the following functions:
* dfbuildersetrfactor() - to control a fraction of the  dataset  used  for
  subsampling
* dfbuildersetrandomvars() - to control number of variables randomly chosen
  for decision rule creation

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   decision forest builder object
    NTrees      -   NTrees>=1, number of trees to train

OUTPUT PARAMETERS:
    DF          -   decision forest. You can compress this forest to  more
                    compact 16-bit representation with dfbinarycompression()
    Rep         -   report, see below for information on its fields.

=== report information produced by forest construction function ==========

Decision forest training report includes following information:
* training set errors
* out-of-bag estimates of errors
* variable importance ratings

Following fields are used to store information:
* training set errors are stored in rep.relclserror, rep.avgce, rep.rmserror,
  rep.avgerror and rep.avgrelerror
* out-of-bag estimates of errors are stored in rep.oobrelclserror, rep.oobavgce,
  rep.oobrmserror, rep.oobavgerror and rep.oobavgrelerror

Variable importance reports, if requested by dfbuildersetimportancegini(),
dfbuildersetimportancetrngini() or dfbuildersetimportancepermutation()
call, are stored in:
* rep.varimportances field stores importance ratings
* rep.topvars stores variable indexes ordered from the most important to
  less important ones

You can find more information about report fields in:
* comments on dfreport structure
* comments on dfbuildersetimportancegini function
* comments on dfbuildersetimportancetrngini function
* comments on dfbuildersetimportancepermutation function

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuilderbuildrandomforest(decisionforestbuilder &s, const ae_int_t ntrees, decisionforest &df, dfreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildercreate` function

/*************************************************************************
This subroutine creates DecisionForestBuilder  object  which  is  used  to
train decision forests.

By default, new builder stores empty dataset and some  reasonable  default
settings. At the very least, you should specify dataset prior to  building
decision forest. You can also tweak settings of  the  forest  construction
algorithm (recommended, although default setting should work well).

Following actions are mandatory:
* calling dfbuildersetdataset() to specify dataset
* calling dfbuilderbuildrandomforest()  to  build  decision  forest  using
  current dataset and default settings

Additionally, you may call:
* dfbuildersetrndvars() or dfbuildersetrndvarsratio() to specify number of
  variables randomly chosen for each split
* dfbuildersetsubsampleratio() to specify fraction of the dataset randomly
  subsampled to build each tree
* dfbuildersetseed() to control random seed chosen for tree construction

INPUT PARAMETERS:
    none

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildercreate(decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildergetprogress` function

/*************************************************************************
This function is an alias for dfbuilderpeekprogress(), left in ALGLIB  for
backward compatibility reasons.

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
double dfbuildergetprogress(const decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuilderpeekprogress` function

/*************************************************************************
This function is used to peek into  decision  forest  construction process
from some other thread and get current progress indicator.

It returns value in [0,1].

INPUT PARAMETERS:
    S           -   decision forest builder object used  to  build  forest
                    in some other thread

RESULT:
    progress value, in [0,1]

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
double dfbuilderpeekprogress(const decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

`dfbuildersetdataset` function

/*************************************************************************
This subroutine adds dense dataset to the internal storage of the  builder
object. Specifying your dataset in the dense format means that  the  dense
version of the forest construction algorithm will be invoked.

INPUT PARAMETERS:
    S           -   decision forest builder object
    XY          -   array[NPoints,NVars+1] (minimum size; actual size  can
                    be larger, only leading part is used anyway), dataset:
                    * first NVars elements of each row store values of the
                      independent variables
                    * last  column  store class number (in 0...NClasses-1)
                      or real value of the dependent variable
    NPoints     -   number of rows in the dataset, NPoints>=1
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   indicates type of the problem being solved:
                    * NClasses>=2 means  that  classification  problem  is
                      solved  (last  column  of  the  dataset stores class
                      number)
                    * NClasses=1 means that regression problem  is  solved
                      (last column of the dataset stores variable value)

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetdataset(decisionforestbuilder &s, const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildersetimportancenone` function

/*************************************************************************
This  function  tells  decision  forest  construction  algorithm  to  skip
variable importance estimation.

INPUT PARAMETERS:
    S           -   decision forest builder object

OUTPUT PARAMETERS:
    S           -   decision forest builder object. Next call to the forest
                    construction function will result in forest being built
                    without variable importance estimation.

  -- ALGLIB --
     Copyright 29.07.2019 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetimportancenone(decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

`dfbuildersetimportanceoobgini` function

/*************************************************************************
This  function  tells  decision  forest  construction  algorithm  to   use
out-of-bag version of Gini variable importance estimation (also  known  as
OOB-MDI).

This version of importance estimation algorithm analyzes mean decrease  in
impurity (MDI) on out-of-bag sample during splits. The result  is  divided
by impurity at the root node in order to produce estimate in [0,1] range.

Such estimates are fast to calculate and resistant to  overfitting  issues
(thanks to the  out-of-bag  estimates  used). However, OOB Gini rating has
following downsides:
* there exist some bias towards continuous and high-cardinality categorical
  variables
* Gini rating allows us to order variables by importance, but it  is  hard
  to define importance of the variable by itself.

NOTE: informally speaking, MDA (permutation importance) rating answers the
      question  "what  part  of  the  model  predictive power is ruined by
      permuting k-th variable?" while MDI tells us "what part of the model
      predictive power was achieved due to usage of k-th variable".

      Thus, MDA rates each variable independently at "0 to 1"  scale while
      MDI (and OOB-MDI too) tends to divide "unit  amount  of  importance"
      between several important variables.

      If  all  variables  are  equally  important,  they  will  have  same
      MDI/OOB-MDI rating, equal (for OOB-MDI: roughly equal)  to  1/NVars.
      However, roughly  same  picture  will  be  produced   for  the  "all
      variables provide information no one is critical" situation  and for
      the "all variables are critical, drop any one, everything is ruined"
      situation.

      Contrary to that, MDA will rate critical variable as ~1.0 important,
      and important but non-critical variable will  have  less  than  unit
      rating.

NOTE: quite an often MDA and MDI return same results. It generally happens
      on problems with low test set error (a few  percents  at  most)  and
      large enough training set to avoid overfitting.

      The difference between MDA, MDI and OOB-MDI becomes  important  only
      on "hard" tasks with high test set error and/or small training set.

INPUT PARAMETERS:
    S           -   decision forest builder object

OUTPUT PARAMETERS:
    S           -   decision forest builder object. Next call to the forest
                    construction function will produce:
                    * importance estimates in rep.varimportances field
                    * variable ranks in rep.topvars field

  -- ALGLIB --
     Copyright 29.07.2019 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetimportanceoobgini(decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

`dfbuildersetimportancepermutation` function

/*************************************************************************
This  function  tells  decision  forest  construction  algorithm  to   use
permutation variable importance estimator (also known as MDA).

This version of importance estimation algorithm analyzes mean increase  in
out-of-bag sum of squared  residuals  after  random  permutation  of  J-th
variable. The result is divided by error computed with all variables being
perturbed in order to produce R-squared-like estimate in [0,1] range.

Such estimate  is  slower to calculate than Gini-based rating  because  it
needs multiple inference runs for each of variables being studied.

ALGLIB uses parallelized and highly  optimized  algorithm  which  analyzes
path through the decision tree and allows  to  handle  most  perturbations
in O(1) time; nevertheless, requesting MDA importances may increase forest
construction time from 10% to 200% (or more,  if  you  have  thousands  of
variables).

However, MDA rating has following benefits over Gini-based ones:
* no bias towards specific variable types
* ability to directly evaluate "absolute" importance of some  variable  at
  "0 to 1" scale (contrary to Gini-based rating, which returns comparative
  importances).

NOTE: informally speaking, MDA (permutation importance) rating answers the
      question  "what  part  of  the  model  predictive power is ruined by
      permuting k-th variable?" while MDI tells us "what part of the model
      predictive power was achieved due to usage of k-th variable".

      Thus, MDA rates each variable independently at "0 to 1"  scale while
      MDI (and OOB-MDI too) tends to divide "unit  amount  of  importance"
      between several important variables.

      If  all  variables  are  equally  important,  they  will  have  same
      MDI/OOB-MDI rating, equal (for OOB-MDI: roughly equal)  to  1/NVars.
      However, roughly  same  picture  will  be  produced   for  the  "all
      variables provide information no one is critical" situation  and for
      the "all variables are critical, drop any one, everything is ruined"
      situation.

      Contrary to that, MDA will rate critical variable as ~1.0 important,
      and important but non-critical variable will  have  less  than  unit
      rating.

NOTE: quite an often MDA and MDI return same results. It generally happens
      on problems with low test set error (a few  percents  at  most)  and
      large enough training set to avoid overfitting.

      The difference between MDA, MDI and OOB-MDI becomes  important  only
      on "hard" tasks with high test set error and/or small training set.

INPUT PARAMETERS:
    S           -   decision forest builder object

OUTPUT PARAMETERS:
    S           -   decision forest builder object. Next call to the forest
                    construction function will produce:
                    * importance estimates in rep.varimportances field
                    * variable ranks in rep.topvars field

  -- ALGLIB --
     Copyright 29.07.2019 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetimportancepermutation(decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

`dfbuildersetimportancetrngini` function

/*************************************************************************
This  function  tells  decision  forest  construction  algorithm  to   use
Gini impurity based variable importance estimation (also known as MDI).

This version of importance estimation algorithm analyzes mean decrease  in
impurity (MDI) on training sample during  splits.  The result  is  divided
by impurity at the root node in order to produce estimate in [0,1] range.

Such estimates are fast to calculate and beautifully  normalized  (sum  to
one) but have following downsides:
* They ALWAYS sum to 1.0, even if output is completely unpredictable. I.e.
  MDI allows to order variables by importance, but does not  tell us about
  "absolute" importances of variables
* there exist some bias towards continuous and high-cardinality categorical
  variables

NOTE: informally speaking, MDA (permutation importance) rating answers the
      question  "what  part  of  the  model  predictive power is ruined by
      permuting k-th variable?" while MDI tells us "what part of the model
      predictive power was achieved due to usage of k-th variable".

      Thus, MDA rates each variable independently at "0 to 1"  scale while
      MDI (and OOB-MDI too) tends to divide "unit  amount  of  importance"
      between several important variables.

      If  all  variables  are  equally  important,  they  will  have  same
      MDI/OOB-MDI rating, equal (for OOB-MDI: roughly equal)  to  1/NVars.
      However, roughly  same  picture  will  be  produced   for  the  "all
      variables provide information no one is critical" situation  and for
      the "all variables are critical, drop any one, everything is ruined"
      situation.

      Contrary to that, MDA will rate critical variable as ~1.0 important,
      and important but non-critical variable will  have  less  than  unit
      rating.

NOTE: quite an often MDA and MDI return same results. It generally happens
      on problems with low test set error (a few  percents  at  most)  and
      large enough training set to avoid overfitting.

      The difference between MDA, MDI and OOB-MDI becomes  important  only
      on "hard" tasks with high test set error and/or small training set.

INPUT PARAMETERS:
    S           -   decision forest builder object

OUTPUT PARAMETERS:
    S           -   decision forest builder object. Next call to the forest
                    construction function will produce:
                    * importance estimates in rep.varimportances field
                    * variable ranks in rep.topvars field

  -- ALGLIB --
     Copyright 29.07.2019 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetimportancetrngini(decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

`dfbuildersetrdfalgo` function

/*************************************************************************
This function sets random decision forest construction algorithm.

As for now, only one decision forest construction algorithm is supported -
a dense "baseline" RDF algorithm.

INPUT PARAMETERS:
    S           -   decision forest builder object
    AlgoType    -   algorithm type:
                    * 0 = baseline dense RDF

OUTPUT PARAMETERS:
    S           -   decision forest builder, see

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetrdfalgo(decisionforestbuilder &s, const ae_int_t algotype, const xparams _xparams = alglib::xdefault);

`dfbuildersetrdfsplitstrength` function

/*************************************************************************
This  function  sets  split  selection  algorithm used by decision  forest
classifier. You may choose several algorithms, with  different  speed  and
quality of the results.

INPUT PARAMETERS:
    S           -   decision forest builder object
    SplitStrength-  split type:
                    * 0 = split at the random position, fastest one
                    * 1 = split at the middle of the range
                    * 2 = strong split at the best point of the range (default)

OUTPUT PARAMETERS:
    S           -   decision forest builder, see

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetrdfsplitstrength(decisionforestbuilder &s, const ae_int_t splitstrength, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildersetrndvars` function

/*************************************************************************
This function sets number  of  variables  (in  [1,NVars]  range)  used  by
decision forest construction algorithm.

The default option is to use roughly sqrt(NVars) variables.

INPUT PARAMETERS:
    S           -   decision forest builder object
    RndVars     -   number of randomly selected variables; values  outside
                    of [1,NVars] range are silently clipped.

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetrndvars(decisionforestbuilder &s, const ae_int_t rndvars, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildersetrndvarsauto` function

/*************************************************************************
This function tells decision forest builder to automatically choose number
of  variables  used  by  decision forest construction  algorithm.  Roughly
sqrt(NVars) variables will be used.

INPUT PARAMETERS:
    S           -   decision forest builder object

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetrndvarsauto(decisionforestbuilder &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildersetrndvarsratio` function

/*************************************************************************
This function sets number of variables used by decision forest construction
algorithm as a fraction of total variable count (0,1) range.

The default option is to use roughly sqrt(NVars) variables.

INPUT PARAMETERS:
    S           -   decision forest builder object
    F           -   round(NVars*F) variables are selected

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetrndvarsratio(decisionforestbuilder &s, const double f, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildersetseed` function

/*************************************************************************
This function sets seed used by internal RNG for  random  subsampling  and
random selection of variable subsets.

By default random seed is used, i.e. every time you build decision forest,
we seed generator with new value  obtained  from  system-wide  RNG.  Thus,
decision forest builder returns non-deterministic results. You can  change
such behavior by specyfing fixed positive seed value.

INPUT PARAMETERS:
    S           -   decision forest builder object
    SeedVal     -   seed value:
                    * positive values are used for seeding RNG with fixed
                      seed, i.e. subsequent runs on same data will return
                      same decision forests
                    * non-positive seed means that random seed is used
                      for every run of builder, i.e. subsequent  runs  on
                      same  datasets  will  return   slightly   different
                      decision forests

OUTPUT PARAMETERS:
    S           -   decision forest builder, see

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetseed(decisionforestbuilder &s, const ae_int_t seedval, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildersetsubsampleratio` function

/*************************************************************************
This function sets size of dataset subsample generated the decision forest
construction algorithm. Size is specified as a fraction of  total  dataset
size.

The default option is to use 50% of the dataset for training, 50% for  the
OOB estimates. You can decrease fraction F down to 10%, 1% or  even  below
in order to reduce overfitting.

INPUT PARAMETERS:
    S           -   decision forest builder object
    F           -   fraction of the dataset to use, in (0,1] range. Values
                    outside of this range will  be  silently  clipped.  At
                    least one element is always selected for the  training
                    set.

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
void dfbuildersetsubsampleratio(decisionforestbuilder &s, const double f, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfbuildrandomdecisionforest` function

/*************************************************************************
This subroutine builds random decision forest.

--------- DEPRECATED VERSION! USE DECISION FOREST BUILDER OBJECT ---------

  -- ALGLIB --
     Copyright 19.02.2009 by Bochkanov Sergey
*************************************************************************/
void dfbuildrandomdecisionforest(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, const ae_int_t ntrees, const double r, ae_int_t &info, decisionforest &df, dfreport &rep, const xparams _xparams = alglib::xdefault);

`dfbuildrandomdecisionforestx1` function

/*************************************************************************
This subroutine builds random decision forest.

--------- DEPRECATED VERSION! USE DECISION FOREST BUILDER OBJECT ---------

  -- ALGLIB --
     Copyright 19.02.2009 by Bochkanov Sergey
*************************************************************************/
void dfbuildrandomdecisionforestx1(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, const ae_int_t ntrees, const ae_int_t nrndvars, const double r, ae_int_t &info, decisionforest &df, dfreport &rep, const xparams _xparams = alglib::xdefault);

`dfclassify` function

/*************************************************************************
This function returns most probable class number for an  input  X.  It  is
same as calling  dfprocess(model,x,y), then determining i=argmax(y[i]) and
returning i.

A class number in [0,NOut) range in returned for classification  problems,
-1 is returned when this function is called for regression problems.

IMPORTANT: this function is thread-unsafe and modifies internal structures
           of the model! You can not use same model  object  for  parallel
           evaluation from several threads.

           Use dftsprocess()  with independent  thread-local  buffers,  if
           you need thread-safe evaluation.

INPUT PARAMETERS:
    Model   -   decision forest model
    X       -   input vector,  array[0..NVars-1].

RESULT:
    class number, -1 for regression tasks

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
ae_int_t dfclassify(decisionforest &model, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfcreatebuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel inference requests.

DF subpackage  provides two sets of computing functions - ones  which  use
internal buffer of DF model  (these  functions are single-threaded because
they use same buffer, which can not  shared  between  threads),  and  ones
which use external buffer.

This function is used to initialize external buffer.

INPUT PARAMETERS
    Model       -   DF model which is associated with newly created buffer

OUTPUT PARAMETERS
    Buf         -   external buffer.


IMPORTANT: buffer object should be used only with model which was used  to
           initialize buffer. Any attempt to  use  buffer  with  different
           object is dangerous - you  may   get  integrity  check  failure
           (exception) because sizes of internal  arrays  do  not  fit  to
           dimensions of the model structure.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void dfcreatebuffer(const decisionforest &model, decisionforestbuffer &buf, const xparams _xparams = alglib::xdefault);

`dfprocess` function

/*************************************************************************
Inference using decision forest

IMPORTANT: this  function  is  thread-unsafe  and  may   modify   internal
           structures of the model! You can not use same model  object for
           parallel evaluation from several threads.

           Use dftsprocess()  with  independent  thread-local  buffers  if
           you need thread-safe evaluation.

INPUT PARAMETERS:
    DF      -   decision forest model
    X       -   input vector,  array[NVars]
    Y       -   possibly preallocated buffer, reallocated if too small

OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

See also DFProcessI.


  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
void dfprocess(const decisionforest &df, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfprocess0` function

/*************************************************************************
This function returns first component of the  inferred  vector  (i.e.  one
with index #0).

It is a convenience wrapper for dfprocess() intended for either:
* 1-dimensional regression problems
* 2-class classification problems

In the former case this function returns inference result as scalar, which
is definitely more convenient that wrapping it as vector.  In  the  latter
case it returns probability of object belonging to class #0.

If you call it for anything different from two cases above, it  will  work
as defined, i.e. return y[0], although it is of less use in such cases.

IMPORTANT: this function is thread-unsafe and modifies internal structures
           of the model! You can not use same model  object  for  parallel
           evaluation from several threads.

           Use dftsprocess() with  independent  thread-local  buffers,  if
           you need thread-safe evaluation.

INPUT PARAMETERS:
    Model   -   DF model
    X       -   input vector,  array[0..NVars-1].

RESULT:
    Y[0]

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double dfprocess0(decisionforest &model, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`dfprocessi` function

/*************************************************************************
'interactive' variant of DFProcess for languages like Python which support
constructs like "Y = DFProcessI(DF,X)" and interactive mode of interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

IMPORTANT: this  function  is  thread-unsafe  and  may   modify   internal
           structures of the model! You can not use same model  object for
           parallel evaluation from several threads.

           Use dftsprocess()  with  independent  thread-local  buffers  if
           you need thread-safe evaluation.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void dfprocessi(const decisionforest &df, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`dfrelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.
    Zero if model solves regression task.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double dfrelclserror(const decisionforest &df, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`dfrmserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    DF      -   decision forest model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.
    Its meaning for regression task is obvious. As for
    classification task, RMS error means error when estimating posterior
    probabilities.

  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
double dfrmserror(const decisionforest &df, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`dfserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void dfserialize(const decisionforest &obj, std::string &s_out);
void dfserialize(const decisionforest &obj, std::ostream &s_out);

`dftsprocess` function

/*************************************************************************
Inference using decision forest

Thread-safe procesing using external buffer for temporaries.

This function is thread-safe (i.e .  you  can  use  same  DF   model  from
multiple threads) as long as you use different buffer objects for different
threads.

INPUT PARAMETERS:
    DF      -   decision forest model
    Buf     -   buffer object, must be  allocated  specifically  for  this
                model with dfcreatebuffer().
    X       -   input vector,  array[NVars]
    Y       -   possibly preallocated buffer, reallocated if too small

OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

See also DFProcessI.


  -- ALGLIB --
     Copyright 16.02.2009 by Bochkanov Sergey
*************************************************************************/
void dftsprocess(const decisionforest &df, decisionforestbuffer &buf, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`dfunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void dfunserialize(const std::string &s_in, decisionforest &obj);
void dfunserialize(const std::istream &s_in, decisionforest &obj);

randomforest_cls example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple classification example: classify points (x,y) in 2D space
        // as ones with x>=0 and ones with x<0 (y is ignored, but our classifier
        // has to find out it).
        //
        // First, we have to create decision forest builder object, load dataset and
        // specify training settings. Our dataset is specified as matrix, which has
        // following format:
        //
        //     x0 y0 class0
        //     x1 y1 class1
        //     x2 y2 class2
        //     ....
        //
        // Here xi and yi can be any values (and in fact you can have any number of
        // independent variables), and classi MUST be integer number in [0,NClasses)
        // range. In our example we denote points with x>=0 as class #0, and
        // ones with negative xi as class #1.
        //
        // NOTE: if you want to solve regression problem, specify NClasses=1. In
        //       this case last column of xy can be any numeric value.
        //
        // For the sake of simplicity, our example includes only 4-point dataset.
        // However, random forests are able to cope with extremely large datasets
        // having millions of examples.
        //
        decisionforestbuilder builder;
        ae_int_t nvars = 2;
        ae_int_t nclasses = 2;
        ae_int_t npoints = 4;
        real_2d_array xy = "[[1,1,0],[1,-1,0],[-1,1,1],[-1,-1,1]]";

        dfbuildercreate(builder);
        dfbuildersetdataset(builder, xy, npoints, nvars, nclasses);

        // in our example we train decision forest using full sample - it allows us
        // to get zero classification error. However, in practical applications smaller
        // values are used: 50%, 25%, 5% or even less.
        dfbuildersetsubsampleratio(builder, 1.0);

        // we train random forest with just one tree; again, in real life situations
        // you typically need from 50 to 500 trees.
        ae_int_t ntrees = 1;
        decisionforest forest;
        dfreport rep;
        dfbuilderbuildrandomforest(builder, ntrees, forest, rep);

        // with such settings (100% of the training set is used) you can expect
        // zero classification error. Beautiful results, but remember - in real life
        // you do not need zero TRAINING SET error, you need good generalization.

        printf("%.4f\n", double(rep.relclserror)); // EXPECTED: 0.0000

        // now, let's perform some simple processing with dfprocess()
        real_1d_array x = "[+1,0]";
        real_1d_array y = "[]";
        dfprocess(forest, x, y);
        printf("%s\n", y.tostring(3).c_str()); // EXPECTED: [+1,0]

        // another option is to use dfprocess0() which returns just first component
        // of the output vector y. ideal for regression problems and binary classifiers.
        double y0;
        y0 = dfprocess0(forest, x);
        printf("%.3f\n", double(y0)); // EXPECTED: 1.000

        // finally, you can use dfclassify() which returns most probable class index (i.e. argmax y[i]).
        ae_int_t i;
        i = dfclassify(forest, x);
        printf("%d\n", int(i)); // EXPECTED: 0
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

randomforest_reg example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple regression example: model f(x,y)=x+y
        //
        // First, we have to create DF builder object, load dataset and specify
        // training settings. Our dataset is specified as matrix, which has following
        // format:
        //
        //     x0 y0 f0
        //     x1 y1 f1
        //     x2 y2 f2
        //     ....
        //
        // Here xi and yi can be any values, and fi is a dependent function value.
        //
        // NOTE: you can also solve classification problems with DF models, see
        //       another example for this unit.
        //
        decisionforestbuilder builder;
        ae_int_t nvars = 2;
        ae_int_t nclasses = 1;
        ae_int_t npoints = 4;
        real_2d_array xy = "[[1,1,+2],[1,-1,0],[-1,1,0],[-1,-1,-2]]";

        dfbuildercreate(builder);
        dfbuildersetdataset(builder, xy, npoints, nvars, nclasses);

        // in our example we train decision forest using full sample - it allows us
        // to get zero classification error. However, in practical applications smaller
        // values are used: 50%, 25%, 5% or even less.
        dfbuildersetsubsampleratio(builder, 1.0);

        // we train random forest with just one tree; again, in real life situations
        // you typically need from 50 to 500 trees.
        ae_int_t ntrees = 1;
        decisionforest model;
        dfreport rep;
        dfbuilderbuildrandomforest(builder, ntrees, model, rep);

        // with such settings (full sample is used) you can expect zero RMS error on the
        // training set. Beautiful results, but remember - in real life you do not
        // need zero TRAINING SET error, you need good generalization.

        printf("%.4f\n", double(rep.rmserror)); // EXPECTED: 0.0000

        // now, let's perform some simple processing with dfprocess()
        real_1d_array x = "[+1,+1]";
        real_1d_array y = "[]";
        dfprocess(model, x, y);
        printf("%s\n", y.tostring(3).c_str()); // EXPECTED: [+2]

        // another option is to use dfprocess0() which returns just first component
        // of the output vector y. ideal for regression problems and binary classifiers.
        double y0;
        y0 = dfprocess0(model, x);
        printf("%.3f\n", double(y0)); // EXPECTED: 2.000

        // there also exist another convenience function, dfclassify(),
        // but it does not work for regression problems - it always returns -1.
        ae_int_t i;
        i = dfclassify(model, x);
        printf("%d\n", int(i)); // EXPECTED: -1
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`directdensesolvers` subpackage

Classes

densesolverlsreport
densesolverreport

Functions

Examples

solve_complex		Solving dense complex linear equations
solve_complex_m		Solving complex matrix equations
solve_hpd		Solving Hermitian positive definite linear equations
solve_ls		Solving dense linear equations in the least squares sense
solve_real		Solving dense linear equations
solve_real_m		Solving dense linear matrix equations
solve_spd		Solving symmetric positive definite linear equations

`densesolverlsreport` class

/*************************************************************************

*************************************************************************/
class densesolverlsreport
{
public:
    densesolverlsreport();
    densesolverlsreport(const densesolverlsreport &rhs);
    densesolverlsreport& operator=(const densesolverlsreport &rhs);
    virtual ~densesolverlsreport();
    ae_int_t terminationtype;
    double r2;
    real_2d_array cx;
    ae_int_t n;
    ae_int_t k;
};

`densesolverreport` class

/*************************************************************************

*************************************************************************/
class densesolverreport
{
public:
    densesolverreport();
    densesolverreport(const densesolverreport &rhs);
    densesolverreport& operator=(const densesolverreport &rhs);
    virtual ~densesolverreport();
    ae_int_t terminationtype;
    double r1;
    double rinf;
};

`cmatrixlusolve` function

/*************************************************************************
Complex dense linear solver for A*x=b with complex N*N A  given  by its LU
decomposition and N*1 vectors x and b. This is  "slow-but-robust"  version
of  the  complex  linear  solver  with  additional  features   which   add
significant performance overhead. Faster version  is  CMatrixLUSolveFast()
function.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation

No iterative refinement is provided because exact form of original matrix
is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! CMatrixLUSolveFast() function.

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixlusolve(const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixlusolve(const complex_2d_array &lua, const integer_1d_array &p, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixlusolvefast` function

/*************************************************************************
Complex dense linear solver for A*x=b with N*N complex A given by  its  LU
decomposition and N*1 vectors x and b. This is  fast  lightweight  version
of solver, which is significantly faster than CMatrixLUSolve(),  but  does
not provide additional information (like condition numbers).

Algorithm features:
* O(N^2) complexity
* no additional time-consuming features, just triangular solver

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

NOTE: unlike  CMatrixLUSolve(),  this   function   does   NOT   check  for
      near-degeneracy of input matrix. It  checks  for  EXACT  degeneracy,
      because this check is easy to do. However,  very  badly  conditioned
      matrices may went unnoticed.


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
bool cmatrixlusolvefast(const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, complex_1d_array &b, const xparams _xparams = alglib::xdefault);
bool cmatrixlusolvefast(const complex_2d_array &lua, const integer_1d_array &p, complex_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixlusolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N complex A given by its  LU  decomposition,
and N*M matrices X and B (multiple right sides).   "Slow-but-feature-rich"
version of the solver.

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation

No iterative refinement  is provided because exact form of original matrix
is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver.
           !
           ! This performance penalty is especially apparent when you  use
           ! ALGLIB parallel capabilities (condition number estimation  is
           ! inherently  sequential).  It   also   becomes significant for
           ! small-scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! CMatrixLUSolveMFast() function.

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixlusolvem(const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_2d_array &b, const ae_int_t m, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixlusolvem(const complex_2d_array &lua, const integer_1d_array &p, const complex_2d_array &b, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixlusolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N complex A given by its  LU  decomposition,
and N*M matrices X and B (multiple  right  sides).  "Fast-but-lightweight"
version of the solver.

Algorithm features:
* O(M*N^2) complexity
* no additional time-consuming features

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    B       -   array[N,M]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
bool cmatrixlusolvemfast(const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, complex_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool cmatrixlusolvemfast(const complex_2d_array &lua, const integer_1d_array &p, complex_2d_array &b, const xparams _xparams = alglib::xdefault);

`cmatrixmixedsolve` function

/*************************************************************************
Dense solver. Same as RMatrixMixedSolve(), but for complex matrices.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixmixedsolve(const complex_2d_array &a, const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixmixedsolve(const complex_2d_array &a, const complex_2d_array &lua, const integer_1d_array &p, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixmixedsolvem` function

/*************************************************************************
Dense solver. Same as RMatrixMixedSolveM(), but for complex matrices.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(M*N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, CMatrixLU result
    P       -   array[0..N-1], pivots array, CMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixmixedsolvem(const complex_2d_array &a, const complex_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const complex_2d_array &b, const ae_int_t m, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixmixedsolvem(const complex_2d_array &a, const complex_2d_array &lua, const integer_1d_array &p, const complex_2d_array &b, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixsolve` function

/*************************************************************************
Complex dense solver for A*x=B with N*N complex matrix A and  N*1  complex
vectors x and b. "Slow-but-feature-rich" version of the solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^3) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, CMatrixSolveFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixsolve(const complex_2d_array &a, const ae_int_t n, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixsolve(const complex_2d_array &a, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixsolvefast` function

/*************************************************************************
Complex dense solver for A*x=B with N*N complex matrix A and  N*1  complex
vectors x and b. "Fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^3) complexity
* no additional time consuming features, just triangular solver

INPUT PARAMETERS:
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS:
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
bool cmatrixsolvefast(const complex_2d_array &a, const ae_int_t n, complex_1d_array &b, const xparams _xparams = alglib::xdefault);
bool cmatrixsolvefast(const complex_2d_array &a, complex_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixsolvem` function

/*************************************************************************
Complex dense solver for A*X=B with N*N  complex  matrix  A,  N*M  complex
matrices  X  and  B.  "Slow-but-feature-rich"   version   which   provides
additional functions, at the cost of slower  performance.  Faster  version
may be invoked with CMatrixSolveMFast() function.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^3+M*N^2) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, CMatrixSolveMFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size
    RFS     -   iterative refinement switch:
                * True - refinement is used.
                  Less performance, more precision.
                * False - refinement is not used.
                  More performance, less precision.

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixsolvem(const complex_2d_array &a, const ae_int_t n, const complex_2d_array &b, const ae_int_t m, const bool rfs, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixsolvem(const complex_2d_array &a, const complex_2d_array &b, const bool rfs, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixsolvemfast` function

/*************************************************************************
Complex dense solver for A*X=B with N*N  complex  matrix  A,  N*M  complex
matrices  X  and  B.  "Fast-but-lightweight" version which  provides  just
triangular solver - and no additional functions like iterative  refinement
or condition number estimation.

Algorithm features:
* O(N^3+M*N^2) complexity
* no additional time consuming functions

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS:
    B       -   array[N,M]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 16.03.2015 by Bochkanov Sergey
*************************************************************************/
bool cmatrixsolvemfast(const complex_2d_array &a, const ae_int_t n, complex_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool cmatrixsolvemfast(const complex_2d_array &a, complex_2d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`hpdmatrixcholeskysolve` function

/*************************************************************************
Dense solver for A*x=b with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition, and N*1 complex vectors x and  b.  This  is
"slow-but-feature-rich" version of the solver  which  estimates  condition
number of the system.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! HPDMatrixCholeskySolveFast() function.

INPUT PARAMETERS
    CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or indefinite matrix
                * rep.r1                condition number in 1-norm
                * rep.rinf              condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void hpdmatrixcholeskysolve(const complex_2d_array &cha, const ae_int_t n, const bool isupper, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixcholeskysolve(const complex_2d_array &cha, const bool isupper, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`hpdmatrixcholeskysolvefast` function

/*************************************************************************
Dense solver for A*x=b with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition, and N*1 complex vectors x and  b.  This  is
"fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time-consuming features

INPUT PARAMETERS
    CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or indefinite system

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
bool hpdmatrixcholeskysolvefast(const complex_2d_array &cha, const ae_int_t n, const bool isupper, complex_1d_array &b, const xparams _xparams = alglib::xdefault);
bool hpdmatrixcholeskysolvefast(const complex_2d_array &cha, const bool isupper, complex_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`hpdmatrixcholeskysolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition and N*M complex matrices X  and  B.  This is
"slow-but-feature-rich" version of the solver which, in  addition  to  the
solution, estimates condition number of the system.

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver. Amount of  overhead  introduced  depends  on  M  (the
           ! larger - the more efficient).
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large Cholesky decomposition.  However,  if  you call
           ! this  function  many  times  for  the same  left  side,  this
           ! overhead BECOMES significant. It  also   becomes  significant
           ! for small-scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! HPDMatrixCholeskySolveMFast() function.


INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                HPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[N,M], right part
    M       -   right part size

OUTPUT PARAMETERS:
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or indefinite matrix
                * rep.r1                condition number in 1-norm
                * rep.rinf              condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void hpdmatrixcholeskysolvem(const complex_2d_array &cha, const ae_int_t n, const bool isupper, const complex_2d_array &b, const ae_int_t m, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixcholeskysolvem(const complex_2d_array &cha, const bool isupper, const complex_2d_array &b, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

`hpdmatrixcholeskysolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N Hermitian positive definite matrix A given
by its Cholesky decomposition and N*M complex matrices X  and  B.  This is
"fast-but-lightweight" version of the solver.

Algorithm features:
* O(M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time-consuming features

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                HPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[N,M], right part
    M       -   right part size

OUTPUT PARAMETERS:
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or indefinite system

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
bool hpdmatrixcholeskysolvemfast(const complex_2d_array &cha, const ae_int_t n, const bool isupper, complex_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool hpdmatrixcholeskysolvemfast(const complex_2d_array &cha, const bool isupper, complex_2d_array &b, const xparams _xparams = alglib::xdefault);

`hpdmatrixsolve` function

/*************************************************************************
Dense solver for A*x=b, with N*N Hermitian positive definite matrix A, and
N*1 complex vectors  x  and  b.  "Slow-but-feature-rich"  version  of  the
solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant   performance   penalty  when
           ! compared with "fast" version  which  just  performs  Cholesky
           ! decomposition and calls triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, HPDMatrixSolveFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   same as in RMatrixSolve
    X       -   same as in RMatrixSolve

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void hpdmatrixsolve(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixsolve(const complex_2d_array &a, const bool isupper, const complex_1d_array &b, complex_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`hpdmatrixsolvefast` function

/*************************************************************************
Dense solver for A*x=b, with N*N Hermitian positive definite matrix A, and
N*1 complex vectors  x  and  b.  "Fast-but-lightweight"  version  of   the
solver without additional functions.

Algorithm features:
* O(N^3) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming functions

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    B       -   array[0..N-1]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or indefinite system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
bool hpdmatrixsolvefast(const complex_2d_array &a, const ae_int_t n, const bool isupper, complex_1d_array &b, const xparams _xparams = alglib::xdefault);
bool hpdmatrixsolvefast(const complex_2d_array &a, const bool isupper, complex_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`hpdmatrixsolvem` function

/*************************************************************************
Dense solver for A*X=B, with N*N Hermitian positive definite matrix A  and
N*M  complex  matrices  X  and  B.  "Slow-but-feature-rich" version of the
solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver.
           !
           ! This performance penalty is especially apparent when you  use
           ! ALGLIB parallel capabilities (condition number estimation  is
           ! inherently  sequential).  It   also   becomes significant for
           ! small-scale problems (N<100).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! HPDMatrixSolveMFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   same as in RMatrixSolve
    X       -   same as in RMatrixSolve

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void hpdmatrixsolvem(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_2d_array &b, const ae_int_t m, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixsolvem(const complex_2d_array &a, const bool isupper, const complex_2d_array &b, complex_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

`hpdmatrixsolvemfast` function

/*************************************************************************
Dense solver for A*X=B, with N*N Hermitian positive definite matrix A  and
N*M complex matrices X and B. "Fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming features like condition number estimation

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    B       -   array[0..N-1]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or indefinite system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
bool hpdmatrixsolvemfast(const complex_2d_array &a, const ae_int_t n, const bool isupper, complex_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool hpdmatrixsolvemfast(const complex_2d_array &a, const bool isupper, complex_2d_array &b, const xparams _xparams = alglib::xdefault);

`rmatrixlusolve` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
real matrix given by its LU decomposition, x and b are real vectors.  This
is "slow-but-robust" version of the linear LU-based solver. Faster version
is RMatrixLUSolveFast() function.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation

No iterative refinement  is provided because exact form of original matrix
is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! RMatrixLUSolveFast() function.

INPUT PARAMETERS
    LUA     -   array[N,N], LU decomposition, RMatrixLU result
    P       -   array[N], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[N], right part

OUTPUT PARAMETERS
    Rep     -   additional report, the following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned matrix
                * rep.r1                condition number in 1-norm
                * rep.rinf              condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros


  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixlusolve(const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixlusolve(const real_2d_array &lua, const integer_1d_array &p, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixlusolvefast` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
real matrix given by its LU decomposition, x and b are real vectors.  This
is "fast-without-any-checks" version of the linear LU-based solver. Slower
but more robust version is RMatrixLUSolve() function.

Algorithm features:
* O(N^2) complexity
* fast algorithm without ANY additional checks, just triangular solver

INPUT PARAMETERS
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system


  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
bool rmatrixlusolvefast(const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, real_1d_array &b, const xparams _xparams = alglib::xdefault);
bool rmatrixlusolvefast(const real_2d_array &lua, const integer_1d_array &p, real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixlusolvem` function

/*************************************************************************
Dense solver.

Similar to RMatrixLUSolve() but solves  task  with  multiple  right  parts
(where b and x are NxM matrices). This  is  "robust-but-slow"  version  of
LU-based solver which performs additional  checks  for  non-degeneracy  of
inputs (condition number estimation). If you need  best  performance,  use
"fast-without-any-checks" version, RMatrixLUSolveMFast().

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation

No iterative refinement  is provided because exact form of original matrix
is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver.
           !
           ! This performance penalty is especially apparent when you  use
           ! ALGLIB parallel capabilities (condition number estimation  is
           ! inherently  sequential).  It   also   becomes significant for
           ! small-scale problems.
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! RMatrixLUSolveMFast() function.

INPUT PARAMETERS
    LUA     -   array[N,N], LU decomposition, RMatrixLU result
    P       -   array[N], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned matrix
                * rep.r1                condition number in 1-norm
                * rep.rinf              condition number in inf-norm
    X       -   array[N,M], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixlusolvem(const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_2d_array &b, const ae_int_t m, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixlusolvem(const real_2d_array &lua, const integer_1d_array &p, const real_2d_array &b, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixlusolvemfast` function

/*************************************************************************
Dense solver.

Similar to RMatrixLUSolve() but solves  task  with  multiple  right parts,
where b and x are NxM matrices.  This is "fast-without-any-checks" version
of LU-based solver. It does not estimate  condition number  of  a  system,
so it is extremely fast. If you need better detection  of  near-degenerate
cases, use RMatrixLUSolveM() function.

Algorithm features:
* O(M*N^2) complexity
* fast algorithm without ANY additional checks, just triangular solver

INPUT PARAMETERS:
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS:
    B       -   array[N,M]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
bool rmatrixlusolvemfast(const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, real_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool rmatrixlusolvemfast(const real_2d_array &lua, const integer_1d_array &p, real_2d_array &b, const xparams _xparams = alglib::xdefault);

`rmatrixmixedsolve` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where BOTH ORIGINAL A AND ITS
LU DECOMPOSITION ARE KNOWN. You can use it if for some  reasons  you  have
both A and its LU decomposition.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixmixedsolve(const real_2d_array &a, const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixmixedsolve(const real_2d_array &a, const real_2d_array &lua, const integer_1d_array &p, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixmixedsolvem` function

/*************************************************************************
Dense solver.

Similar to RMatrixMixedSolve() but  solves task with multiple right  parts
(where b and x are NxM matrices).

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(M*N^2) complexity

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    LUA     -   array[0..N-1,0..N-1], LU decomposition, RMatrixLU result
    P       -   array[0..N-1], pivots array, RMatrixLU result
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixmixedsolvem(const real_2d_array &a, const real_2d_array &lua, const integer_1d_array &p, const ae_int_t n, const real_2d_array &b, const ae_int_t m, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixmixedsolvem(const real_2d_array &a, const real_2d_array &lua, const integer_1d_array &p, const real_2d_array &b, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixsolve` function

/*************************************************************************
Dense solver for A*x=b with N*N real matrix A and N*1 real vectorx  x  and
b. This is "slow-but-feature rich" version of the  linear  solver.  Faster
version is RMatrixSolveFast() function.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* iterative refinement
* O(N^3) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations. It is also very significant on small matrices.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, RMatrixSolveFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, the following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned matrix
                * rep.r1                condition number in 1-norm
                * rep.rinf              condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixsolve(const real_2d_array &a, const ae_int_t n, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixsolve(const real_2d_array &a, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixsolvefast` function

/*************************************************************************
Dense solver.

This  subroutine  solves  a  system  A*x=b,  where A is NxN non-denegerate
real matrix, x  and  b  are  vectors.  This is a "fast" version of  linear
solver which does NOT provide  any  additional  functions  like  condition
number estimation or iterative refinement.

Algorithm features:
* efficient algorithm O(N^3) complexity
* no performance overhead from additional functionality

If you need condition number estimation or iterative refinement, use  more
feature-rich version - RMatrixSolve().

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 16.03.2015 by Bochkanov Sergey
*************************************************************************/
bool rmatrixsolvefast(const real_2d_array &a, const ae_int_t n, real_1d_array &b, const xparams _xparams = alglib::xdefault);
bool rmatrixsolvefast(const real_2d_array &a, real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixsolvels` function

/*************************************************************************
Dense solver.

This subroutine finds solution of the linear system A*X=B with non-square,
possibly degenerate A.  System  is  solved in the least squares sense, and
general least squares solution  X = X0 + CX*y  which  minimizes |A*X-B| is
returned. If A is non-degenerate, solution in the usual sense is returned.

Algorithm features:
* automatic detection (and correct handling!) of degenerate cases
* iterative refinement
* O(N^3) complexity

INPUT PARAMETERS
    A       -   array[0..NRows-1,0..NCols-1], system matrix
    NRows   -   vertical size of A
    NCols   -   horizontal size of A
    B       -   array[0..NCols-1], right part
    Threshold-  a number in [0,1]. Singular values  beyond  Threshold*Largest are
                considered  zero.  Set  it to 0.0, if you don't understand
                what it means, so the solver will choose good value on its
                own.

OUTPUT PARAMETERS
    Rep     -   solver report, see below for more info
    X       -   array[0..N-1,0..M-1], it contains:
                * solution of A*X=B (even for singular A)
                * zeros, if SVD subroutine failed

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* TerminationType is set to:
            * -4 for SVD failure
            * >0 for success
* R2        reciprocal of condition number: 1/cond(A), 2-norm.
* N         = NCols
* K         dim(Null(A))
* CX        array[0..N-1,0..K-1], kernel of A.
            Columns of CX store such vectors that A*CX[i]=0.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 24.08.2009 by Bochkanov Sergey
*************************************************************************/
void rmatrixsolvels(const real_2d_array &a, const ae_int_t nrows, const ae_int_t ncols, const real_1d_array &b, const double threshold, real_1d_array &x, densesolverlsreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixsolvels(const real_2d_array &a, const real_1d_array &b, const double threshold, real_1d_array &x, densesolverlsreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixsolvem` function

/*************************************************************************
Dense solver.

Similar to RMatrixSolve() but solves task with multiple right parts (where
b and x are NxM matrices). This is  "slow-but-robust"  version  of  linear
solver with additional functionality  like  condition  number  estimation.
There also exists faster version - RMatrixSolveMFast().

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* optional iterative refinement
* O(N^3+M*N^2) complexity

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear  system
           ! and  performs  iterative   refinement,   which   results   in
           ! significant performance penalty  when  compared  with  "fast"
           ! version  which  just  performs  LU  decomposition  and  calls
           ! triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations. It also very significant on small matrices.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, RMatrixSolveMFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size
    RFS     -   iterative refinement switch:
                * True - refinement is used.
                  Less performance, more precision.
                * False - refinement is not used.
                  More performance, less precision.

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or exactly singular matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixsolvem(const real_2d_array &a, const ae_int_t n, const real_2d_array &b, const ae_int_t m, const bool rfs, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixsolvem(const real_2d_array &a, const real_2d_array &b, const bool rfs, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixsolvemfast` function

/*************************************************************************
Dense solver.

Similar to RMatrixSolve() but solves task with multiple right parts (where
b and x are NxM matrices). This is "fast" version of linear  solver  which
does NOT offer additional functions like condition  number  estimation  or
iterative refinement.

Algorithm features:
* O(N^3+M*N^2) complexity
* no additional functionality, highest performance

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size
    RFS     -   iterative refinement switch:
                * True - refinement is used.
                  Less performance, more precision.
                * False - refinement is not used.
                  More performance, less precision.

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True for well-conditioned matrix
    False for extremely badly conditioned or exactly singular problem

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
bool rmatrixsolvemfast(const real_2d_array &a, const ae_int_t n, real_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool rmatrixsolvemfast(const real_2d_array &a, real_2d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spdmatrixcholeskysolve` function

/*************************************************************************
Dense solver for A*x=b with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*1 real vectors x and b. This is "slow-
but-feature-rich"  version  of  the  solver  which,  in  addition  to  the
solution, performs condition number estimation.

Algorithm features:
* automatic detection of degenerate cases
* O(N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which results in 10-15x  performance  penalty  when  compared
           ! with "fast" version which just calls triangular solver.
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! SPDMatrixCholeskySolveFast() function.

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[N], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or indefinite matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N]:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void spdmatrixcholeskysolve(const real_2d_array &cha, const ae_int_t n, const bool isupper, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskysolve(const real_2d_array &cha, const bool isupper, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spdmatrixcholeskysolvefast` function

/*************************************************************************
Dense solver for A*x=b with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*1 real vectors x and b. This is "fast-
but-lightweight" version of the solver.

Algorithm features:
* O(N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional features

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of A
    IsUpper -   what half of CHA is provided
    B       -   array[N], right part

OUTPUT PARAMETERS
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
bool spdmatrixcholeskysolvefast(const real_2d_array &cha, const ae_int_t n, const bool isupper, real_1d_array &b, const xparams _xparams = alglib::xdefault);
bool spdmatrixcholeskysolvefast(const real_2d_array &cha, const bool isupper, real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spdmatrixcholeskysolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*M vectors X and B. It  is  "slow-but-
feature-rich" version of the solver which estimates  condition  number  of
the system.

Algorithm features:
* automatic detection of degenerate cases
* O(M*N^2) complexity
* condition number estimation
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant  performance   penalty   when
           ! compared with "fast"  version  which  just  calls  triangular
           ! solver. Amount of  overhead  introduced  depends  on  M  (the
           ! larger - the more efficient).
           !
           ! This performance penalty is insignificant  when compared with
           ! cost of large LU decomposition.  However,  if you  call  this
           ! function many times for the same  left  side,  this  overhead
           ! BECOMES significant. It  also  becomes significant for small-
           ! scale problems (N<50).
           !
           ! In such cases we strongly recommend you to use faster solver,
           ! SPDMatrixCholeskySolveMFast() function.

INPUT PARAMETERS
    CHA     -   array[0..N-1,0..N-1], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or indefinite matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N]:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void spdmatrixcholeskysolvem(const real_2d_array &cha, const ae_int_t n, const bool isupper, const real_2d_array &b, const ae_int_t m, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskysolvem(const real_2d_array &cha, const bool isupper, const real_2d_array &b, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskysolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A given
by its Cholesky decomposition, and N*M vectors X and B. It  is  "fast-but-
lightweight" version of  the  solver  which  just  solves  linear  system,
without any additional functions.

Algorithm features:
* O(M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional functionality

INPUT PARAMETERS
    CHA     -   array[N,N], Cholesky decomposition,
                SPDMatrixCholesky result
    N       -   size of CHA
    IsUpper -   what half of CHA is provided
    B       -   array[N,M], right part
    M       -   right part size

OUTPUT PARAMETERS
    B       -   array[N]:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  -- ALGLIB --
     Copyright 18.03.2015 by Bochkanov Sergey
*************************************************************************/
bool spdmatrixcholeskysolvemfast(const real_2d_array &cha, const ae_int_t n, const bool isupper, real_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool spdmatrixcholeskysolvemfast(const real_2d_array &cha, const bool isupper, real_2d_array &b, const xparams _xparams = alglib::xdefault);

`spdmatrixsolve` function

/*************************************************************************
Dense linear solver for A*x=b with N*N real  symmetric  positive  definite
matrix A,  N*1 vectors x and b.  "Slow-but-feature-rich"  version  of  the
solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant   performance   penalty  when
           ! compared with "fast" version  which  just  performs  Cholesky
           ! decomposition and calls triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, SPDMatrixSolveFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or indefinite matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void spdmatrixsolve(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixsolve(const real_2d_array &a, const bool isupper, const real_1d_array &b, real_1d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spdmatrixsolvefast` function

/*************************************************************************
Dense linear solver for A*x=b with N*N real  symmetric  positive  definite
matrix A,  N*1 vectors x and  b.  "Fast-but-lightweight"  version  of  the
solver.

Algorithm features:
* O(N^3) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming features like condition number estimation

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    B       -   array[N], it contains:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
bool spdmatrixsolvefast(const real_2d_array &a, const ae_int_t n, const bool isupper, real_1d_array &b, const xparams _xparams = alglib::xdefault);
bool spdmatrixsolvefast(const real_2d_array &a, const bool isupper, real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spdmatrixsolvem` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A,  and
N*M vectors X and B. It is "slow-but-feature-rich" version of the solver.

Algorithm features:
* automatic detection of degenerate cases
* condition number estimation
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle

No iterative refinement is provided because such partial representation of
matrix does not allow efficient calculation of extra-precise  matrix-vector
products for large matrices. Use RMatrixSolve or RMatrixMixedSolve  if  you
need iterative refinement.

IMPORTANT: ! this function is NOT the most efficient linear solver provided
           ! by ALGLIB. It estimates condition  number  of  linear system,
           ! which  results  in  significant   performance   penalty  when
           ! compared with "fast" version  which  just  performs  Cholesky
           ! decomposition and calls triangular solver.
           !
           ! This  performance  penalty  is  especially  visible  in   the
           ! multithreaded mode, because both condition number  estimation
           ! and   iterative    refinement   are   inherently   sequential
           ! calculations.
           !
           ! Thus, if you need high performance and if you are pretty sure
           ! that your system is well conditioned, we  strongly  recommend
           ! you to use faster solver, SPDMatrixSolveMFast() function.

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    Rep     -   additional report, following fields are set:
                * rep.terminationtype   >0 for success
                                        -3 for badly conditioned or indefinite matrix
                * rep.r1    condition number in 1-norm
                * rep.rinf  condition number in inf-norm
    X       -   array[N,M], it contains:
                * rep.terminationtype>0  => solution
                * rep.terminationtype=-3 => filled by zeros

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 27.01.2010 by Bochkanov Sergey
*************************************************************************/
void spdmatrixsolvem(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_2d_array &b, const ae_int_t m, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixsolvem(const real_2d_array &a, const bool isupper, const real_2d_array &b, real_2d_array &x, densesolverreport &rep, const xparams _xparams = alglib::xdefault);

`spdmatrixsolvemfast` function

/*************************************************************************
Dense solver for A*X=B with N*N symmetric positive definite matrix A,  and
N*M vectors X and B. It is "fast-but-lightweight" version of the solver.

Algorithm features:
* O(N^3+M*N^2) complexity
* matrix is represented by its upper or lower triangle
* no additional time consuming features

INPUT PARAMETERS
    A       -   array[0..N-1,0..N-1], system matrix
    N       -   size of A
    IsUpper -   what half of A is provided
    B       -   array[0..N-1,0..M-1], right part
    M       -   right part size

OUTPUT PARAMETERS
    B       -   array[N,M], it contains:
                * result=true    =>  overwritten by solution
                * result=false   =>  filled by zeros

RETURNS:
    True, if the system was solved
    False, for an extremely badly conditioned or exactly singular system

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 17.03.2015 by Bochkanov Sergey
*************************************************************************/
bool spdmatrixsolvemfast(const real_2d_array &a, const ae_int_t n, const bool isupper, real_2d_array &b, const ae_int_t m, const xparams _xparams = alglib::xdefault);
bool spdmatrixsolvemfast(const real_2d_array &a, const bool isupper, real_2d_array &b, const xparams _xparams = alglib::xdefault);

solve_complex example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates solution of a complex linear system
        //
        complex_1d_array x;
        integer_1d_array pivots;
        densesolverreport rep;

        //
        // First, solve A*x=b with a feature-rich cmatrixsolve() which supports iterative improvement
        // and condition number estimation
        //
        complex_2d_array a = "[[-4,2i],[-1i,3]]";
        complex_1d_array b = "[8i,5]";
        cmatrixsolve(a, b, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [-1.0000i, 2.0000]

        //
        // Then, solve C*x=d with cmatrixsolvefast() which has lower overhead
        //
        complex_2d_array c = "[[3i,1],[2i,4]]";
        complex_1d_array d = "[2,-2]";
        cmatrixsolvefast(c, d);
        printf("%s\n", d.tostring(4).c_str()); // EXPECTED: [-1.0000i, -1.0000]

        //
        // Sometimes you have LU decomposition of the system matrix readily
        // available. In such cases it is possible to save a lot of time by
        // passing precomputed LU factors to cmatrixlusolve(). The only
        // downside of such approach is that iterative refinement is unavailable
        // because original (unmodified) form of the system matrix is unknown
        // to ALGLIB.
        //
        // However, if you have BOTH original matrix and its LU decomposition,
        // it is possible to use cmatrixmixedsolve() which accepts both matrix
        // itself and its factors, and uses original matrix to refine solution
        // obtained with LU factors.
        //
        complex_2d_array e = "[[-3,4i],[2i,4]]";
        complex_2d_array lue = "[[-3,4i],[2i,4]]";
        complex_1d_array f = "[2i,0]";
        cmatrixlu(lue, pivots);
        cmatrixlusolve(lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [-2.0000i, -1.0000]

        cmatrixmixedsolve(e, lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [-2.0000i, -1.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

solve_complex_m example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates solution of a dense complex matrix system
        //
        complex_2d_array x;
        integer_1d_array pivots;
        densesolverreport rep;

        //
        // First, solve A*X=B with a feature-rich cmatrixsolvem() which supports
        // iterative improvement and condition number estimation. Here A is
        // an N*N matrix, X is an N*M matrix, B is an N*M matrix.
        //
        complex_2d_array a = "[[4i,-2],[-1,3i]]";
        complex_2d_array b = "[[8i,10i,4i],[5,1,-1]]";
        cmatrixsolvem(a, b, true, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [[1.0000, 2.0000,1.0000],[-2.0000i,-1.0000i,0.0000]]

        //
        // Then, solve C*X=D with cmatrixsolvemfast() which has lower overhead
        // due to condition number estimation and iterative refinement parts
        // being dropped.
        //
        complex_2d_array c = "[[3,1],[2,4]]";
        complex_2d_array d = "[[2,1],[-2,4]]";
        cmatrixsolvemfast(c, d);
        printf("%s\n", d.tostring(4).c_str()); // EXPECTED: [[1.0000,0.0000],[-1.0000,1.0000]]

        //
        // Sometimes you have LU decomposition of the system matrix readily
        // available. In such cases it is possible to save a lot of time by
        // passing precomputed LU factors to cmatrixlusolve(). The only
        // downside of such approach is that iterative refinement is unavailable
        // because original (unmodified) form of the system matrix is unknown
        // to ALGLIB.
        //
        // However, if you have BOTH original matrix and its LU decomposition,
        // it is possible to use cmatrixmixedsolve() which accepts both matrix
        // itself and its factors, and uses original matrix to refine solution
        // obtained with LU factors.
        //
        complex_2d_array e = "[[3,4],[2,4]]";
        complex_2d_array lue = "[[3,4],[2,4]]";
        complex_2d_array f = "[[2,5],[0,6]]";
        cmatrixlu(lue, pivots);
        cmatrixlusolvem(lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [[2.0000,-1.0000],[-1.0000,2.0000]]

        cmatrixmixedsolvem(e, lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [[2.0000,-1.0000],[-1.0000,2.0000]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

solve_hpd example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates solution of a Hermitian positive definite complex system
        //
        complex_1d_array x;
        densesolverreport rep;
        bool isupper = true;

        //
        // First, solve A*x=b with a feature-rich hpdmatrixsolve() which supports iterative improvement
        // and condition number estimation
        //
        complex_2d_array a = "[[4,1i],[-1i,4]]";
        complex_1d_array b = "[6,-9i]";
        hpdmatrixsolve(a, isupper, b, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, -2.0000i]

        //
        // Then, solve C*x=d with hpdmatrixsolvefast() which has lower overhead
        //
        complex_2d_array c = "[[3,-1i],[1i,3]]";
        complex_1d_array d = "[-2i,-2]";
        hpdmatrixsolvefast(c, isupper, d);
        printf("%s\n", d.tostring(4).c_str()); // EXPECTED: [-1.0000i, -1.0000]

        //
        // Sometimes you have Cholesky decomposition of the system matrix readily
        // available. In such cases it is possible to save a lot of time by
        // passing precomputed Cholesky factor to hpdmatrixcholeskysolve(). The only
        // downside of such approach is that iterative refinement is unavailable
        // because original (unmodified) form of the system matrix is unknown
        // to ALGLIB.
        //
        complex_2d_array e = "[[3,2],[2,3]]";
        complex_1d_array f = "[4,1]";
        hpdmatrixcholesky(e, isupper);
        hpdmatrixcholeskysolve(e, isupper, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [2.0000, -1.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

solve_ls example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_1d_array x;
        densesolverlsreport rep;
        real_2d_array a = "[[4,2],[-1,3],[6,5]]";
        real_1d_array b = "[8,5,16]";
        rmatrixsolvels(a, b, 0.0, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 2.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

solve_real example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates solution of a dense real linear system
        //
        real_1d_array x;
        integer_1d_array pivots;
        densesolverreport rep;

        //
        // First, solve A*x=b with a feature-rich rmatrixsolve() which supports iterative improvement
        // and condition number estimation
        //
        real_2d_array a = "[[4,2],[-1,3]]";
        real_1d_array b = "[8,5]";
        rmatrixsolve(a, b, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 2.0000]

        //
        // Then, solve C*x=d with rmatrixsolvefast() which has lower overhead
        //
        real_2d_array c = "[[3,1],[2,4]]";
        real_1d_array d = "[2,-2]";
        rmatrixsolvefast(c, d);
        printf("%s\n", d.tostring(4).c_str()); // EXPECTED: [1.0000, -1.0000]

        //
        // Sometimes you have LU decomposition of the system matrix readily
        // available. In such cases it is possible to save a lot of time by
        // passing precomputed LU factors to rmatrixlusolve(). The only
        // downside of such approach is that iterative refinement is unavailable
        // because original (unmodified) form of the system matrix is unknown
        // to ALGLIB.
        //
        // However, if you have BOTH original matrix and its LU decomposition,
        // it is possible to use rmatrixmixedsolve() which accepts both matrix
        // itself and its factors, and uses original matrix to refine solution
        // obtained with LU factors.
        //
        real_2d_array e = "[[3,4],[2,4]]";
        real_2d_array lue = "[[3,4],[2,4]]";
        real_1d_array f = "[2,0]";
        rmatrixlu(lue, pivots);
        rmatrixlusolve(lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [2.0000, -1.0000]

        rmatrixmixedsolve(e, lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [2.0000, -1.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

solve_real_m example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates solution of a dense real matrix system
        //
        real_2d_array x;
        integer_1d_array pivots;
        densesolverreport rep;

        //
        // First, solve A*X=B with a feature-rich rmatrixsolvem() which supports
        // iterative improvement and condition number estimation. Here A is
        // an N*N matrix, X is an N*M matrix, B is an N*M matrix.
        //
        real_2d_array a = "[[4,2],[-1,3]]";
        real_2d_array b = "[[8,10,4],[5,1,-1]]";
        rmatrixsolvem(a, b, true, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [[1.0000, 2.0000,1.0000],[2.0000,1.0000,0.0000]]

        //
        // Then, solve C*X=D with rmatrixsolvemfast() which has lower overhead
        // due to condition number estimation and iterative refinement parts
        // being dropped.
        //
        real_2d_array c = "[[3,1],[2,4]]";
        real_2d_array d = "[[2,1],[-2,4]]";
        rmatrixsolvemfast(c, d);
        printf("%s\n", d.tostring(4).c_str()); // EXPECTED: [[1.0000,0.0000],[-1.0000,1.0000]]

        //
        // Sometimes you have LU decomposition of the system matrix readily
        // available. In such cases it is possible to save a lot of time by
        // passing precomputed LU factors to rmatrixlusolve(). The only
        // downside of such approach is that iterative refinement is unavailable
        // because original (unmodified) form of the system matrix is unknown
        // to ALGLIB.
        //
        // However, if you have BOTH original matrix and its LU decomposition,
        // it is possible to use rmatrixmixedsolve() which accepts both matrix
        // itself and its factors, and uses original matrix to refine solution
        // obtained with LU factors.
        //
        real_2d_array e = "[[3,4],[2,4]]";
        real_2d_array lue = "[[3,4],[2,4]]";
        real_2d_array f = "[[2,5],[0,6]]";
        rmatrixlu(lue, pivots);
        rmatrixlusolvem(lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [[2.0000,-1.0000],[-1.0000,2.0000]]

        rmatrixmixedsolvem(e, lue, pivots, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [[2.0000,-1.0000],[-1.0000,2.0000]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

solve_spd example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates solution of a symmetric positive definite real system
        //
        real_1d_array x;
        densesolverreport rep;
        bool isupper = true;

        //
        // First, solve A*x=b with a feature-rich spdmatrixsolve() which supports iterative improvement
        // and condition number estimation
        //
        real_2d_array a = "[[4,1],[1,4]]";
        real_1d_array b = "[6,9]";
        spdmatrixsolve(a, isupper, b, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 2.0000]

        //
        // Then, solve C*x=d with spdmatrixsolvefast() which has lower overhead
        //
        real_2d_array c = "[[3,1],[1,3]]";
        real_1d_array d = "[2,-2]";
        spdmatrixsolvefast(c, isupper, d);
        printf("%s\n", d.tostring(4).c_str()); // EXPECTED: [1.0000, -1.0000]

        //
        // Sometimes you have Cholesky decomposition of the system matrix readily
        // available. In such cases it is possible to save a lot of time by
        // passing precomputed Cholesky factor to spdmatrixcholeskysolve(). The only
        // downside of such approach is that iterative refinement is unavailable
        // because original (unmodified) form of the system matrix is unknown
        // to ALGLIB.
        //
        real_2d_array e = "[[3,2],[2,3]]";
        real_1d_array f = "[4,1]";
        spdmatrixcholesky(e, isupper);
        spdmatrixcholeskysolve(e, isupper, f, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [2.0000, -1.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`directsparsesolvers` subpackage

Functions

sparselusolve
sparsesolve
sparsesolvelsreg
sparsespdcholeskysolve
sparsespdsolve
sparsespdsolvesks

Examples

solvesks_d_1		Solving low profile positive definite sparse systems with Skyline (SKS) solver
sparse_solve		Solving general sparse linear systems
sparse_solve_cholesky		Solving positive definite sparse linear systems with the supernodal Cholesky solver

`sparselusolve` function

/*************************************************************************
Sparse linear solver for A*x=b with general (nonsymmetric) N*N sparse real
matrix A given by its LU factorization, N*1 vectors x and b.

IMPORTANT: this solver requires input matrix  to  be  in  the  CRS  sparse
           storage format. An exception will  be  generated  if  you  pass
           matrix in some other format (HASH or SKS).

INPUT PARAMETERS
    A       -   LU factorization of the sparse matrix, must be NxN exactly
                in CRS storage format
    P, Q    -   pivot indexes from LU factorization
    N       -   size of A, N>0
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate system).

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparselusolve(const sparsematrix &a, const integer_1d_array &p, const integer_1d_array &q, const real_1d_array &b, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

`sparsesolve` function

/*************************************************************************
Sparse linear solver for A*x=b with general (nonsymmetric) N*N sparse real
matrix A, N*1 vectors x and b.

This function internally uses several solvers:
* supernodal solver with static pivoting applied to  a  2N*2N  regularized
  augmented  system, followed by iterative refinement. This solver  is   a
  recommended option because it provides the best speed and has the lowest
  memory requirements.
* sparse LU with dynamic pivoting for stability. Provides better  accuracy
  at the cost of a significantly lower performance. Recommended  only  for
  extremely unstable problems.

INPUT PARAMETERS
    A       -   sparse matrix, must be NxN exactly, any storage format
    B       -   array[N], right part
    SolverType- solver type to use:
                * 0     use the best solver. It is augmented system in the
                        current version, but may change in future releases
                * 10    use 'default profile' of the supernodal solver with
                        static   pivoting.   The  'default'   profile   is
                        intended for systems with plenty of memory; it  is
                        optimized for the best convergence at the cost  of
                        increased RAM usage. Recommended option.
                * 11    use 'limited memory'  profile  of  the  supernodal
                        solver with static  pivoting.  The  limited-memory
                        profile is intended for problems with millions  of
                        variables.  On  most  systems  it  has  the   same
                        convergence as the default profile, having somewhat
                        worse results only for ill-conditioned systems.
                * 20    use sparse LU with dynamic pivoting for stability.
                        Not intended for large-scale problems.

OUTPUT PARAMETERS
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate system).

  -- ALGLIB --
     Copyright 18.11.2023 by Bochkanov Sergey
*************************************************************************/
void sparsesolve(const sparsematrix &a, const real_1d_array &b, const ae_int_t solvertype, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);
void sparsesolve(const sparsematrix &a, const real_1d_array &b, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparsesolvelsreg` function

/*************************************************************************
Sparse linear least squares solver for A*x=b with  general  (nonsymmetric)
N*N sparse real matrix A, N*1 vectors x and b.

This function solves a regularized linear least squares problem of the form

        (                      )
    min ( |Ax-b|^2 + reg*|x|^2 ), with reg>=sqrt(MachineAccuracy)
        (                      )

The function internally uses supernodal  solver  to  solve  an  augmented-
regularized sparse system. The solver, which was initially used  to  solve
sparse square system, can also  be  used  to  solve  rectangular  systems,
provided that the system is regularized with regularizing  coefficient  at
least sqrt(MachineAccuracy), which is ~10^8 (double precision).

It can be used to solve both full rank and rank deficient systems.

INPUT PARAMETERS
    A       -   sparse MxN matrix, any storage format
    B       -   array[M], right part
    Reg     -   regularization coefficient, Reg>=sqrt(MachineAccuracy),
                lower values will be silently increased.
    SolverType- solver type to use:
                * 0     use the best solver. It is augmented system in the
                        current version, but may change in future releases
                * 10    use 'default profile' of the supernodal solver with
                        static   pivoting.   The  'default'   profile   is
                        intended for systems with plenty of memory; it  is
                        optimized for the best convergence at the cost  of
                        increased RAM usage. Recommended option.
                * 11    use 'limited memory'  profile  of  the  supernodal
                        solver with static  pivoting.  The  limited-memory
                        profile is intended for problems with millions  of
                        variables.  On  most  systems  it  has  the   same
                        convergence as the default profile, having somewhat
                        worse results only for ill-conditioned systems.

OUTPUT PARAMETERS
    X       -   array[N], least squares solution
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success.

                  Present version of the solver does NOT returns negative
                  completion codes because  it  does  not  fail.  However,
                  future ALGLIB versions may include solvers which  return
                  negative completion codes.

  -- ALGLIB --
     Copyright 18.11.2023 by Bochkanov Sergey
*************************************************************************/
void sparsesolvelsreg(const sparsematrix &a, const real_1d_array &b, const double reg, const ae_int_t solvertype, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);
void sparsesolvelsreg(const sparsematrix &a, const real_1d_array &b, const double reg, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

`sparsespdcholeskysolve` function

/*************************************************************************
Sparse linear solver for A*x=b with N*N real  symmetric  positive definite
matrix A given by its Cholesky decomposition, and N*1 vectors x and b.

IMPORTANT: this solver requires input matrix to be in  the  SKS  (Skyline)
           or CRS (compressed row storage) format. An  exception  will  be
           generated if you pass matrix in some other format.

INPUT PARAMETERS
    A       -   sparse NxN matrix stored in CRs or SKS format, must be NxN
                exactly
    IsUpper -   which half of A is provided (another half is ignored)
    B       -   array[N], right part

OUTPUT PARAMETERS
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate or non-SPD system).

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparsespdcholeskysolve(const sparsematrix &a, const bool isupper, const real_1d_array &b, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparsespdsolve` function

/*************************************************************************
Sparse linear solver for A*x=b with N*N  sparse  real  symmetric  positive
definite matrix A, N*1 vectors x and b.

This solver  converts  input  matrix  to  CRS  format,  performs  Cholesky
factorization using supernodal Cholesky  decomposition  with  permutation-
reducing ordering and uses sparse triangular solver to get solution of the
original system.

INPUT PARAMETERS
    A       -   sparse matrix, must be NxN exactly.
                Can be stored in any sparse storage format, CRS is preferred.
    IsUpper -   which half of A is provided (another half is ignored).
                It is better to store the lower triangle because it allows
                us to avoid one transposition during internal conversion.
    B       -   array[N], right part

OUTPUT PARAMETERS
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate or non-SPD system).

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparsespdsolve(const sparsematrix &a, const bool isupper, const real_1d_array &b, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparsespdsolvesks` function

/*************************************************************************
Sparse linear solver for A*x=b with N*N  sparse  real  symmetric  positive
definite matrix A, N*1 vectors x and b.

This solver  converts  input  matrix  to  SKS  format,  performs  Cholesky
factorization using  SKS  Cholesky  subroutine  (works  well  for  limited
bandwidth matrices) and uses sparse triangular solvers to get solution  of
the original system.

IMPORTANT: this  function  is  intended  for  low  profile (variable band)
           linear systems with dense or nearly-dense bands. Only  in  such
           cases  it  provides  some  performance  improvement  over  more
           general sparsrspdsolve(). If your  system  has  high  bandwidth
           or sparse band, the general sparsrspdsolve() is  likely  to  be
           more efficient.

INPUT PARAMETERS
    A       -   sparse matrix, must be NxN exactly
    IsUpper -   which half of A is provided (another half is ignored)
    B       -   array[0..N-1], right part

OUTPUT PARAMETERS
    X       -   array[N], it contains:
                * rep.terminationtype>0    =>  solution
                * rep.terminationtype=-3   =>  filled by zeros
    Rep     -   solver report, following fields are set:
                * rep.terminationtype - solver status; >0 for success,
                  set to -3 on failure (degenerate or non-SPD system).

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparsespdsolvesks(const sparsematrix &a, const bool isupper, const real_1d_array &b, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

solvesks_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates creation/initialization of the sparse matrix
        // in the SKS (Skyline) storage format and solution using SKS-based direct
        // solver.
        //
        // NOTE: the SKS solver is intended for 'easy' tasks, i.e. low-profile positive
        //       definite systems (e.g. matrices with average bandwidth as low as 3),
        //       where it can avoid some overhead associated with more powerful supernodal
        //       Cholesky solver with AMD ordering.
        //
        //       It is recommended to use more powerful solvers for more difficult problems:
        //       * sparsespdsolve() for larger sparse positive definite systems
        //       * sparsesolve() for general (nonsymmetric) linear systems
        //
        // First, we have to create matrix and initialize it. Matrix is created
        // in the SKS format, using fixed bandwidth initialization function.
        // Several points should be noted:
        //
        // 1. SKS sparse storage format also allows variable bandwidth matrices;
        //    we just do not want to overcomplicate this example.
        //
        // 2. SKS format requires you to specify matrix geometry prior to
        //    initialization of its elements with sparseset(). If you specified
        //    bandwidth=1, you can not change your mind afterwards and call
        //    sparseset() for non-existent elements.
        // 
        // 3. Because SKS solver need just one triangle of SPD matrix, we can
        //    omit initialization of the lower triangle of our matrix.
        //
        ae_int_t n = 4;
        ae_int_t bandwidth = 1;
        sparsematrix s;
        sparsecreatesksband(n, n, bandwidth, s);
        sparseset(s, 0, 0, 2.0);
        sparseset(s, 0, 1, 1.0);
        sparseset(s, 1, 1, 3.0);
        sparseset(s, 1, 2, 1.0);
        sparseset(s, 2, 2, 3.0);
        sparseset(s, 2, 3, 1.0);
        sparseset(s, 3, 3, 2.0);

        //
        // Now we have symmetric positive definite 4x4 system width bandwidth=1:
        //
        //     [ 2 1     ]   [ x0]]   [  4 ]
        //     [ 1 3 1   ]   [ x1 ]   [ 10 ]
        //     [   1 3 1 ] * [ x2 ] = [ 15 ]
        //     [     1 2 ]   [ x3 ]   [ 11 ]
        //
        // After successful creation we can call SKS solver.
        //
        real_1d_array b = "[4,10,15,11]";
        sparsesolverreport rep;
        real_1d_array x;
        bool isuppertriangle = true;
        sparsespdsolvesks(s, isuppertriangle, b, x, rep);
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 2.0000, 3.0000, 4.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

sparse_solve example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates creation/initialization of a sparse matrix and linear
        // system solution using a direct solver. This solver can handle any problem sizes
        // - from several tens of variables to millions of variables.
        //
        // First, we create a sparse matrix in the flexible hash table-based storage format,
        // initialize it and convert to the CRS format.
        //
        ae_int_t n = 4;
        sparsematrix s;
        sparsecreate(n, n, 0, s);
        sparseset(s, 0, 0, 2.0);
        sparseset(s, 0, 1, 1.0);
        sparseset(s, 1, 0, 1.0);
        sparseset(s, 1, 1, 3.0);
        sparseset(s, 1, 2, -1.0);
        sparseset(s, 2, 2, 3.0);
        sparseset(s, 2, 3, 1.0);
        sparseset(s, 3, 2, 1.0);
        sparseset(s, 3, 3, 2.0);

        //
        // Now we have symmetric positive definite 4x4 system
        //
        //     [ 2 1     ]   [ x0]]   [ 3 ]
        //     [ 1 3 -1  ]   [ x1 ]   [ 2 ]
        //     [     3 1 ] * [ x2 ] = [ 8 ]
        //     [     1 2 ]   [ x3 ]   [ 6 ]
        //
        // Now, it is time to call the solver. The sparsesolve() function supports several
        // solvers, our recommendation is to choose the default one. In the current version
        // it is a supernodal solver with static pivoting, followed by the iterative refinement.
        //
        real_1d_array b = "[3,2,8,6]";
        sparsesolverreport rep;
        real_1d_array x;
        ae_int_t solvertype = 0;
        sparsesolve(s, b, solvertype, x, rep);
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 1.0000, 2.0000, 2.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

sparse_solve_cholesky example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates creation/initialization of a sparse matrix and linear
        // system solution using Cholesky-based direct solver. This solver can handle any
        // problem sizes - from several tens of variables to millions of variables.
        //
        // First, we create a sparse matrix in the flexible hash table-based storage format,
        // initialize it and convert to the CRS format. Because the matrix is symmetric,
        // it is enough to specify only one triangle. The example below initializes the
        // lower one.
        //
        ae_int_t n = 4;
        sparsematrix s;
        sparsecreate(n, n, 0, s);
        sparseset(s, 0, 0, 2.0);
        sparseset(s, 1, 0, 1.0);
        sparseset(s, 1, 1, 3.0);
        sparseset(s, 2, 1, 1.0);
        sparseset(s, 2, 2, 3.0);
        sparseset(s, 3, 2, 1.0);
        sparseset(s, 3, 3, 2.0);

        //
        // Now we have symmetric positive definite 4x4 system
        //
        //     [ 2 1     ]   [ x0]]   [  4 ]
        //     [ 1 3 1   ]   [ x1 ]   [ 10 ]
        //     [   1 3 1 ] * [ x2 ] = [ 15 ]
        //     [     1 2 ]   [ x3 ]   [ 11 ]
        //
        // Now, it is time to call the solver.
        //
        real_1d_array b = "[4,10,15,11]";
        sparsesolverreport rep;
        real_1d_array x;
        bool isuppertriangle = false;
        sparsespdsolve(s, isuppertriangle, b, x, rep);
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [1.0000, 2.0000, 3.0000, 4.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`elliptic` subpackage

Functions

ellipticintegrale
ellipticintegralk
ellipticintegralkhighprecision
incompleteellipticintegrale
incompleteellipticintegralk

Examples

`ellipticintegrale` function

/*************************************************************************
Complete elliptic integral of the second kind

Approximates the integral


           pi/2
            -
           | |                 2
E(m)  =    |    sqrt( 1 - m sin t ) dt
         | |
          -
           0

using the approximation

     P(x)  -  x log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0, 1       10000       2.1e-16     7.3e-17

Cephes Math Library, Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegrale(const double m, const xparams _xparams = alglib::xdefault);

`ellipticintegralk` function

/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

using the approximation

    P(x)  -  log x Q(x).

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegralk(const double m, const xparams _xparams = alglib::xdefault);

`ellipticintegralkhighprecision` function

/*************************************************************************
Complete elliptic integral of the first kind

Approximates the integral



           pi/2
            -
           | |
           |           dt
K(m)  =    |    ------------------
           |                   2
         | |    sqrt( 1 - m sin t )
          -
           0

where m = 1 - m1, using the approximation

    P(x)  -  log x Q(x).

The argument m1 is used rather than m so that the logarithmic
singularity at m = 1 will be shifted to the origin; this
preserves maximum accuracy.

K(0) = pi/2.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,1        30000       2.5e-16     6.8e-17

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double ellipticintegralkhighprecision(const double m1, const xparams _xparams = alglib::xdefault);

`incompleteellipticintegrale` function

/*************************************************************************
Incomplete elliptic integral of the second kind

Approximates the integral


               phi
                -
               | |
               |                   2
E(phi_\m)  =    |    sqrt( 1 - m sin t ) dt
               |
             | |
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.

ACCURACY:

Tested at random arguments with phi in [-10, 10] and m in
[0, 1].
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10      150000       3.3e-15     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegrale(const double phi, const double m, const xparams _xparams = alglib::xdefault);

`incompleteellipticintegralk` function

/*************************************************************************
Incomplete elliptic integral of the first kind F(phi|m)

Approximates the integral



               phi
                -
               | |
               |           dt
F(phi_\m)  =    |    ------------------
               |                   2
             | |    sqrt( 1 - m sin t )
              -
               0

of amplitude phi and modulus m, using the arithmetic -
geometric mean algorithm.




ACCURACY:

Tested at random points with m in [0, 1] and phi as indicated.

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -10,10       200000      7.4e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompleteellipticintegralk(const double phi, const double m, const xparams _xparams = alglib::xdefault);

`evd` subpackage

Classes

eigsubspacereport
eigsubspacestate

Functions

eigsubspacecreate
eigsubspacecreatebuf
eigsubspaceooccontinue
eigsubspaceoocgetrequestdata
eigsubspaceoocgetrequestinfo
eigsubspaceoocsendresult
eigsubspaceoocstart
eigsubspaceoocstop
eigsubspacesetcond
eigsubspacesetwarmstart
eigsubspacesolvedenses
eigsubspacesolvesparses
hmatrixevd
hmatrixevdi
hmatrixevdr
rmatrixevd
smatrixevd
smatrixevdi
smatrixevdr
smatrixtdevd
smatrixtdevdi
smatrixtdevdr

Examples

`eigsubspacereport` class

/*************************************************************************
This object stores state of the subspace iteration algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class eigsubspacereport
{
public:
    eigsubspacereport();
    eigsubspacereport(const eigsubspacereport &rhs);
    eigsubspacereport& operator=(const eigsubspacereport &rhs);
    virtual ~eigsubspacereport();
    ae_int_t iterationscount;
};

`eigsubspacestate` class

/*************************************************************************
This object stores state of the subspace iteration algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class eigsubspacestate
{
public:
    eigsubspacestate();
    eigsubspacestate(const eigsubspacestate &rhs);
    eigsubspacestate& operator=(const eigsubspacestate &rhs);
    virtual ~eigsubspacestate();
};

`eigsubspacecreate` function

/*************************************************************************
This function initializes subspace iteration solver. This solver  is  used
to solve symmetric real eigenproblems where just a few (top K) eigenvalues
and corresponding eigenvectors is required.

This solver can be significantly faster than  complete  EVD  decomposition
in the following case:
* when only just a small fraction  of  top  eigenpairs  of dense matrix is
  required. When K approaches N, this solver is slower than complete dense
  EVD
* when problem matrix is sparse (and/or is not known explicitly, i.e. only
  matrix-matrix product can be performed)

USAGE (explicit dense/sparse matrix):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
   or other functions
3. User  calls  eigsubspacesolvedense() or eigsubspacesolvesparse() methods,
   which take algorithm state and 2D array or alglib.sparsematrix object.

USAGE (out-of-core mode):
1. User initializes algorithm state with eigsubspacecreate() call
2. [optional] User tunes solver parameters by calling eigsubspacesetcond()
   or other functions
3. User activates out-of-core mode of  the  solver  and  repeatedly  calls
   communication functions in a loop like below:
   > alglib.eigsubspaceoocstart(state)
   > while alglib.eigsubspaceooccontinue(state) do
   >     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
   >     alglib.eigsubspaceoocgetrequestdata(state, out X)
   >     [calculate  Y=A*X, with X=R^NxM]
   >     alglib.eigsubspaceoocsendresult(state, in Y)
   > alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    N       -   problem dimensionality, N>0
    K       -   number of top eigenvector to calculate, 0<K<=N.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: if you solve many similar EVD problems you may  find  it  useful  to
      reuse previous subspace as warm-start point for new EVD problem.  It
      can be done with eigsubspacesetwarmstart() function.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreate(const ae_int_t n, const ae_int_t k, eigsubspacestate &state, const xparams _xparams = alglib::xdefault);

`eigsubspacecreatebuf` function

/*************************************************************************
Buffered version of constructor which aims to reuse  previously  allocated
memory as much as possible.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacecreatebuf(const ae_int_t n, const ae_int_t k, eigsubspacestate &state, const xparams _xparams = alglib::xdefault);

`eigsubspaceooccontinue` function

/*************************************************************************
This function performs subspace iteration  in  the  out-of-core  mode.  It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
bool eigsubspaceooccontinue(eigsubspacestate &state, const xparams _xparams = alglib::xdefault);

`eigsubspaceoocgetrequestdata` function

/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by solver to user code: matrix X (array[N,RequestSize) which have  to
be multiplied by out-of-core matrix A in a product A*X.

This function returns just request data; in order to get size of  the data
prior to processing requestm, use eigsubspaceoocgetrequestinfo().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    X               -   possibly  preallocated   storage;  reallocated  if
                        needed, left unchanged, if large enough  to  store
                        request data.

OUTPUT PARAMETERS:
    X               -   array[N,RequestSize] or larger, leading  rectangle
                        is filled with dense matrix X.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocgetrequestdata(eigsubspacestate &state, real_2d_array &x, const xparams _xparams = alglib::xdefault);

`eigsubspaceoocgetrequestinfo` function

/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by solver to user code: request type (current version  of  the solver
sends only requests for matrix-matrix products) and request size (size  of
the matrices being multiplied).

This function returns just request metrics; in order  to  get contents  of
the matrices being multiplied, use eigsubspaceoocgetrequestdata().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode

OUTPUT PARAMETERS:
    RequestType     -   type of the request to process:
                        * 0 - for matrix-matrix product A*X, with A  being
                          NxN matrix whose eigenvalues/vectors are needed,
                          and X being NxREQUESTSIZE one which is  returned
                          by the eigsubspaceoocgetrequestdata().
    RequestSize     -   size of the X matrix (number of columns),  usually
                        it is several times larger than number of  vectors
                        K requested by user.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocgetrequestinfo(eigsubspacestate &state, ae_int_t &requesttype, ae_int_t &requestsize, const xparams _xparams = alglib::xdefault);

`eigsubspaceoocsendresult` function

/*************************************************************************
This function is used to send user reply to out-of-core  request  sent  by
solver. Usually it is product A*X for returned by solver matrix X.

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    AX              -   array[N,RequestSize] or larger, leading  rectangle
                        is filled with product A*X.


  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocsendresult(eigsubspacestate &state, const real_2d_array &ax, const xparams _xparams = alglib::xdefault);

`eigsubspaceoocstart` function

/*************************************************************************
This  function  initiates  out-of-core  mode  of  subspace eigensolver. It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State       -   solver object
    MType       -   matrix type and solver mode:

                    * 0 =   real symmetric matrix A, products  of the form
                            A*X are computed. At every step  the  basis of
                            the  invariant  subspace  is  reorthogonalized
                            with LQ decomposition  which  makes  the  algo
                            more robust.

                            The first mode introduced in ALGLIB, the  most
                            precise and robust. However, it is  suboptimal
                            for easy problems which can be solved  in  3-5
                            iterations without LQ step.

                    * 1 =   real symmetric matrix A, products  of the form
                            A*X are computed. The  invariant  subspace  is
                            NOT reorthogonalized,  no  error  checks.  The
                            solver  stops  after   specified   number   of
                            iterations which should be small, 5 at most.

                            This mode is intended for easy  problems  with
                            extremely fast convergence.

                    Future versions of ALGLIB may  introduce  support  for
                    other  matrix   types;   for   now,   only   symmetric
                    eigenproblems are supported.


  -- ALGLIB --
     Copyright 07.06.2023 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocstart(eigsubspacestate &state, const ae_int_t mtype, const xparams _xparams = alglib::xdefault);

`eigsubspaceoocstop` function

/*************************************************************************
This  function  finalizes out-of-core  mode  of  subspace eigensolver.  It
should be used in conjunction with other out-of-core-related functions  of
this subspackage in a loop like below:

> alglib.eigsubspaceoocstart(state)
> while alglib.eigsubspaceooccontinue(state) do
>     alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M)
>     alglib.eigsubspaceoocgetrequestdata(state, out X)
>     [calculate  Y=A*X, with X=R^NxM]
>     alglib.eigsubspaceoocsendresult(state, in Y)
> alglib.eigsubspaceoocstop(state, out W, out Z, out Report)

INPUT PARAMETERS:
    State       -   solver state

OUTPUT PARAMETERS:
    W           -   array[K], depending on solver settings:
                    * top  K  eigenvalues ordered  by  descending   -   if
                      eigenvectors are returned in Z
                    * zeros - if invariant subspace is returned in Z
    Z           -   array[N,K], depending on solver settings either:
                    * matrix of eigenvectors found
                    * orthogonal basis of K-dimensional invariant subspace
    Rep         -   report with additional parameters

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspaceoocstop(eigsubspacestate &state, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);

`eigsubspacesetcond` function

/*************************************************************************
This function sets stopping critera for the solver:
* error in eigenvector/value allowed by solver
* maximum number of iterations to perform

INPUT PARAMETERS:
    State       -   solver structure
    Eps         -   eps>=0,  with non-zero value used to tell solver  that
                    it can  stop  after  all  eigenvalues  converged  with
                    error  roughly  proportional  to  eps*MAX(LAMBDA_MAX),
                    where LAMBDA_MAX is a maximum eigenvalue.
                    Zero  value  means  that  no  check  for  precision is
                    performed.
    MaxIts      -   maxits>=0,  with non-zero value used  to  tell  solver
                    that it can stop after maxits  steps  (no  matter  how
                    precise current estimate is)

NOTE: passing  eps=0  and  maxits=0  results  in  automatic  selection  of
      moderate eps as stopping criteria (1.0E-6 in current implementation,
      but it may change without notice).

NOTE: very small values of eps are possible (say, 1.0E-12),  although  the
      larger problem you solve (N and/or K), the  harder  it  is  to  find
      precise eigenvectors because rounding errors tend to accumulate.

NOTE: passing non-zero eps results in  some performance  penalty,  roughly
      equal to 2N*(2K)^2 FLOPs per iteration. These additional computations
      are required in order to estimate current error in  eigenvalues  via
      Rayleigh-Ritz process.
      Most of this additional time is  spent  in  construction  of  ~2Kx2K
      symmetric  subproblem  whose  eigenvalues  are  checked  with  exact
      eigensolver.
      This additional time is negligible if you search for eigenvalues  of
      the large dense matrix, but may become noticeable on  highly  sparse
      EVD problems, where cost of matrix-matrix product is low.
      If you set eps to exactly zero,  Rayleigh-Ritz  phase  is completely
      turned off.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesetcond(eigsubspacestate &state, const double eps, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`eigsubspacesetwarmstart` function

/*************************************************************************
This function sets warm-start mode of the solver: next call to the  solver
will reuse previous subspace as warm-start  point.  It  can  significantly
speed-up convergence when you solve many similar eigenproblems.

INPUT PARAMETERS:
    State       -   solver structure
    UseWarmStart-   either True or False

  -- ALGLIB --
     Copyright 12.11.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesetwarmstart(eigsubspacestate &state, const bool usewarmstart, const xparams _xparams = alglib::xdefault);

`eigsubspacesolvedenses` function

/*************************************************************************
This  function runs subspace eigensolver for dense NxN symmetric matrix A,
given by its upper or lower triangle.

This function can not process nonsymmetric matrices.

INPUT PARAMETERS:
    State       -   solver state
    A           -   array[N,N], symmetric NxN matrix given by one  of  its
                    triangles
    IsUpper     -   whether upper or lower triangle of  A  is  given  (the
                    other one is not referenced at all).

OUTPUT PARAMETERS:
    W           -   array[K], top  K  eigenvalues ordered  by   descending
                    of their absolute values
    Z           -   array[N,K], matrix of eigenvectors found
    Rep         -   report with additional parameters

NOTE: internally this function allocates a copy of NxN dense A. You should
      take it into account when working with very large matrices occupying
      almost all RAM.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesolvedenses(eigsubspacestate &state, const real_2d_array &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);

`eigsubspacesolvesparses` function

/*************************************************************************
This  function runs eigensolver for dense NxN symmetric matrix A, given by
upper or lower triangle.

This function can not process nonsymmetric matrices.

INPUT PARAMETERS:
    State       -   solver state
    A           -   NxN symmetric matrix given by one of its triangles
    IsUpper     -   whether upper or lower triangle of  A  is  given  (the
                    other one is not referenced at all).

OUTPUT PARAMETERS:
    W           -   array[K], top  K  eigenvalues ordered  by   descending
                    of their absolute values
    Z           -   array[N,K], matrix of eigenvectors found
    Rep         -   report with additional parameters

  -- ALGLIB --
     Copyright 16.01.2017 by Bochkanov Sergey
*************************************************************************/
void eigsubspacesolvesparses(eigsubspacestate &state, const sparsematrix &a, const bool isupper, real_1d_array &w, real_2d_array &z, eigsubspacereport &rep, const xparams _xparams = alglib::xdefault);

`hmatrixevd` function

/*************************************************************************
Finding the eigenvalues and eigenvectors of a Hermitian matrix

The algorithm finds eigen pairs of a Hermitian matrix by  reducing  it  to
real tridiagonal form and using the QL/QR algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   Hermitian matrix which is given  by  its  upper  or  lower
                triangular part.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format.
    ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                not. If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.

Output parameters:
    D       -   eigenvalues in ascending order.
                Array whose index ranges within [0..N-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains the eigenvectors.
                Array whose indexes range within [0..N-1, 0..N-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

Note:
    eigenvectors of Hermitian matrix are defined up to  multiplication  by
    a complex number L, such that |L|=1.

  -- ALGLIB --
     Copyright 2005, 23 March 2007 by Bochkanov Sergey
*************************************************************************/
bool hmatrixevd(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, real_1d_array &d, complex_2d_array &z, const xparams _xparams = alglib::xdefault);

`hmatrixevdi` function

/*************************************************************************
Subroutine for finding the eigenvalues and  eigenvectors  of  a  Hermitian
matrix with given indexes by using bisection and inverse iteration methods

Input parameters:
    A       -   Hermitian matrix which is given  by  its  upper  or  lower
                triangular part.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                not. If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    I1, I2 -    index interval for searching (from I1 to I2).
                0 <= I1 <= I2 <= N-1.

Output parameters:
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..I2-I1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..I2-I1].
                In  that  case,  the eigenvectors are stored in the matrix
                columns.

Result:
    True, if successful. W contains the eigenvalues, Z contains the
    eigenvectors (if needed).

    False, if the bisection method subroutine  wasn't  able  to  find  the
    eigenvalues  in  the  given  interval  or  if  the  inverse  iteration
    subroutine wasn't able to find  all  the  corresponding  eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned.

Note:
    eigen vectors of Hermitian matrix are defined up to multiplication  by
    a complex number L, such as |L|=1.

  -- ALGLIB --
     Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
*************************************************************************/
bool hmatrixevdi(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const ae_int_t i1, const ae_int_t i2, real_1d_array &w, complex_2d_array &z, const xparams _xparams = alglib::xdefault);

`hmatrixevdr` function

/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of  a  Hermitian
matrix  in  a  given half-interval (A, B] by using a bisection and inverse
iteration

Input parameters:
    A       -   Hermitian matrix which is given  by  its  upper  or  lower
                triangular  part.  Array  whose   indexes   range   within
                [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors  are  needed  or
                not. If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    B1, B2 -    half-interval (B1, B2] to search eigenvalues in.

Output parameters:
    M       -   number of eigenvalues found in a given half-interval, M>=0
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..M-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if successful. M contains the number of eigenvalues in the given
    half-interval (could be equal to 0), W contains the eigenvalues,
    Z contains the eigenvectors (if needed).

    False, if the bisection method subroutine  wasn't  able  to  find  the
    eigenvalues  in  the  given  interval  or  if  the  inverse  iteration
    subroutine  wasn't  able  to  find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned, M  is
    equal to 0.

Note:
    eigen vectors of Hermitian matrix are defined up to multiplication  by
    a complex number L, such as |L|=1.

  -- ALGLIB --
     Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey.
*************************************************************************/
bool hmatrixevdr(const complex_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const double b1, const double b2, ae_int_t &m, real_1d_array &w, complex_2d_array &z, const xparams _xparams = alglib::xdefault);

`rmatrixevd` function

/*************************************************************************
Finding eigenvalues and eigenvectors of a general (unsymmetric) matrix

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The algorithm finds eigenvalues and eigenvectors of a general matrix by
using the QR algorithm with multiple shifts. The algorithm can find
eigenvalues and both left and right eigenvectors.

The right eigenvector is a vector x such that A*x = w*x, and the left
eigenvector is a vector y such that y'*A = w*y' (here y' implies a complex
conjugate transposition of vector y).

Input parameters:
    A       -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    VNeeded -   flag controlling whether eigenvectors are needed or not.
                If VNeeded is equal to:
                 * 0, eigenvectors are not returned;
                 * 1, right eigenvectors are returned;
                 * 2, left eigenvectors are returned;
                 * 3, both left and right eigenvectors are returned.

Output parameters:
    WR      -   real parts of eigenvalues.
                Array whose index ranges within [0..N-1].
    WR      -   imaginary parts of eigenvalues.
                Array whose index ranges within [0..N-1].
    VL, VR  -   arrays of left and right eigenvectors (if they are needed).
                If WI[i]=0, the respective eigenvalue is a real number,
                and it corresponds to the column number I of matrices VL/VR.
                If WI[i]>0, we have a pair of complex conjugate numbers with
                positive and negative imaginary parts:
                    the first eigenvalue WR[i] + sqrt(-1)*WI[i];
                    the second eigenvalue WR[i+1] + sqrt(-1)*WI[i+1];
                    WI[i]>0
                    WI[i+1] = -WI[i] < 0
                In that case, the eigenvector  corresponding to the first
                eigenvalue is located in i and i+1 columns of matrices
                VL/VR (the column number i contains the real part, and the
                column number i+1 contains the imaginary part), and the vector
                corresponding to the second eigenvalue is a complex conjugate to
                the first vector.
                Arrays whose indexes range within [0..N-1, 0..N-1].

Result:
    True, if the algorithm has converged.
    False, if the algorithm has not converged.

Note 1:
    Some users may ask the following question: what if WI[N-1]>0?
    WI[N] must contain an eigenvalue which is complex conjugate to the
    N-th eigenvalue, but the array has only size N?
    The answer is as follows: such a situation cannot occur because the
    algorithm finds a pairs of eigenvalues, therefore, if WI[i]>0, I is
    strictly less than N-1.

Note 2:
    The algorithm performance depends on the value of the internal parameter
    NS of the InternalSchurDecomposition subroutine which defines the number
    of shifts in the QR algorithm (similarly to the block width in block-matrix
    algorithms of linear algebra). If you require maximum performance
    on your machine, it is recommended to adjust this parameter manually.


See also the InternalTREVC subroutine.

The algorithm is based on the LAPACK 3.0 library.
*************************************************************************/
bool rmatrixevd(const real_2d_array &a, const ae_int_t n, const ae_int_t vneeded, real_1d_array &wr, real_1d_array &wi, real_2d_array &vl, real_2d_array &vr, const xparams _xparams = alglib::xdefault);

`smatrixevd` function

/*************************************************************************
Finding the eigenvalues and eigenvectors of a symmetric matrix

The algorithm finds eigen pairs of a symmetric matrix by reducing it to
tridiagonal form and using the QL/QR algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpper -   storage format.

Output parameters:
    D       -   eigenvalues in ascending order.
                Array whose index ranges within [0..N-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains the eigenvectors.
                Array whose indexes range within [0..N-1, 0..N-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
bool smatrixevd(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, real_1d_array &d, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`smatrixevdi` function

/*************************************************************************
Subroutine for finding the eigenvalues and  eigenvectors  of  a  symmetric
matrix with given indexes by using bisection and inverse iteration methods.

Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part. Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    I1, I2 -    index interval for searching (from I1 to I2).
                0 <= I1 <= I2 <= N-1.

Output parameters:
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..I2-I1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..I2-I1].
                In that case, the eigenvectors are stored in the matrix columns.

Result:
    True, if successful. W contains the eigenvalues, Z contains the
    eigenvectors (if needed).

    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned.

  -- ALGLIB --
     Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixevdi(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const ae_int_t i1, const ae_int_t i2, real_1d_array &w, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`smatrixevdr` function

/*************************************************************************
Subroutine for finding the eigenvalues (and eigenvectors) of  a  symmetric
matrix  in  a  given half open interval (A, B] by using  a  bisection  and
inverse iteration

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   symmetric matrix which is given by its upper or lower
                triangular part. Array [0..N-1, 0..N-1].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not returned;
                 * 1, the eigenvectors are returned.
    IsUpperA -  storage format of matrix A.
    B1, B2 -    half open interval (B1, B2] to search eigenvalues in.

Output parameters:
    M       -   number of eigenvalues found in a given half-interval (M>=0).
    W       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains eigenvectors.
                Array whose indexes range within [0..N-1, 0..M-1].
                The eigenvectors are stored in the matrix columns.

Result:
    True, if successful. M contains the number of eigenvalues in the given
    half-interval (could be equal to 0), W contains the eigenvalues,
    Z contains the eigenvectors (if needed).

    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors.
    In that case, the eigenvalues and eigenvectors are not returned,
    M is equal to 0.

  -- ALGLIB --
     Copyright 07.01.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixevdr(const real_2d_array &a, const ae_int_t n, const ae_int_t zneeded, const bool isupper, const double b1, const double b2, ae_int_t &m, real_1d_array &w, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`smatrixtdevd` function

/*************************************************************************
Finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix

The algorithm finds the eigen pairs of a tridiagonal symmetric matrix by
using an QL/QR algorithm with implicit shifts.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    D       -   the main diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-1].
    E       -   the secondary diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix A.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not needed;
                 * 1, the eigenvectors of a tridiagonal matrix
                   are multiplied by the square matrix Z. It is used if the
                   tridiagonal matrix is obtained by the similarity
                   transformation of a symmetric matrix;
                 * 2, the eigenvectors of a tridiagonal matrix replace the
                   square matrix Z;
                 * 3, matrix Z contains the first row of the eigenvectors
                   matrix.
    Z       -   if ZNeeded=1, Z contains the square matrix by which the
                eigenvectors are multiplied.
                Array whose indexes range within [0..N-1, 0..N-1].

Output parameters:
    D       -   eigenvalues in ascending order.
                Array whose index ranges within [0..N-1].
    Z       -   if ZNeeded is equal to:
                 * 0, Z hasn't changed;
                 * 1, Z contains the product of a given matrix (from the left)
                   and the eigenvectors matrix (from the right);
                 * 2, Z contains the eigenvectors.
                 * 3, Z contains the first row of the eigenvectors matrix.
                If ZNeeded<3, Z is the array whose indexes range within [0..N-1, 0..N-1].
                In that case, the eigenvectors are stored in the matrix columns.
                If ZNeeded=3, Z is the array whose indexes range within [0..0, 0..N-1].

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
bool smatrixtdevd(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`smatrixtdevdi` function

/*************************************************************************
Subroutine for finding tridiagonal matrix eigenvalues/vectors with given
indexes (in ascending order) by using the bisection and inverse iteraion.

Input parameters:
    D       -   the main diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-1].
    E       -   the secondary diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix. N>=0.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not needed;
                 * 1, the eigenvectors of a tridiagonal matrix are multiplied
                   by the square matrix Z. It is used if the
                   tridiagonal matrix is obtained by the similarity transformation
                   of a symmetric matrix.
                 * 2, the eigenvectors of a tridiagonal matrix replace
                   matrix Z.
    I1, I2  -   index interval for searching (from I1 to I2).
                0 <= I1 <= I2 <= N-1.
    Z       -   if ZNeeded is equal to:
                 * 0, Z isn't used and remains unchanged;
                 * 1, Z contains the square matrix (array whose indexes range within [0..N-1, 0..N-1])
                   which reduces the given symmetric matrix to  tridiagonal form;
                 * 2, Z isn't used (but changed on the exit).

Output parameters:
    D       -   array of the eigenvalues found.
                Array whose index ranges within [0..I2-I1].
    Z       -   if ZNeeded is equal to:
                 * 0, doesn't contain any information;
                 * 1, contains the product of a given NxN matrix Z (from the left) and
                   Nx(I2-I1) matrix of the eigenvectors found (from the right).
                   Array whose indexes range within [0..N-1, 0..I2-I1].
                 * 2, contains the matrix of the eigenvalues found.
                   Array whose indexes range within [0..N-1, 0..I2-I1].


Result:

    True, if successful. In that case, D contains the eigenvalues,
    Z contains the eigenvectors (if needed).
    It should be noted that the subroutine changes the size of arrays D and Z.

    False, if the bisection method subroutine wasn't able to find the eigenvalues
    in the given interval or if the inverse iteration subroutine wasn't able
    to find all the corresponding eigenvectors. In that case, the eigenvalues
    and eigenvectors are not returned.

  -- ALGLIB --
     Copyright 25.12.2005 by Bochkanov Sergey
*************************************************************************/
bool smatrixtdevdi(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, const ae_int_t i1, const ae_int_t i2, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`smatrixtdevdr` function

/*************************************************************************
Subroutine for finding the tridiagonal matrix eigenvalues/vectors in a
given half-interval (A, B] by using bisection and inverse iteration.

Input parameters:
    D       -   the main diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-1].
    E       -   the secondary diagonal of a tridiagonal matrix.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix, N>=0.
    ZNeeded -   flag controlling whether the eigenvectors are needed or not.
                If ZNeeded is equal to:
                 * 0, the eigenvectors are not needed;
                 * 1, the eigenvectors of a tridiagonal matrix are multiplied
                   by the square matrix Z. It is used if the tridiagonal
                   matrix is obtained by the similarity transformation
                   of a symmetric matrix.
                 * 2, the eigenvectors of a tridiagonal matrix replace matrix Z.
    A, B    -   half-interval (A, B] to search eigenvalues in.
    Z       -   if ZNeeded is equal to:
                 * 0, Z isn't used and remains unchanged;
                 * 1, Z contains the square matrix (array whose indexes range
                   within [0..N-1, 0..N-1]) which reduces the given symmetric
                   matrix to tridiagonal form;
                 * 2, Z isn't used (but changed on the exit).

Output parameters:
    D       -   array of the eigenvalues found.
                Array whose index ranges within [0..M-1].
    M       -   number of eigenvalues found in the given half-interval (M>=0).
    Z       -   if ZNeeded is equal to:
                 * 0, doesn't contain any information;
                 * 1, contains the product of a given NxN matrix Z (from the
                   left) and NxM matrix of the eigenvectors found (from the
                   right). Array whose indexes range within [0..N-1, 0..M-1].
                 * 2, contains the matrix of the eigenvectors found.
                   Array whose indexes range within [0..N-1, 0..M-1].

Result:

    True, if successful. In that case, M contains the number of eigenvalues
    in the given half-interval (could be equal to 0), D contains the eigenvalues,
    Z contains the eigenvectors (if needed).
    It should be noted that the subroutine changes the size of arrays D and Z.

    False, if the bisection method subroutine wasn't able to find the
    eigenvalues in the given interval or if the inverse iteration subroutine
    wasn't able to find all the corresponding eigenvectors. In that case,
    the eigenvalues and eigenvectors are not returned, M is equal to 0.

  -- ALGLIB --
     Copyright 31.03.2008 by Bochkanov Sergey
*************************************************************************/
bool smatrixtdevdr(real_1d_array &d, const real_1d_array &e, const ae_int_t n, const ae_int_t zneeded, const double a, const double b, ae_int_t &m, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`expintegrals` subpackage

Functions

exponentialintegralei
exponentialintegralen

Examples

`exponentialintegralei` function

/*************************************************************************
Exponential integral Ei(x)

              x
               -     t
              | |   e
   Ei(x) =   -|-   ---  dt .
            | |     t
             -
            -inf

Not defined for x <= 0.
See also expn.c.



ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE       0,100       50000      8.6e-16     1.3e-16

Cephes Math Library Release 2.8:  May, 1999
Copyright 1999 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralei(const double x, const xparams _xparams = alglib::xdefault);

`exponentialintegralen` function

/*************************************************************************
Exponential integral En(x)

Evaluates the exponential integral

                inf.
                  -
                 | |   -xt
                 |    e
     E (x)  =    |    ----  dt.
      n          |      n
               | |     t
                -
                 1


Both n and x must be nonnegative.

The routine employs either a power series, a continued
fraction, or an asymptotic formula depending on the
relative values of n and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0, 30       10000       1.7e-15     3.6e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 2000 by Stephen L. Moshier
*************************************************************************/
double exponentialintegralen(const double x, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`fdistr` subpackage

Functions

fcdistribution
fdistribution
invfdistribution

Examples

`fcdistribution` function

/*************************************************************************
Complemented F distribution

Returns the area from x to infinity under the F density
function (also known as Snedcor's density or the
variance ratio density).


                     inf.
                      -
             1       | |  a-1      b-1
1-P(x)  =  ------    |   t    (1-t)    dt
           B(a,b)  | |
                    -
                     x


The incomplete beta integral is used, according to the
formula

P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ).


ACCURACY:

Tested at random points (a,b,x) in the indicated intervals.
               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    1,100       100000      3.7e-14     5.9e-16
   IEEE      1,5    1,100       100000      8.0e-15     1.6e-15
   IEEE      0,1    1,10000     100000      1.8e-11     3.5e-13
   IEEE      1,5    1,10000     100000      2.0e-11     3.0e-12

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fcdistribution(const ae_int_t a, const ae_int_t b, const double x, const xparams _xparams = alglib::xdefault);

`fdistribution` function

/*************************************************************************
F distribution

Returns the area from zero to x under the F density
function (also known as Snedcor's density or the
variance ratio density).  This is the density
of x = (u1/df1)/(u2/df2), where u1 and u2 are random
variables having Chi square distributions with df1
and df2 degrees of freedom, respectively.
The incomplete beta integral is used, according to the
formula

P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ).


The arguments a and b are greater than zero, and x is
nonnegative.

ACCURACY:

Tested at random points (a,b,x).

               x     a,b                     Relative error:
arithmetic  domain  domain     # trials      peak         rms
   IEEE      0,1    0,100       100000      9.8e-15     1.7e-15
   IEEE      1,5    0,100       100000      6.5e-15     3.5e-16
   IEEE      0,1    1,10000     100000      2.2e-11     3.3e-12
   IEEE      1,5    1,10000     100000      1.1e-11     1.7e-13

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double fdistribution(const ae_int_t a, const ae_int_t b, const double x, const xparams _xparams = alglib::xdefault);

`invfdistribution` function

/*************************************************************************
Inverse of complemented F distribution

Finds the F density argument x such that the integral
from x to infinity of the F density is equal to the
given probability p.

This is accomplished using the inverse beta integral
function and the relations

     z = incbi( df2/2, df1/2, p )
     x = df2 (1-z) / (df1 z).

Note: the following relations hold for the inverse of
the uncomplemented F distribution:

     z = incbi( df1/2, df2/2, p )
     x = df2 z / (df1 (1-z)).

ACCURACY:

Tested at random points (a,b,p).

             a,b                     Relative error:
arithmetic  domain     # trials      peak         rms
 For p between .001 and 1:
   IEEE     1,100       100000      8.3e-15     4.7e-16
   IEEE     1,10000     100000      2.1e-11     1.4e-13
 For p between 10^-6 and 10^-3:
   IEEE     1,100        50000      1.3e-12     8.4e-15
   IEEE     1,10000      50000      3.0e-12     4.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invfdistribution(const ae_int_t a, const ae_int_t b, const double y, const xparams _xparams = alglib::xdefault);

`fft` subpackage

Functions

fftc1d
fftc1dinv
fftr1d
fftr1dbuf
fftr1dinv
fftr1dinvbuf

Examples

fft_complex_d1		Complex FFT: simple example
fft_complex_d2		Complex FFT: advanced example
fft_real_d1		Real FFT: simple example
fft_real_d2		Real FFT: advanced example

`fftc1d` function

/*************************************************************************
1-dimensional complex FFT.

Array size N may be arbitrary number (composite or prime).  Composite  N's
are handled with cache-oblivious variation of  a  Cooley-Tukey  algorithm.
Small prime-factors are transformed using hard coded  codelets (similar to
FFTW codelets, but without low-level  optimization),  large  prime-factors
are handled with Bluestein's algorithm.

Fastests transforms are for smooth N's (prime factors are 2, 3,  5  only),
most fast for powers of 2. When N have prime factors  larger  than  these,
but orders of magnitude smaller than N, computations will be about 4 times
slower than for nearby highly composite N's. When N itself is prime, speed
will be 6 times lower.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - complex function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   DFT of a input array, array[0..N-1]
            A_out[j] = SUM(A_in[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1)


  -- ALGLIB --
     Copyright 29.05.2009 by Bochkanov Sergey
*************************************************************************/
void fftc1d(complex_1d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
void fftc1d(complex_1d_array &a, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`fftc1dinv` function

/*************************************************************************
1-dimensional complex inverse FFT.

Array size N may be arbitrary number (composite or prime).  Algorithm  has
O(N*logN) complexity for any N (composite or prime).

See FFTC1D() description for more information about algorithm performance.

INPUT PARAMETERS
    A   -   array[0..N-1] - complex array to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   inverse DFT of a input array, array[0..N-1]
            A_out[j] = SUM(A_in[k]/N*exp(+2*pi*sqrt(-1)*j*k/N), k = 0..N-1)


  -- ALGLIB --
     Copyright 29.05.2009 by Bochkanov Sergey
*************************************************************************/
void fftc1dinv(complex_1d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
void fftc1dinv(complex_1d_array &a, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`fftr1d` function

/*************************************************************************
1-dimensional real FFT.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - real function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    F   -   DFT of a input array, array[0..N-1]
            F[j] = SUM(A[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1)

NOTE: there is a buffered version  of  this  function, FFTR1DBuf(),  which
      reuses memory previously allocated for A as much as possible.

NOTE:
    F[] satisfies symmetry property F[k] = conj(F[N-k]),  so just one half
of  array  is  usually needed. But for convinience subroutine returns full
complex array (with frequencies above N/2), so its result may be  used  by
other FFT-related subroutines.


  -- ALGLIB --
     Copyright 01.06.2009 by Bochkanov Sergey
*************************************************************************/
void fftr1d(const real_1d_array &a, const ae_int_t n, complex_1d_array &f, const xparams _xparams = alglib::xdefault);
void fftr1d(const real_1d_array &a, complex_1d_array &f, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`fftr1dbuf` function

/*************************************************************************
1-dimensional real FFT, a buffered function which does not reallocate  F[]
if its length is enough to store the result  (i.e.  it  reuses  previously
allocated memory as much as possible).

  -- ALGLIB --
     Copyright 01.06.2009 by Bochkanov Sergey
*************************************************************************/
void fftr1dbuf(const real_1d_array &a, const ae_int_t n, complex_1d_array &f, const xparams _xparams = alglib::xdefault);
void fftr1dbuf(const real_1d_array &a, complex_1d_array &f, const xparams _xparams = alglib::xdefault);

`fftr1dinv` function

/*************************************************************************
1-dimensional real inverse FFT.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    F   -   array[0..floor(N/2)] - frequencies from forward real FFT
    N   -   problem size

OUTPUT PARAMETERS
    A   -   inverse DFT of a input array, array[0..N-1]

NOTE: there is a buffered version of this function, FFTR1DInvBuf(),  which
      reuses memory previously allocated for A as much as possible.

NOTE:
    F[] should satisfy symmetry property F[k] = conj(F[N-k]), so just  one
half of frequencies array is needed - elements from 0 to floor(N/2).  F[0]
is ALWAYS real. If N is even F[floor(N/2)] is real too. If N is odd,  then
F[floor(N/2)] has no special properties.

Relying on properties noted above, FFTR1DInv subroutine uses only elements
from 0th to floor(N/2)-th. It ignores imaginary part of F[0],  and in case
N is even it ignores imaginary part of F[floor(N/2)] too.

When you call this function using full arguments list - "FFTR1DInv(F,N,A)"
- you can pass either either frequencies array with N elements or  reduced
array with roughly N/2 elements - subroutine will  successfully  transform
both.

If you call this function using reduced arguments list -  "FFTR1DInv(F,A)"
- you must pass FULL array with N elements (although higher  N/2 are still
not used) because array size is used to automatically determine FFT length

  -- ALGLIB --
     Copyright 01.06.2009 by Bochkanov Sergey
*************************************************************************/
void fftr1dinv(const complex_1d_array &f, const ae_int_t n, real_1d_array &a, const xparams _xparams = alglib::xdefault);
void fftr1dinv(const complex_1d_array &f, real_1d_array &a, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`fftr1dinvbuf` function

/*************************************************************************
1-dimensional real inverse FFT, buffered version, which does not reallocate
A[] if its length is enough to store the result (i.e. it reuses previously
allocated memory as much as possible).

  -- ALGLIB --
     Copyright 01.06.2009 by Bochkanov Sergey
*************************************************************************/
void fftr1dinvbuf(const complex_1d_array &f, const ae_int_t n, real_1d_array &a, const xparams _xparams = alglib::xdefault);
void fftr1dinvbuf(const complex_1d_array &f, real_1d_array &a, const xparams _xparams = alglib::xdefault);

fft_complex_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // first we demonstrate forward FFT:
        // [1i,1i,1i,1i] is converted to [4i, 0, 0, 0]
        //
        complex_1d_array z = "[1i,1i,1i,1i]";
        fftc1d(z);
        printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [4i,0,0,0]

        //
        // now we convert [4i, 0, 0, 0] back to [1i,1i,1i,1i]
        // with backward FFT
        //
        fftc1dinv(z);
        printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [1i,1i,1i,1i]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

fft_complex_d2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // first we demonstrate forward FFT:
        // [0,1,0,1i] is converted to [1+1i, -1-1i, -1-1i, 1+1i]
        //
        complex_1d_array z = "[0,1,0,1i]";
        fftc1d(z);
        printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [1+1i, -1-1i, -1-1i, 1+1i]

        //
        // now we convert result back with backward FFT
        //
        fftc1dinv(z);
        printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [0,1,0,1i]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

fft_real_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // first we demonstrate forward FFT:
        // [1,1,1,1] is converted to [4, 0, 0, 0]
        //
        real_1d_array x = "[1,1,1,1]";
        complex_1d_array f;
        real_1d_array x2;
        fftr1d(x, f);
        printf("%s\n", f.tostring(3).c_str()); // EXPECTED: [4,0,0,0]

        //
        // now we convert [4, 0, 0, 0] back to [1,1,1,1]
        // with backward FFT
        //
        fftr1dinv(f, x2);
        printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,1,1,1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

fft_real_d2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // first we demonstrate forward FFT:
        // [1,2,3,4] is converted to [10, -2+2i, -2, -2-2i]
        //
        // note that output array is self-adjoint:
        // * f[0] = conj(f[0])
        // * f[1] = conj(f[3])
        // * f[2] = conj(f[2])
        //
        real_1d_array x = "[1,2,3,4]";
        complex_1d_array f;
        real_1d_array x2;
        fftr1d(x, f);
        printf("%s\n", f.tostring(3).c_str()); // EXPECTED: [10, -2+2i, -2, -2-2i]

        //
        // now we convert [10, -2+2i, -2, -2-2i] back to [1,2,3,4]
        //
        fftr1dinv(f, x2);
        printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,2,3,4]

        //
        // remember that F is self-adjoint? It means that we can pass just half
        // (slightly larger than half) of F to inverse real FFT and still get our result.
        //
        // I.e. instead [10, -2+2i, -2, -2-2i] we pass just [10, -2+2i, -2] and everything works!
        //
        // NOTE: in this case we should explicitly pass array length (which is 4) to ALGLIB;
        // if not, it will automatically use array length to determine FFT size and
        // will erroneously make half-length FFT.
        //
        f = "[10, -2+2i, -2]";
        fftr1dinv(f, 4, x2);
        printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,2,3,4]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`fht` subpackage

Functions

fhtr1d
fhtr1dinv

Examples

`fhtr1d` function

/*************************************************************************
1-dimensional Fast Hartley Transform.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - real function to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   FHT of a input array, array[0..N-1],
            A_out[k] = sum(A_in[j]*(cos(2*pi*j*k/N)+sin(2*pi*j*k/N)), j=0..N-1)


  -- ALGLIB --
     Copyright 04.06.2009 by Bochkanov Sergey
*************************************************************************/
void fhtr1d(real_1d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`fhtr1dinv` function

/*************************************************************************
1-dimensional inverse FHT.

Algorithm has O(N*logN) complexity for any N (composite or prime).

INPUT PARAMETERS
    A   -   array[0..N-1] - complex array to be transformed
    N   -   problem size

OUTPUT PARAMETERS
    A   -   inverse FHT of a input array, array[0..N-1]


  -- ALGLIB --
     Copyright 29.05.2009 by Bochkanov Sergey
*************************************************************************/
void fhtr1dinv(real_1d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`filters` subpackage

Functions

filterema
filterlrma
filtersma

Examples

filters_d_ema		EMA(alpha) filter
filters_d_lrma		LRMA(k) filter
filters_d_sma		SMA(k) filter

`filterema` function

/*************************************************************************
Filters: exponential moving averages.

This filter replaces array by results of EMA(alpha) filter. EMA(alpha) is
defined as filter which replaces X[] by S[]:
    S[0] = X[0]
    S[t] = alpha*X[t] + (1-alpha)*S[t-1]

INPUT PARAMETERS:
    X           -   array[N], array to process. It can be larger than N,
                    in this case only first N points are processed.
    N           -   points count, N>=0
    alpha       -   0<alpha<=1, smoothing parameter.

OUTPUT PARAMETERS:
    X           -   array, whose first N elements were processed
                    with EMA(alpha)

NOTE 1: this function uses efficient in-place  algorithm  which  does not
        allocate temporary arrays.

NOTE 2: this algorithm uses BOTH previous points and  current  one,  i.e.
        new value of X[i] depends on BOTH previous point and X[i] itself.

NOTE 3: technical analytis users quite often work  with  EMA  coefficient
        expressed in DAYS instead of fractions. If you want to  calculate
        EMA(N), where N is a number of days, you can use alpha=2/(N+1).

  -- ALGLIB --
     Copyright 25.10.2011 by Bochkanov Sergey
*************************************************************************/
void filterema(real_1d_array &x, const ae_int_t n, const double alpha, const xparams _xparams = alglib::xdefault);
void filterema(real_1d_array &x, const double alpha, const xparams _xparams = alglib::xdefault);

Examples: [1]

`filterlrma` function

/*************************************************************************
Filters: linear regression moving averages.

This filter replaces array by results of LRMA(K) filter.

LRMA(K) is defined as filter which, for each data  point,  builds  linear
regression  model  using  K  prevous  points (point itself is included in
these K points) and calculates value of this linear model at the point in
question.

INPUT PARAMETERS:
    X           -   array[N], array to process. It can be larger than N,
                    in this case only first N points are processed.
    N           -   points count, N>=0
    K           -   K>=1 (K can be larger than N ,  such  cases  will  be
                    correctly handled). Window width. K=1 corresponds  to
                    identity transformation (nothing changes).

OUTPUT PARAMETERS:
    X           -   array, whose first N elements were processed with LRMA(K)

NOTE 1: this function uses efficient in-place  algorithm  which  does not
        allocate temporary arrays.

NOTE 2: this algorithm makes only one pass through array and uses running
        sum  to speed-up calculation of the averages. Additional measures
        are taken to ensure that running sum on a long sequence  of  zero
        elements will be correctly reset to zero even in the presence  of
        round-off error.

NOTE 3: this  is  unsymmetric version of the algorithm,  which  does  NOT
        averages points after the current one. Only X[i], X[i-1], ... are
        used when calculating new value of X[i]. We should also note that
        this algorithm uses BOTH previous points and  current  one,  i.e.
        new value of X[i] depends on BOTH previous point and X[i] itself.

  -- ALGLIB --
     Copyright 25.10.2011 by Bochkanov Sergey
*************************************************************************/
void filterlrma(real_1d_array &x, const ae_int_t n, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void filterlrma(real_1d_array &x, const ae_int_t k, const xparams _xparams = alglib::xdefault);

Examples: [1]

`filtersma` function

/*************************************************************************
Filters: simple moving averages (unsymmetric).

This filter replaces array by results of SMA(K) filter. SMA(K) is defined
as filter which averages at most K previous points (previous - not points
AROUND central point) - or less, in case of the first K-1 points.

INPUT PARAMETERS:
    X           -   array[N], array to process. It can be larger than N,
                    in this case only first N points are processed.
    N           -   points count, N>=0
    K           -   K>=1 (K can be larger than N ,  such  cases  will  be
                    correctly handled). Window width. K=1 corresponds  to
                    identity transformation (nothing changes).

OUTPUT PARAMETERS:
    X           -   array, whose first N elements were processed with SMA(K)

NOTE 1: this function uses efficient in-place  algorithm  which  does not
        allocate temporary arrays.

NOTE 2: this algorithm makes only one pass through array and uses running
        sum  to speed-up calculation of the averages. Additional measures
        are taken to ensure that running sum on a long sequence  of  zero
        elements will be correctly reset to zero even in the presence  of
        round-off error.

NOTE 3: this  is  unsymmetric version of the algorithm,  which  does  NOT
        averages points after the current one. Only X[i], X[i-1], ... are
        used when calculating new value of X[i]. We should also note that
        this algorithm uses BOTH previous points and  current  one,  i.e.
        new value of X[i] depends on BOTH previous point and X[i] itself.

  -- ALGLIB --
     Copyright 25.10.2011 by Bochkanov Sergey
*************************************************************************/
void filtersma(real_1d_array &x, const ae_int_t n, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void filtersma(real_1d_array &x, const ae_int_t k, const xparams _xparams = alglib::xdefault);

Examples: [1]

filters_d_ema example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate EMA(0.5) filtering for time series.
        //
        real_1d_array x = "[5,6,7,8]";

        //
        // Apply filter.
        // We should get [5, 5.5, 6.25, 7.125] as result
        //
        filterema(x, 0.5);
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [5,5.5,6.25,7.125]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

filters_d_lrma example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate LRMA(3) filtering for time series.
        //
        real_1d_array x = "[7,8,8,9,12,12]";

        //
        // Apply filter.
        // We should get [7.0000, 8.0000, 8.1667, 8.8333, 11.6667, 12.5000] as result
        //    
        filterlrma(x, 3);
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [7.0000,8.0000,8.1667,8.8333,11.6667,12.5000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

filters_d_sma example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate SMA(k) filtering for time series.
        //
        real_1d_array x = "[5,6,7,8]";

        //
        // Apply filter.
        // We should get [5, 5.5, 6.5, 7.5] as result
        //
        filtersma(x, 2);
        printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [5,5.5,6.5,7.5]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`fitsphere` subpackage

Functions

fitspherels
fitspheremc
fitspheremi
fitspheremz
fitspherex

Examples

`fitspherels` function

/*************************************************************************
Fits least squares (LS) circle (or NX-dimensional sphere) to data  (a  set
of points in NX-dimensional space).

Least squares circle minimizes sum of squared deviations between distances
from points to the center and  some  "candidate"  radius,  which  is  also
fitted to the data.

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    R       -   radius

  -- ALGLIB --
     Copyright 07.05.2018 by Bochkanov Sergey
*************************************************************************/
void fitspherels(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &r, const xparams _xparams = alglib::xdefault);

`fitspheremc` function

/*************************************************************************
Fits minimum circumscribed (MC) circle (or NX-dimensional sphere) to  data
(a set of points in NX-dimensional space).

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RHi     -   radius

NOTE: this function is an easy-to-use wrapper around more powerful "expert"
      function fitspherex().

      This  wrapper  is optimized  for  ease of use and stability - at the
      cost of somewhat lower  performance  (we  have  to  use  very  tight
      stopping criteria for inner optimizer because we want to  make  sure
      that it will converge on any dataset).

      If you are ready to experiment with settings of  "expert"  function,
      you can achieve ~2-4x speedup over standard "bulletproof" settings.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void fitspheremc(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &rhi, const xparams _xparams = alglib::xdefault);

`fitspheremi` function

/*************************************************************************
Fits maximum inscribed circle (or NX-dimensional sphere) to data (a set of
points in NX-dimensional space).

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RLo     -   radius

NOTE: this function is an easy-to-use wrapper around more powerful "expert"
      function fitspherex().

      This  wrapper  is optimized  for  ease of use and stability - at the
      cost of somewhat lower  performance  (we  have  to  use  very  tight
      stopping criteria for inner optimizer because we want to  make  sure
      that it will converge on any dataset).

      If you are ready to experiment with settings of  "expert"  function,
      you can achieve ~2-4x speedup over standard "bulletproof" settings.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void fitspheremi(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &rlo, const xparams _xparams = alglib::xdefault);

`fitspheremz` function

/*************************************************************************
Fits minimum zone circle (or NX-dimensional sphere)  to  data  (a  set  of
points in NX-dimensional space).

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RLo     -   radius of inscribed circle
    RHo     -   radius of circumscribed circle

NOTE: this function is an easy-to-use wrapper around more powerful "expert"
      function fitspherex().

      This  wrapper  is optimized  for  ease of use and stability - at the
      cost of somewhat lower  performance  (we  have  to  use  very  tight
      stopping criteria for inner optimizer because we want to  make  sure
      that it will converge on any dataset).

      If you are ready to experiment with settings of  "expert"  function,
      you can achieve ~2-4x speedup over standard "bulletproof" settings.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void fitspheremz(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &rlo, double &rhi, const xparams _xparams = alglib::xdefault);

`fitspherex` function

/*************************************************************************
Fitting minimum circumscribed, maximum inscribed or minimum  zone  circles
(or NX-dimensional spheres)  to  data  (a  set of points in NX-dimensional
space).

This  is  expert  function  which  allows  to  tweak  many  parameters  of
underlying nonlinear solver:
* stopping criteria for inner iterations
* number of outer iterations

You may tweak all these parameters or only some  of  them,  leaving  other
ones at their default state - just specify zero  value,  and  solver  will
fill it with appropriate default one.

These comments also include some discussion of  approach  used  to  handle
such unusual fitting problem,  its  stability,  drawbacks  of  alternative
methods, and convergence properties.

INPUT PARAMETERS:
    XY      -   array[NPoints,NX] (or larger), contains dataset.
                One row = one point in NX-dimensional space.
    NPoints -   dataset size, NPoints>0
    NX      -   space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on)
    ProblemType-used to encode problem type:
                * 0 for least squares circle
                * 1 for minimum circumscribed circle/sphere fitting (MC)
                * 2 for  maximum inscribed circle/sphere fitting (MI)
                * 3 for minimum zone circle fitting (difference between
                    Rhi and Rlo is minimized), denoted as MZ
    EpsX    -   stopping condition for NLC optimizer:
                * must be non-negative
                * use 0 to choose default value (1.0E-12 is used by default)
                * you may specify larger values, up to 1.0E-6, if you want
                  to   speed-up   solver;   NLC   solver  performs several
                  preconditioned  outer  iterations,   so   final   result
                  typically has precision much better than EpsX.
    AULIts  -   number of outer iterations performed by NLC optimizer:
                * must be non-negative
                * use 0 to choose default value (20 is used by default)
                * you may specify values smaller than 20 if you want to
                  speed up solver; 10 often results in good combination of
                  precision and speed; sometimes you may get good results
                  with just 6 outer iterations.
                Ignored for ProblemType=0.

OUTPUT PARAMETERS:
    CX      -   central point for a sphere
    RLo     -   radius:
                * for ProblemType=2,3, radius of the inscribed sphere
                * for ProblemType=0 - radius of the least squares sphere
                * for ProblemType=1 - zero
    RHo     -   radius:
                * for ProblemType=1,3, radius of the circumscribed sphere
                * for ProblemType=0 - radius of the least squares sphere
                * for ProblemType=2 - zero

NOTE: ON THE UNIQUENESS OF SOLUTIONS

ALGLIB provides solution to several related circle fitting  problems:   MC
(minimum circumscribed), MI (maximum inscribed)   and   MZ  (minimum zone)
fitting, LS (least squares) fitting.

It  is  important  to  note  that  among these problems only MC and LS are
convex and have unique solution independently from starting point.

As  for MI,  it  may (or  may  not, depending on dataset properties)  have
multiple solutions, and it always  has  one degenerate solution C=infinity
which corresponds to infinitely large radius. Thus, there are no guarantees
that solution to  MI returned by this solver will be the best one (and  no
one can provide you with such guarantee because problem is  NP-hard).  The
only guarantee you have is that this solution is locally optimal, i.e.  it
can not be improved by infinitesimally small tweaks in the parameters.

It  is  also  possible  to "run away" to infinity when  started  from  bad
initial point located outside of point cloud (or when point cloud does not
span entire circumference/surface of the sphere).

Finally,  MZ (minimum zone circle) stands somewhere between MC  and  MI in
stability. It is somewhat regularized by "circumscribed" term of the merit
function; however, solutions to  MZ may be non-unique, and in some unlucky
cases it is also possible to "run away to infinity".


NOTE: ON THE NONLINEARLY CONSTRAINED PROGRAMMING APPROACH

The problem formulation for MC  (minimum circumscribed   circle;  for  the
sake of simplicity we omit MZ and MI here) is:

        [     [         ]2 ]
    min [ max [ XY[i]-C ]  ]
     C  [  i  [         ]  ]

i.e. it is unconstrained nonsmooth optimization problem of finding  "best"
central point, with radius R being unambiguously  determined  from  C.  In
order to move away from non-smoothness we use following reformulation:

        [   ]                  [         ]2
    min [ R ] subject to R>=0, [ XY[i]-C ]  <= R^2
    C,R [   ]                  [         ]

i.e. it becomes smooth quadratically constrained optimization problem with
linear target function. Such problem statement is 100% equivalent  to  the
original nonsmooth one, but much easier  to  approach.  We solve  it  with
MinNLC solver provided by ALGLIB.


NOTE: ON INSTABILITY OF SEQUENTIAL LINEARIZATION APPROACH

ALGLIB  has  nonlinearly  constrained  solver which proved to be stable on
such problems. However, some authors proposed to linearize constraints  in
the vicinity of current approximation (Ci,Ri) and to get next  approximate
solution (Ci+1,Ri+1) as solution to linear programming problem. Obviously,
LP problems are easier than nonlinearly constrained ones.

Indeed,  such approach  to   MC/MI/MZ   resulted   in  ~10-20x increase in
performance (when compared with NLC solver). However, it turned  out  that
in some cases linearized model fails to predict correct direction for next
step and tells us that we converged to solution even when we are still 2-4
digits of precision away from it.

It is important that it is not failure of LP solver - it is failure of the
linear model;  even  when  solved  exactly,  it  fails  to  handle  subtle
nonlinearities which arise near the solution. We validated it by comparing
results returned by ALGLIB linear solver with that of MATLAB.

In our experiments with linearization:
* MC failed most often, at both realistic and synthetic datasets
* MI sometimes failed, but sometimes succeeded
* MZ often  succeeded; our guess is that presence of two independent  sets
  of constraints (one set for Rlo and another one for Rhi) and  two  terms
  in the target function (Rlo and Rhi) regularizes task,  so  when  linear
  model fails to handle nonlinearities from Rlo, it uses  Rhi  as  a  hint
  (and vice versa).

Because linearization approach failed to achieve stable results, we do not
include it in ALGLIB.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void fitspherex(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, const ae_int_t problemtype, const double epsx, const ae_int_t aulits, real_1d_array &cx, double &rlo, double &rhi, const xparams _xparams = alglib::xdefault);

`fresnel` subpackage

Functions

fresnelintegral

Examples

`fresnelintegral` function

/*************************************************************************
Fresnel integral

Evaluates the Fresnel integrals

          x
          -
         | |
C(x) =   |   cos(pi/2 t**2) dt,
       | |
        -
         0

          x
          -
         | |
S(x) =   |   sin(pi/2 t**2) dt.
       | |
        -
         0


The integrals are evaluated by a power series for x < 1.
For x >= 1 auxiliary functions f(x) and g(x) are employed
such that

C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 )
S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 )



ACCURACY:

 Relative error.

Arithmetic  function   domain     # trials      peak         rms
  IEEE       S(x)      0, 10       10000       2.0e-15     3.2e-16
  IEEE       C(x)      0, 10       10000       1.8e-15     3.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
*************************************************************************/
void fresnelintegral(const double x, double &c, double &s, const xparams _xparams = alglib::xdefault);

`gammafunc` subpackage

Functions

gammafunction
lngamma

Examples

`gammafunction` function

/*************************************************************************
Gamma function

Input parameters:
    X   -   argument

Domain:
    0 < X < 171.6
    -170 < X < 0, X is not an integer.

Relative error:
 arithmetic   domain     # trials      peak         rms
    IEEE    -170,-33      20000       2.3e-15     3.3e-16
    IEEE     -33,  33     20000       9.4e-16     2.2e-16
    IEEE      33, 171.6   20000       2.3e-15     3.2e-16

Cephes Math Library Release 2.8:  June, 2000
Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double gammafunction(const double x, const xparams _xparams = alglib::xdefault);

`lngamma` function

/*************************************************************************
Natural logarithm of gamma function

Input parameters:
    X       -   argument

Result:
    logarithm of the absolute value of the Gamma(X).

Output parameters:
    SgnGam  -   sign(Gamma(X))

Domain:
    0 < X < 2.55e305
    -2.55e305 < X < 0, X is not an integer.

ACCURACY:
arithmetic      domain        # trials     peak         rms
   IEEE    0, 3                 28000     5.4e-16     1.1e-16
   IEEE    2.718, 2.556e305     40000     3.5e-16     8.3e-17
The error criterion was relative when the function magnitude
was greater than one but absolute when it was less than one.

The following test used the relative error criterion, though
at certain points the relative error could be much higher than
indicated.
   IEEE    -200, -4             10000     4.8e-16     1.3e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007).
*************************************************************************/
double lngamma(const double x, double &sgngam, const xparams _xparams = alglib::xdefault);

`gkq` subpackage

Functions

gkqgenerategaussjacobi
gkqgenerategausslegendre
gkqgeneraterec
gkqlegendrecalc
gkqlegendretbl

Examples

`gkqgenerategaussjacobi` function

/*************************************************************************
Returns   Gauss   and   Gauss-Kronrod   nodes/weights   for   Gauss-Jacobi
quadrature on [-1,1] with weight function

    W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.
    Alpha       -   power-law coefficient, Alpha>-1
    Beta        -   power-law coefficient, Beta>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -5    no real and positive Gauss-Kronrod formula can
                            be created for such a weight function  with  a
                            given number of nodes.
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha or  Beta  are  too  close
                            to -1 to obtain weights/nodes with high enough
                            accuracy, or, may be, N is too large.  Try  to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
                    * +2    OK, but quadrature rule have exterior  nodes,
                            x[0]<-1 or x[n-1]>+1
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqgenerategaussjacobi(const ae_int_t n, const double alpha, const double beta, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss, const xparams _xparams = alglib::xdefault);

`gkqgenerategausslegendre` function

/*************************************************************************
Returns   Gauss   and   Gauss-Kronrod   nodes/weights  for  Gauss-Legendre
quadrature with N points.

GKQLegendreCalc (calculation) or  GKQLegendreTbl  (precomputed  table)  is
used depending on machine precision and number of nodes.

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqgenerategausslegendre(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss, const xparams _xparams = alglib::xdefault);

`gkqgeneraterec` function

/*************************************************************************
Computation of nodes and weights of a Gauss-Kronrod quadrature formula

The algorithm generates the N-point Gauss-Kronrod quadrature formula  with
weight  function  given  by  coefficients  alpha  and beta of a recurrence
relation which generates a system of orthogonal polynomials:

    P-1(x)   =  0
    P0(x)    =  1
    Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zero moment Mu0

    Mu0 = integral(W(x)dx,a,b)


INPUT PARAMETERS:
    Alpha       -   alpha coefficients, array[0..floor(3*K/2)].
    Beta        -   beta coefficients,  array[0..ceil(3*K/2)].
                    Beta[0] is not used and may be arbitrary.
                    Beta[I]>0.
    Mu0         -   zeroth moment of the weight function.
    N           -   number of nodes of the Gauss-Kronrod quadrature formula,
                    N >= 3,
                    N =  2*K+1.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -5    no real and positive Gauss-Kronrod formula can
                            be created for such a weight function  with  a
                            given number of nodes.
                    * -4    N is too large, task may be ill  conditioned -
                            x[i]=x[i+1] found.
                    * -3    internal eigenproblem solver hasn't converged
                    * -2    Beta[i]<=0
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).

  -- ALGLIB --
     Copyright 08.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqgeneraterec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss, const xparams _xparams = alglib::xdefault);

`gkqlegendrecalc` function

/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.

Reduction to tridiagonal eigenproblem is used.

INPUT PARAMETERS:
    N           -   number of Kronrod nodes, must be odd number, >=3.

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).

  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqlegendrecalc(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss, const xparams _xparams = alglib::xdefault);

`gkqlegendretbl` function

/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N  points  using
pre-calculated table. Nodes/weights were  computed  with  accuracy  up  to
1.0E-32 (if MPFR version of ALGLIB is used). In standard double  precision
accuracy reduces to something about 2.0E-16 (depending  on your compiler's
handling of long floating point constants).

INPUT PARAMETERS:
    N           -   number of Kronrod nodes.
                    N can be 15, 21, 31, 41, 51, 61.

OUTPUT PARAMETERS:
    X           -   array[0..N-1] - array of quadrature nodes, ordered in
                    ascending order.
    WKronrod    -   array[0..N-1] - Kronrod weights
    WGauss      -   array[0..N-1] - Gauss weights (interleaved with zeros
                    corresponding to extended Kronrod nodes).


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gkqlegendretbl(const ae_int_t n, real_1d_array &x, real_1d_array &wkronrod, real_1d_array &wgauss, double &eps, const xparams _xparams = alglib::xdefault);

`gq` subpackage

Functions

gqgenerategausshermite
gqgenerategaussjacobi
gqgenerategausslaguerre
gqgenerategausslegendre
gqgenerategausslobattorec
gqgenerategaussradaurec
gqgeneraterec

Examples

`gqgenerategausshermite` function

/*************************************************************************
Returns  nodes/weights  for  Gauss-Hermite  quadrature on (-inf,+inf) with
weight function W(x)=Exp(-x*x)

INPUT PARAMETERS:
    N           -   number of nodes, >=1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes.  May be, N is too large. Try to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausshermite(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`gqgenerategaussjacobi` function

/*************************************************************************
Returns  nodes/weights  for  Gauss-Jacobi quadrature on [-1,1] with weight
function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).

INPUT PARAMETERS:
    N           -   number of nodes, >=1
    Alpha       -   power-law coefficient, Alpha>-1
    Beta        -   power-law coefficient, Beta>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha or  Beta  are  too  close
                            to -1 to obtain weights/nodes with high enough
                            accuracy, or, may be, N is too large.  Try  to
                            use multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha/Beta was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategaussjacobi(const ae_int_t n, const double alpha, const double beta, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`gqgenerategausslaguerre` function

/*************************************************************************
Returns  nodes/weights  for  Gauss-Laguerre  quadrature  on  [0,+inf) with
weight function W(x)=Power(x,Alpha)*Exp(-x)

INPUT PARAMETERS:
    N           -   number of nodes, >=1
    Alpha       -   power-law coefficient, Alpha>-1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error  was   detected   when   calculating
                            weights/nodes. Alpha is too  close  to  -1  to
                            obtain weights/nodes with high enough accuracy
                            or, may  be,  N  is  too  large.  Try  to  use
                            multiple precision version.
                    * -3    internal eigenproblem solver hasn't converged
                    * -1    incorrect N/Alpha was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausslaguerre(const ae_int_t n, const double alpha, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`gqgenerategausslegendre` function

/*************************************************************************
Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
nodes.

INPUT PARAMETERS:
    N           -   number of nodes, >=1

OUTPUT PARAMETERS:
    Info        -   error code:
                    * -4    an  error   was   detected   when  calculating
                            weights/nodes.  N  is  too  large   to  obtain
                            weights/nodes  with  high   enough   accuracy.
                            Try  to   use   multiple   precision  version.
                    * -3    internal eigenproblem solver hasn't  converged
                    * -1    incorrect N was passed
                    * +1    OK
    X           -   array[0..N-1] - array of quadrature nodes,
                    in ascending order.
    W           -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausslegendre(const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`gqgenerategausslobattorec` function

/*************************************************************************
Computation of nodes and weights for a Gauss-Lobatto quadrature formula

The algorithm generates the N-point Gauss-Lobatto quadrature formula  with
weight function given by coefficients alpha and beta of a recurrence which
generates a system of orthogonal polynomials.

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha   -   array[0..N-2], alpha coefficients
    Beta    -   array[0..N-2], beta coefficients.
                Zero-indexed element is not used, may be arbitrary.
                Beta[I]>0
    Mu0     -   zeroth moment of the weighting function.
    A       -   left boundary of the integration interval.
    B       -   right boundary of the integration interval.
    N       -   number of nodes of the quadrature formula, N>=3
                (including the left and right boundary nodes).

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.

  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategausslobattorec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const double a, const double b, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`gqgenerategaussradaurec` function

/*************************************************************************
Computation of nodes and weights for a Gauss-Radau quadrature formula

The algorithm generates the N-point Gauss-Radau  quadrature  formula  with
weight function given by the coefficients alpha and  beta  of a recurrence
which generates a system of orthogonal polynomials.

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha   -   array[0..N-2], alpha coefficients.
    Beta    -   array[0..N-1], beta coefficients
                Zero-indexed element is not used.
                Beta[I]>0
    Mu0     -   zeroth moment of the weighting function.
    A       -   left boundary of the integration interval.
    N       -   number of nodes of the quadrature formula, N>=2
                (including the left boundary node).

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.


  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void gqgenerategaussradaurec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const double a, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`gqgeneraterec` function

/*************************************************************************
Computation of nodes and weights for a Gauss quadrature formula

The algorithm generates the N-point Gauss quadrature formula  with  weight
function given by coefficients alpha and beta  of  a  recurrence  relation
which generates a system of orthogonal polynomials:

P-1(x)   =  0
P0(x)    =  1
Pn+1(x)  =  (x-alpha(n))*Pn(x)  -  beta(n)*Pn-1(x)

and zeroth moment Mu0

Mu0 = integral(W(x)dx,a,b)

INPUT PARAMETERS:
    Alpha   -   array[0..N-1], alpha coefficients
    Beta    -   array[0..N-1], beta coefficients
                Zero-indexed element is not used and may be arbitrary.
                Beta[I]>0.
    Mu0     -   zeroth moment of the weight function.
    N       -   number of nodes of the quadrature formula, N>=1

OUTPUT PARAMETERS:
    Info    -   error code:
                * -3    internal eigenproblem solver hasn't converged
                * -2    Beta[i]<=0
                * -1    incorrect N was passed
                *  1    OK
    X       -   array[0..N-1] - array of quadrature nodes,
                in ascending order.
    W       -   array[0..N-1] - array of quadrature weights.

  -- ALGLIB --
     Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
void gqgeneraterec(const real_1d_array &alpha, const real_1d_array &beta, const double mu0, const ae_int_t n, ae_int_t &info, real_1d_array &x, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`hermite` subpackage

Functions

hermitecalculate
hermitecoefficients
hermitesum

Examples

`hermitecalculate` function

/*************************************************************************
Calculation of the value of the Hermite polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial Hn at x
*************************************************************************/
double hermitecalculate(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`hermitecoefficients` function

/*************************************************************************
Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void hermitecoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`hermitesum` function

/*************************************************************************
Summation of Hermite polynomials using Clenshaw's recurrence formula.

This routine calculates
    c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Hermite polynomial at x
*************************************************************************/
double hermitesum(const real_1d_array &c, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`hqrnd` subpackage

Classes

hqrndstate

Functions

hqrndcontinuous
hqrnddiscrete
hqrndexponential
hqrndnormal
hqrndnormal2
hqrndnormalm
hqrndnormalv
hqrndrandomize
hqrndseed
hqrnduniformi
hqrnduniformr
hqrndunit2

Examples

`hqrndstate` class

/*************************************************************************
Portable high quality random number generator state.
Initialized with HQRNDRandomize() or HQRNDSeed().

Fields:
    S1, S2      -   seed values
    V           -   precomputed value
    MagicV      -   'magic' value used to determine whether State structure
                    was correctly initialized.
*************************************************************************/
class hqrndstate
{
public:
    hqrndstate();
    hqrndstate(const hqrndstate &rhs);
    hqrndstate& operator=(const hqrndstate &rhs);
    virtual ~hqrndstate();
};

`hqrndcontinuous` function

/*************************************************************************
This function generates random number from continuous  distribution  given
by finite sample X.

INPUT PARAMETERS
    State   -   high quality random number generator, must be
                initialized with HQRNDRandomize() or HQRNDSeed().
        X   -   finite sample, array[N] (can be larger, in this  case only
                leading N elements are used). THIS ARRAY MUST BE SORTED BY
                ASCENDING.
        N   -   number of elements to use, N>=1

RESULT
    this function returns random number from continuous distribution which
    tries to approximate X as mush as possible. min(X)<=Result<=max(X).

  -- ALGLIB --
     Copyright 08.11.2011 by Bochkanov Sergey
*************************************************************************/
double hqrndcontinuous(hqrndstate &state, const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`hqrnddiscrete` function

/*************************************************************************
This function generates  random number from discrete distribution given by
finite sample X.

INPUT PARAMETERS
    State   -   high quality random number generator, must be
                initialized with HQRNDRandomize() or HQRNDSeed().
        X   -   finite sample
        N   -   number of elements to use, N>=1

RESULT
    this function returns one of the X[i] for random i=0..N-1

  -- ALGLIB --
     Copyright 08.11.2011 by Bochkanov Sergey
*************************************************************************/
double hqrnddiscrete(hqrndstate &state, const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`hqrndexponential` function

/*************************************************************************
Random number generator: exponential distribution

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 11.08.2007 by Bochkanov Sergey
*************************************************************************/
double hqrndexponential(hqrndstate &state, const double lambdav, const xparams _xparams = alglib::xdefault);

`hqrndnormal` function

/*************************************************************************
Random number generator: normal numbers

This function generates one random number from normal distribution.
Its performance is equal to that of HQRNDNormal2()

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double hqrndnormal(hqrndstate &state, const xparams _xparams = alglib::xdefault);

`hqrndnormal2` function

/*************************************************************************
Random number generator: normal numbers

This function generates two independent random numbers from normal
distribution. Its performance is equal to that of HQRNDNormal()

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void hqrndnormal2(hqrndstate &state, double &x1, double &x2, const xparams _xparams = alglib::xdefault);

`hqrndnormalm` function

/*************************************************************************
Random number generator: matrix with random entries (normal distribution)

This function generates MxN random matrix.

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void hqrndnormalm(hqrndstate &state, const ae_int_t m, const ae_int_t n, real_2d_array &x, const xparams _xparams = alglib::xdefault);

`hqrndnormalv` function

/*************************************************************************
Random number generator: vector with random entries (normal distribution)

This function generates N random numbers from normal distribution.

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void hqrndnormalv(hqrndstate &state, const ae_int_t n, real_1d_array &x, const xparams _xparams = alglib::xdefault);

`hqrndrandomize` function

/*************************************************************************
HQRNDState  initialization  with  random  values  which come from standard
RNG.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void hqrndrandomize(hqrndstate &state, const xparams _xparams = alglib::xdefault);

`hqrndseed` function

/*************************************************************************
HQRNDState initialization with seed values

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void hqrndseed(const ae_int_t s1, const ae_int_t s2, hqrndstate &state, const xparams _xparams = alglib::xdefault);

`hqrnduniformi` function

/*************************************************************************
This function generates random integer number in [0, N)

1. State structure must be initialized with HQRNDRandomize() or HQRNDSeed()
2. N can be any positive number except for very large numbers:
   * close to 2^31 on 32-bit systems
   * close to 2^62 on 64-bit systems
   An exception will be generated if N is too large.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
ae_int_t hqrnduniformi(hqrndstate &state, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`hqrnduniformr` function

/*************************************************************************
This function generates random real number in (0,1),
not including interval boundaries

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double hqrnduniformr(hqrndstate &state, const xparams _xparams = alglib::xdefault);

`hqrndunit2` function

/*************************************************************************
Random number generator: random X and Y such that X^2+Y^2=1

State structure must be initialized with HQRNDRandomize() or HQRNDSeed().

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void hqrndunit2(hqrndstate &state, double &x, double &y, const xparams _xparams = alglib::xdefault);

`ibetaf` subpackage

Functions

incompletebeta
invincompletebeta

Examples

`incompletebeta` function

/*************************************************************************
Incomplete beta integral

Returns incomplete beta integral of the arguments, evaluated
from zero to x.  The function is defined as

                 x
    -            -
   | (a+b)      | |  a-1     b-1
 -----------    |   t   (1-t)   dt.
  -     -     | |
 | (a) | (b)   -
                0

The domain of definition is 0 <= x <= 1.  In this
implementation a and b are restricted to positive values.
The integral from x to 1 may be obtained by the symmetry
relation

   1 - incbet( a, b, x )  =  incbet( b, a, 1-x ).

The integral is evaluated by a continued fraction expansion
or, when b*x is small, by a power series.

ACCURACY:

Tested at uniformly distributed random points (a,b,x) with a and b
in "domain" and x between 0 and 1.
                                       Relative error
arithmetic   domain     # trials      peak         rms
   IEEE      0,5         10000       6.9e-15     4.5e-16
   IEEE      0,85       250000       2.2e-13     1.7e-14
   IEEE      0,1000      30000       5.3e-12     6.3e-13
   IEEE      0,10000    250000       9.3e-11     7.1e-12
   IEEE      0,100000    10000       8.7e-10     4.8e-11
Outputs smaller than the IEEE gradual underflow threshold
were excluded from these statistics.

Cephes Math Library, Release 2.8:  June, 2000
Copyright 1984, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletebeta(const double a, const double b, const double x, const xparams _xparams = alglib::xdefault);

`invincompletebeta` function

/*************************************************************************
Inverse of imcomplete beta integral

Given y, the function finds x such that

 incbet( a, b, x ) = y .

The routine performs interval halving or Newton iterations to find the
root of incbet(a,b,x) - y = 0.


ACCURACY:

                     Relative error:
               x     a,b
arithmetic   domain  domain  # trials    peak       rms
   IEEE      0,1    .5,10000   50000    5.8e-12   1.3e-13
   IEEE      0,1   .25,100    100000    1.8e-13   3.9e-15
   IEEE      0,1     0,5       50000    1.1e-12   5.5e-15
With a and b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   50000    5.8e-12   1.1e-13
   IEEE      0,1    .5,100    100000    1.7e-14   7.9e-16
With a = .5, b constrained to half-integer or integer values:
   IEEE      0,1    .5,10000   10000    8.3e-11   1.0e-11

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1996, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletebeta(const double a, const double b, const double y, const xparams _xparams = alglib::xdefault);

`idw` subpackage

Classes

idwbuilder
idwcalcbuffer
idwmodel
idwreport

Functions

idwbuildercreate
idwbuildersetalgomstab
idwbuildersetalgotextbookmodshepard
idwbuildersetalgotextbookshepard
idwbuildersetconstterm
idwbuildersetnlayers
idwbuildersetpoints
idwbuildersetuserterm
idwbuildersetzeroterm
idwcalc
idwcalc1
idwcalc2
idwcalc3
idwcalcbuf
idwcreatecalcbuffer
idwfit
idwgridcalc2v
idwgridcalc2vsubset
idwpeekprogress
idwserialize
idwtscalcbuf
idwunserialize

Examples

idw_d_mstab		Simple model built with IDW-MSTAB algorithm
idw_d_serialize		IDW model serialization/unserialization

`idwbuilder` class

/*************************************************************************
Builder object used to generate IDW (Inverse Distance Weighting) model.
*************************************************************************/
class idwbuilder
{
public:
    idwbuilder();
    idwbuilder(const idwbuilder &rhs);
    idwbuilder& operator=(const idwbuilder &rhs);
    virtual ~idwbuilder();
};

`idwcalcbuffer` class

/*************************************************************************
Buffer  object  which  is  used  to  perform  evaluation  requests  in  the
multithreaded mode (multiple threads working with same IDW object).

This object should be created with idwcreatecalcbuffer().
*************************************************************************/
class idwcalcbuffer
{
public:
    idwcalcbuffer();
    idwcalcbuffer(const idwcalcbuffer &rhs);
    idwcalcbuffer& operator=(const idwcalcbuffer &rhs);
    virtual ~idwcalcbuffer();
};

`idwmodel` class

/*************************************************************************
IDW (Inverse Distance Weighting) model object.
*************************************************************************/
class idwmodel
{
public:
    idwmodel();
    idwmodel(const idwmodel &rhs);
    idwmodel& operator=(const idwmodel &rhs);
    virtual ~idwmodel();
};

`idwreport` class

/*************************************************************************
IDW fitting report:
    rmserror        RMS error
    avgerror        average error
    maxerror        maximum error
    r2              coefficient of determination,  R-squared, 1-RSS/TSS
*************************************************************************/
class idwreport
{
public:
    idwreport();
    idwreport(const idwreport &rhs);
    idwreport& operator=(const idwreport &rhs);
    virtual ~idwreport();
    double rmserror;
    double avgerror;
    double maxerror;
    double r2;
};

`idwbuildercreate` function

/*************************************************************************
This subroutine creates builder object used  to  generate IDW  model  from
irregularly sampled (scattered) dataset.  Multidimensional  scalar/vector-
-valued are supported.

Builder object is used to fit model to data as follows:
* builder object is created with idwbuildercreate() function
* dataset is added with idwbuildersetpoints() function
* one of the modern IDW algorithms is chosen with either:
  * idwbuildersetalgomstab()            - Multilayer STABilized algorithm (interpolation)
  Alternatively, one of the textbook algorithms can be chosen (not recommended):
  * idwbuildersetalgotextbookshepard()  - textbook Shepard algorithm
  * idwbuildersetalgotextbookmodshepard()-textbook modified Shepard algorithm
* finally, model construction is performed with idwfit() function.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    NX  -   dimensionality of the argument, NX>=1
    NY  -   dimensionality of the function being modeled, NY>=1;
            NY=1 corresponds to classic scalar function, NY>=1 corresponds
            to vector-valued function.

OUTPUT PARAMETERS:
    State-  builder object

  -- ALGLIB PROJECT --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildercreate(const ae_int_t nx, const ae_int_t ny, idwbuilder &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwbuildersetalgomstab` function

/*************************************************************************
This function sets IDW model  construction  algorithm  to  the  Multilayer
Stabilized IDW method (IDW-MSTAB), a  latest  incarnation  of  the inverse
distance weighting interpolation which fixes shortcomings of  the original
and modified Shepard's variants.

The distinctive features of IDW-MSTAB are:
1) exact interpolation  is  pursued  (as  opposed  to  fitting  and  noise
   suppression)
2) improved robustness when compared with that of other algorithms:
   * MSTAB shows almost no strange  fitting  artifacts  like  ripples  and
     sharp spikes (unlike N-dimensional splines and HRBFs)
   * MSTAB does not return function values far from the  interval  spanned
     by the dataset; say, if all your points have |f|<=1, you  can be sure
     that model value won't deviate too much from [-1,+1]
3) good model construction time competing with that of HRBFs  and  bicubic
   splines
4) ability to work with any number of dimensions, starting from NX=1

The drawbacks of IDW-MSTAB (and all IDW algorithms in general) are:
1) dependence of the model evaluation time on the search radius
2) bad extrapolation properties, models built by this method  are  usually
   conservative in their predictions

Thus, IDW-MSTAB is  a  good  "default"  option  if  you  want  to  perform
scattered multidimensional interpolation. Although it has  its  drawbacks,
it is easy to use and robust, which makes it a good first step.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    State   -   builder object
    SRad    -   initial search radius, SRad>0 is required. A model  value
                is obtained by "smart" averaging  of  the  dataset  points
                within search radius.

NOTE 1: IDW interpolation can  correctly  handle  ANY  dataset,  including
        datasets with non-distinct points. In case non-distinct points are
        found, an average value for this point will be calculated.

NOTE 2: the memory requirements for model storage are O(NPoints*NLayers).
        The model construction needs twice as much memory as model storage.

NOTE 3: by default 16 IDW layers are built which is enough for most cases.
        You can change this parameter with idwbuildersetnlayers()  method.
        Larger values may be necessary if you need to reproduce  extrafine
        details at distances smaller than SRad/65536.  Smaller value   may
        be necessary if you have to save memory and  computing  time,  and
        ready to sacrifice some model quality.


ALGORITHM DESCRIPTION

ALGLIB implementation of IDW is somewhat similar to the modified Shepard's
method (one with search radius R) but overcomes several of its  drawbacks,
namely:
1) a tendency to show stepwise behavior for uniform datasets
2) a tendency to show terrible interpolation properties for highly
   nonuniform datasets which often arise in geospatial tasks
  (function values are densely sampled across multiple separated
  "tracks")

IDW-MSTAB method performs several passes over dataset and builds a sequence
of progressively refined IDW models  (layers),  which starts from one with
largest search radius SRad  and continues to smaller  search  radii  until
required number of  layers  is  built.  Highest  layers  reproduce  global
behavior of the target function at larger distances  whilst  lower  layers
reproduce fine details at smaller distances.

Each layer is an IDW model built with following modifications:
* weights go to zero when distance approach to the current search radius
* an additional regularizing term is added to the distance: w=1/(d^2+lambda)
* an additional fictional term with unit weight and zero function value is
  added in order to promote continuity  properties  at  the  isolated  and
  boundary points

By default, 16 layers is built, which is enough for most  cases.  You  can
change this parameter with idwbuildersetnlayers() method.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetalgomstab(idwbuilder &state, const double srad, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwbuildersetalgotextbookmodshepard` function

/*************************************************************************
This function sets  IDW  model  construction  algorithm  to the 'textbook'
modified Shepard's algorithm with user-specified search radius.

IMPORTANT: we do NOT recommend using textbook IDW algorithms because  they
           have terrible interpolation properties. Use MSTAB in all cases.

INPUT PARAMETERS:
    State   -   builder object
    R       -   search radius

NOTE 1: IDW interpolation can  correctly  handle  ANY  dataset,  including
        datasets with non-distinct points. In case non-distinct points are
        found, an average value for this point will be calculated.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetalgotextbookmodshepard(idwbuilder &state, const double r, const xparams _xparams = alglib::xdefault);

`idwbuildersetalgotextbookshepard` function

/*************************************************************************
This function sets  IDW  model  construction  algorithm  to  the  textbook
Shepard's algorithm with custom (user-specified) power parameter.

IMPORTANT: we do NOT recommend using textbook IDW algorithms because  they
           have terrible interpolation properties. Use MSTAB in all cases.

INPUT PARAMETERS:
    State   -   builder object
    P       -   power parameter, P>0; good value to start with is 2.0

NOTE 1: IDW interpolation can  correctly  handle  ANY  dataset,  including
        datasets with non-distinct points. In case non-distinct points are
        found, an average value for this point will be calculated.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetalgotextbookshepard(idwbuilder &state, const double p, const xparams _xparams = alglib::xdefault);

`idwbuildersetconstterm` function

/*************************************************************************
This function sets constant prior term (model value at infinity).

Constant prior term is determined as mean value over dataset.

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 29.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetconstterm(idwbuilder &state, const xparams _xparams = alglib::xdefault);

`idwbuildersetnlayers` function

/*************************************************************************
This function changes number of layers used by IDW-MSTAB algorithm.

The more layers you have, the finer details can  be  reproduced  with  IDW
model. The less layers you have, the less memory and CPU time is  consumed
by the model.

Memory consumption grows linearly with layers count,  running  time  grows
sub-linearly.

The default number of layers is 16, which allows you to reproduce  details
at distance down to SRad/65536. You will rarely need to change it.

INPUT PARAMETERS:
    State   -   builder object
    NLayers -   NLayers>=1, the number of layers used by the model.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetnlayers(idwbuilder &state, const ae_int_t nlayers, const xparams _xparams = alglib::xdefault);

`idwbuildersetpoints` function

/*************************************************************************
This function adds dataset to the builder object.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

INPUT PARAMETERS:
    State   -   builder object
    XY      -   points, array[N,NX+NY]. One row  corresponds to  one point
                in the dataset. First NX elements  are  coordinates,  next
                NY elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset, N>=0.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetpoints(idwbuilder &state, const real_2d_array &xy, const ae_int_t n, const xparams _xparams = alglib::xdefault);
void idwbuildersetpoints(idwbuilder &state, const real_2d_array &xy, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwbuildersetuserterm` function

/*************************************************************************
This function sets prior term (model value at infinity) as  user-specified
value.

INPUT PARAMETERS:
    S       -   spline builder
    V       -   value for user-defined prior

NOTE: for vector-valued models all components of the prior are set to same
      user-specified value

  -- ALGLIB --
     Copyright 29.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetuserterm(idwbuilder &state, const double v, const xparams _xparams = alglib::xdefault);

`idwbuildersetzeroterm` function

/*************************************************************************
This function sets zero prior term (model value at infinity).

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 29.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwbuildersetzeroterm(idwbuilder &state, const xparams _xparams = alglib::xdefault);

`idwcalc` function

/*************************************************************************
This function calculates values of the IDW model at the given point.

This is general function which can be used for arbitrary NX (dimension  of
the space of arguments) and NY (dimension of the function itself). However
when  you  have  NY=1  you  may  find more convenient to  use  idwcalc1(),
idwcalc2() or idwcalc3().

NOTE: this function modifies internal temporaries of the  IDW  model, thus
      IT IS NOT  THREAD-SAFE!  If  you  want  to  perform  parallel  model
      evaluation from the multiple threads, use idwtscalcbuf()  with  per-
      thread buffer object.

INPUT PARAMETERS:
    S       -   IDW model
    X       -   coordinates, array[NX]. X may have more than NX  elements,
                in this case only leading NX will be used.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is out-parameter and will  be
                reallocated after call to this function. In case you  want
                to reuse previously allocated Y, you may use idwcalcbuf(),
                which reallocates Y only when it is too small.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwcalc(idwmodel &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwcalc1` function

/*************************************************************************
IDW interpolation: scalar target, 1-dimensional argument

NOTE: this function modifies internal temporaries of the  IDW  model, thus
      IT IS NOT  THREAD-SAFE!  If  you  want  to  perform  parallel  model
      evaluation from the multiple threads, use idwtscalcbuf()  with  per-
      thread buffer object.

INPUT PARAMETERS:
    S   -   IDW interpolant built with IDW builder
    X0  -   argument value

Result:
    IDW interpolant S(X0)

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
double idwcalc1(idwmodel &s, const double x0, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwcalc2` function

/*************************************************************************
IDW interpolation: scalar target, 2-dimensional argument

NOTE: this function modifies internal temporaries of the  IDW  model, thus
      IT IS NOT  THREAD-SAFE!  If  you  want  to  perform  parallel  model
      evaluation from the multiple threads, use idwtscalcbuf()  with  per-
      thread buffer object.

INPUT PARAMETERS:
    S       -   IDW interpolant built with IDW builder
    X0, X1  -   argument value

Result:
    IDW interpolant S(X0,X1)

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
double idwcalc2(idwmodel &s, const double x0, const double x1, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwcalc3` function

/*************************************************************************
IDW interpolation: scalar target, 3-dimensional argument

NOTE: this function modifies internal temporaries of the  IDW  model, thus
      IT IS NOT  THREAD-SAFE!  If  you  want  to  perform  parallel  model
      evaluation from the multiple threads, use idwtscalcbuf()  with  per-
      thread buffer object.

INPUT PARAMETERS:
    S       -   IDW interpolant built with IDW builder
    X0,X1,X2-   argument value

Result:
    IDW interpolant S(X0,X1,X2)

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
double idwcalc3(idwmodel &s, const double x0, const double x1, const double x2, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwcalcbuf` function

/*************************************************************************
This function calculates values of the IDW model at the given point.

Same as idwcalc(), but does not reallocate Y when in is large enough to
store function values.

NOTE: this function modifies internal temporaries of the  IDW  model, thus
      IT IS NOT  THREAD-SAFE!  If  you  want  to  perform  parallel  model
      evaluation from the multiple threads, use idwtscalcbuf()  with  per-
      thread buffer object.

INPUT PARAMETERS:
    S       -   IDW model
    X       -   coordinates, array[NX]. X may have more than NX  elements,
                in this case only leading NX will be used.
    Y       -   possibly preallocated array

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is not reallocated when it
                is larger than NY.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwcalcbuf(idwmodel &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`idwcreatecalcbuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel  IDW  model  evaluations  (with  one  IDW  model  instance  being
used from multiple threads, as long as  different  threads  use  different
instances of buffer).

This buffer object can be used with  idwtscalcbuf()  function  (here  "ts"
stands for "thread-safe", "buf" is a suffix which denotes  function  which
reuses previously allocated output space).

How to use it:
* create IDW model structure or load it from file
* call idwcreatecalcbuffer(), once per thread working with IDW model  (you
  should call this function only AFTER model initialization, see below for
  more information)
* call idwtscalcbuf() from different threads,  with  each  thread  working
  with its own copy of buffer object.

INPUT PARAMETERS
    S           -   IDW model

OUTPUT PARAMETERS
    Buf         -   external buffer.


IMPORTANT: buffer object should be used only with  IDW model object  which
           was used to initialize buffer. Any attempt to use buffer   with
           different object is dangerous - you may  get  memory  violation
           error because sizes of internal arrays do not fit to dimensions
           of the IDW structure.

IMPORTANT: you  should  call  this function only for model which was built
           with model builder (or unserialized from file). Sizes  of  some
           internal structures are determined only after model  is  built,
           so buffer object created before model construction  stage  will
           be useless (and any attempt to use it will result in exception).

  -- ALGLIB --
     Copyright 22.10.2018 by Sergey Bochkanov
*************************************************************************/
void idwcreatecalcbuffer(const idwmodel &s, idwcalcbuffer &buf, const xparams _xparams = alglib::xdefault);

`idwfit` function

/*************************************************************************
This function fits IDW model to the dataset using current IDW construction
algorithm. A model being built and fitting report are returned.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    State   -   builder object

OUTPUT PARAMETERS:
    Model   -   an IDW model built with current algorithm
    Rep     -   model fitting report, fields of this structure contain
                information about average fitting errors.

NOTE: although IDW-MSTAB algorithm is an  interpolation  method,  i.e.  it
      tries to fit the model exactly, it can  handle  datasets  with  non-
      distinct points which can not be fit exactly; in such  cases  least-
      squares fitting is performed.

  -- ALGLIB --
     Copyright 22.10.2018 by Bochkanov Sergey
*************************************************************************/
void idwfit(idwbuilder &state, idwmodel &model, idwreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`idwgridcalc2v` function

/*************************************************************************
This function calculates values  of  an  IDW  model  at  a  regular  grid,
which  has  N0*N1 points, with Point[I,J] = (X0[I], X1[J]).  Vector-valued
IDW models are supported.

This function returns 0.0 when:
* the model is not initialized
* NX<>2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (MSTAB) IDW's.

INPUT PARAMETERS:
    S       -   IDW model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension, N0>=1

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension, N1>=1

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1], where NY is a  number of
                "output" vector values (this  function   supports  vector-
                valued IDW models). Y is out-variable and  is  reallocated
                by this function.
                Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]), for:
                *  K=0...NY-1
                * I0=0...N0-1
                * I1=0...N1-1

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  idwmodel
      structure in multiple threads calling  this function  for  different
      grids.

NOTE: if you need function values on some subset  of  regular  grid, which
      may be described as "several compact and  dense  islands",  you  may
      use idwgridcalc2vsubset().

  -- ALGLIB --
     Copyright 24.11.2023 by Bochkanov Sergey
*************************************************************************/
void idwgridcalc2v(const idwmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`idwgridcalc2vsubset` function

/*************************************************************************
This function calculates values of an  IDW  model  at  some  subset  of  a
regular grid:
* the grid has N0*N1 points, with Point[I,J] = (X0[I], X1[J])
* only values at some subset of the grid are required
Vector-valued IDW models are supported.

This function returns 0.0 when:
* the model is not initialized
* NX<>2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel processing is implemented only for modern (MSTAB) IDW's.

INPUT PARAMETERS:
    S       -   IDW model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension, N0>=1

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension, N1>=1

    FlagY   -   array[N0*N1]:
                * Y[I0+I1*N0] corresponds to node (X0[I0],X1[I1])
                * it is a "bitmap" array which contains  False  for  nodes
                  which are NOT calculated, and True for nodes  which  are
                  required.

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued IDW models):
                * Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]),
                  for K=0...NY-1, I0=0...N0-1, I1=0...N1-1.
                * elements of Y[] which correspond  to  FlagY[]=True   are
                  loaded by model values (which may be  exactly  zero  for
                  some nodes).
                * elements of Y[] which correspond to FlagY[]=False MAY be
                  initialized by zeros OR may  be  calculated.  Generally,
                  they  are   not   calculated,  but  future  SIMD-capable
                  versions may compute several elements in a batch.

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  idwmodel
      structure in multiple threads calling  this function  for  different
      grids.

  -- ALGLIB --
     Copyright 24.11.2023 by Bochkanov Sergey
*************************************************************************/
void idwgridcalc2vsubset(const idwmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, const boolean_1d_array &flagy, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`idwpeekprogress` function

/*************************************************************************
This function is used to peek into the IDW construction  process from some
other thread and get the progress indicator. It returns value in [0,1].

IMPORTANT: only MSTAB algorithm supports peeking into progress  indicator.
           Legacy versions of the Shepard's method do  not support it. You
           will always get zero as the result.

INPUT PARAMETERS:
    S           -   RBF model object

RESULT:
    progress value, in [0,1]

  -- ALGLIB --
     Copyright 27.11.2023 by Bochkanov Sergey
*************************************************************************/
double idwpeekprogress(const idwbuilder &s, const xparams _xparams = alglib::xdefault);

`idwserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void idwserialize(const idwmodel &obj, std::string &s_out);
void idwserialize(const idwmodel &obj, std::ostream &s_out);

`idwtscalcbuf` function

/*************************************************************************
This function calculates values of the IDW model at the given point, using
external  buffer  object  (internal  temporaries  of  IDW  model  are  not
modified).

This function allows to use same IDW model object  in  different  threads,
assuming  that  different   threads  use different instances of the buffer
structure.

INPUT PARAMETERS:
    S       -   IDW model, may be shared between different threads
    Buf     -   buffer object created for this particular instance of  IDW
                model with idwcreatecalcbuffer().
    X       -   coordinates, array[NX]. X may have more than NX  elements,
                in this case only  leading NX will be used.
    Y       -   possibly preallocated array

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is not reallocated when it
                is larger than NY.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void idwtscalcbuf(const idwmodel &s, idwcalcbuffer &buf, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`idwunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void idwunserialize(const std::string &s_in, idwmodel &obj);
void idwunserialize(const std::istream &s_in, idwmodel &obj);

idw_d_mstab example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example illustrates basic concepts of the IDW models:
        // creation and evaluation.
        // 
        // Suppose that we have set of 2-dimensional points with associated
        // scalar function values, and we want to build an IDW model using
        // our data.
        // 
        // NOTE: we can work with N-dimensional models and vector-valued functions too :)
        // 
        // Typical sequence of steps is given below:
        // 1. we create IDW builder object
        // 2. we attach our dataset to the IDW builder and tune algorithm settings
        // 3. we generate IDW model
        // 4. we use IDW model instance (evaluate, serialize, etc.)
        //
        double v;

        //
        // Step 1: IDW builder creation.
        //
        // We have to specify dimensionality of the space (2 or 3) and
        // dimensionality of the function (scalar or vector).
        //
        // New builder object is empty - it has not dataset and uses
        // default model construction settings
        //
        idwbuilder builder;
        idwbuildercreate(2, 1, builder);

        //
        // Step 2: dataset addition
        //
        // XY contains two points - x0=(-1,0) and x1=(+1,0) -
        // and two function values f(x0)=2, f(x1)=3.
        //
        real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
        idwbuildersetpoints(builder, xy);

        //
        // Step 3: choose IDW algorithm and generate model
        //
        // We use modified stabilized IDW algorithm with following parameters:
        // * SRad - set to 5.0 (search radius must be large enough)
        //
        // IDW-MSTAB algorithm is a state-of-the-art implementation of IDW which
        // is competitive with RBFs and bicubic splines. See comments on the
        // idwbuildersetalgomstab() function for more information.
        //
        idwmodel model;
        idwreport rep;
        idwbuildersetalgomstab(builder, 5.0);
        idwfit(builder, model, rep);

        //
        // Step 4: model was built, evaluate its value
        //
        v = idwcalc2(model, 1.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 3.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

idw_d_serialize example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example shows how to serialize and unserialize IDW model.
        // 
        // Suppose that we have set of 2-dimensional points with associated
        // scalar function values, and we have built an IDW model using
        // our data.
        //
        // This model can be serialized to string or stream. ALGLIB supports
        // flexible (un)serialization, i.e. you can move serialized model
        // representation between different machines (32-bit or 64-bit),
        // different CPU architectures (x86/64, ARM) or even different
        // programming languages supported by ALGLIB (C#, C++, ...).
        //
        // Our first step is to build model, evaluate it at point (1,0),
        // and serialize it to string.
        //
        std::string s;
        double v;
        real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
        idwbuilder builder;
        idwmodel model;
        idwmodel model2;
        idwreport rep;
        idwbuildercreate(2, 1, builder);
        idwbuildersetpoints(builder, xy);
        idwbuildersetalgomstab(builder, 5.0);
        idwfit(builder, model, rep);
        v = idwcalc2(model, 1.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 3.000

        //
        // Serialization + unserialization to a different instance
        // of the model class.
        //
        alglib::idwserialize(model, s);
        alglib::idwunserialize(s, model2);

        //
        // Evaluate unserialized model at the same point
        //
        v = idwcalc2(model2, 1.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 3.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`igammaf` subpackage

Functions

incompletegamma
incompletegammac
invincompletegammac

Examples

`incompletegamma` function

/*************************************************************************
Incomplete gamma integral

The function is defined by

                          x
                           -
                  1       | |  -t  a-1
 igam(a,x)  =   -----     |   e   t   dt.
                 -      | |
                | (a)    -
                          0


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,30       200000       3.6e-14     2.9e-15
   IEEE      0,100      300000       9.9e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegamma(const double a, const double x, const xparams _xparams = alglib::xdefault);

`incompletegammac` function

/*************************************************************************
Complemented incomplete gamma integral

The function is defined by


 igamc(a,x)   =   1 - igam(a,x)

                           inf.
                             -
                    1       | |  -t  a-1
              =   -----     |   e   t   dt.
                   -      | |
                  | (a)    -
                            x


In this implementation both arguments must be positive.
The integral is evaluated by either a power series or
continued fraction expansion, depending on the relative
values of a and x.

ACCURACY:

Tested at random a, x.
               a         x                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,100      200000       1.9e-14     1.7e-15
   IEEE     0.01,0.5  0,100      200000       1.4e-13     1.6e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1985, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
double incompletegammac(const double a, const double x, const xparams _xparams = alglib::xdefault);

`invincompletegammac` function

/*************************************************************************
Inverse of complemented imcomplete gamma integral

Given p, the function finds x such that

 igamc( a, x ) = p.

Starting with the approximate value

        3
 x = a t

 where

 t = 1 - d - ndtri(p) sqrt(d)

and

 d = 1/9a,

the routine performs up to 10 Newton iterations to find the
root of igamc(a,x) - p = 0.

ACCURACY:

Tested at random a, p in the intervals indicated.

               a        p                      Relative error:
arithmetic   domain   domain     # trials      peak         rms
   IEEE     0.5,100   0,0.5       100000       1.0e-14     1.7e-15
   IEEE     0.01,0.5  0,0.5       100000       9.0e-14     3.4e-15
   IEEE    0.5,10000  0,0.5        20000       2.3e-13     3.8e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invincompletegammac(const double a, const double y0, const xparams _xparams = alglib::xdefault);

`intcomp` subpackage

Functions

nsfitspheremcc
nsfitspheremic
nsfitspheremzc
nsfitspherex
spline1dfitcubic
spline1dfithermite
spline1dfitpenalized
spline1dfitpenalizedw

Examples

`nsfitspheremcc` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspheremc() instead.


  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void nsfitspheremcc(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &rhi, const xparams _xparams = alglib::xdefault);

`nsfitspheremic` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspheremi() instead.

  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void nsfitspheremic(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &rlo, const xparams _xparams = alglib::xdefault);

`nsfitspheremzc` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspheremz() instead.

  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void nsfitspheremzc(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, real_1d_array &cx, double &rlo, double &rhi, const xparams _xparams = alglib::xdefault);

`nsfitspherex` function

/*************************************************************************
This function is left for backward compatibility.
Use fitspherex() instead.

  -- ALGLIB --
     Copyright 14.04.2017 by Bochkanov Sergey
*************************************************************************/
void nsfitspherex(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nx, const ae_int_t problemtype, const double epsx, const ae_int_t aulits, const double penalty, real_1d_array &cx, double &rlo, double &rhi, const xparams _xparams = alglib::xdefault);

`spline1dfitcubic` function

/*************************************************************************
Deprecated fitting function with O(N*M^2+M^3) running time. Superseded  by
spline1dfit().

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfitcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

`spline1dfithermite` function

/*************************************************************************
Deprecated fitting function with O(N*M^2+M^3) running time. Superseded  by
spline1dfit().

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermite(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfithermite(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

`spline1dfitpenalized` function

/*************************************************************************
This function is an obsolete and deprecated version of fitting by
penalized cubic spline.

It was superseded by spline1dfit(), which is an orders of magnitude faster
and more memory-efficient implementation.

Do NOT use this function in the new code!

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalized(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfitpenalized(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

`spline1dfitpenalizedw` function

/*************************************************************************
This function is an obsolete and deprecated version of fitting by
penalized cubic spline.

It was superseded by spline1dfit(), which is an orders of magnitude faster
and more memory-efficient implementation.

Do NOT use this function in the new code!

  -- ALGLIB PROJECT --
     Copyright 19.10.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dfitpenalizedw(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfitpenalizedw(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t m, const double rho, ae_int_t &info, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

`inverseupdate` subpackage

Functions

rmatrixinvupdatecolumn
rmatrixinvupdaterow
rmatrixinvupdatesimple
rmatrixinvupdateuv

Examples

`rmatrixinvupdatecolumn` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm updates matrix A^-1 when adding a vector to a column
of matrix A.

Input parameters:
    InvA        -   inverse of matrix A.
                    Array whose indexes range within [0..N-1, 0..N-1].
    N           -   size of matrix A.
    UpdColumn   -   the column of A whose vector U was added.
                    0 <= UpdColumn <= N-1
    U           -   the vector to be added to a column.
                    Array whose index ranges within [0..N-1].

Output parameters:
    InvA        -   inverse of modified matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdatecolumn(real_2d_array &inva, const ae_int_t n, const ae_int_t updcolumn, const real_1d_array &u, const xparams _xparams = alglib::xdefault);

`rmatrixinvupdaterow` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm updates matrix A^-1 when adding a vector to a row
of matrix A.

Input parameters:
    InvA    -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    UpdRow  -   the row of A whose vector V was added.
                0 <= Row <= N-1
    V       -   the vector to be added to a row.
                Array whose index ranges within [0..N-1].

Output parameters:
    InvA    -   inverse of modified matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdaterow(real_2d_array &inva, const ae_int_t n, const ae_int_t updrow, const real_1d_array &v, const xparams _xparams = alglib::xdefault);

`rmatrixinvupdatesimple` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm updates matrix A^-1 when adding a number to an element
of matrix A.

Input parameters:
    InvA    -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    UpdRow  -   row where the element to be updated is stored.
    UpdColumn - column where the element to be updated is stored.
    UpdVal  -   a number to be added to the element.


Output parameters:
    InvA    -   inverse of modified matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdatesimple(real_2d_array &inva, const ae_int_t n, const ae_int_t updrow, const ae_int_t updcolumn, const double updval, const xparams _xparams = alglib::xdefault);

`rmatrixinvupdateuv` function

/*************************************************************************
Inverse matrix update by the Sherman-Morrison formula

The algorithm computes the inverse of matrix A+u*v' by using the given matrix
A^-1 and the vectors u and v.

Input parameters:
    InvA    -   inverse of matrix A.
                Array whose indexes range within [0..N-1, 0..N-1].
    N       -   size of matrix A.
    U       -   the vector modifying the matrix.
                Array whose index ranges within [0..N-1].
    V       -   the vector modifying the matrix.
                Array whose index ranges within [0..N-1].

Output parameters:
    InvA - inverse of matrix A + u*v'.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void rmatrixinvupdateuv(real_2d_array &inva, const ae_int_t n, const real_1d_array &u, const real_1d_array &v, const xparams _xparams = alglib::xdefault);

`iterativesparse` subpackage

Classes

sparsesolverreport
sparsesolverstate

Functions

sparsesolvegmres
sparsesolvercreate
sparsesolverooccontinue
sparsesolveroocgetrequestdata
sparsesolveroocgetrequestdata1
sparsesolveroocgetrequestinfo
sparsesolveroocsendresult
sparsesolveroocstart
sparsesolveroocstop
sparsesolverrequesttermination
sparsesolverresults
sparsesolversetalgogmres
sparsesolversetcond
sparsesolversetstartingpoint
sparsesolversetxrep
sparsesolversolve
sparsesolversolvesymmetric
sparsesolvesymmetricgmres

Examples

`sparsesolverreport` class

/*************************************************************************
This structure is a sparse solver report (both direct and iterative solvers
use this structure).

Following fields can be accessed by users:
* TerminationType (specific error codes depend on the solver  being  used,
  with positive values ALWAYS signaling  that something useful is returned
  in X, and negative values ALWAYS meaning critical failures.
* NMV - number of matrix-vector products performed (0 for direct solvers)
* IterationsCount - inner iterations count (0 for direct solvers)
* R2 - squared residual
*************************************************************************/
class sparsesolverreport
{
public:
    sparsesolverreport();
    sparsesolverreport(const sparsesolverreport &rhs);
    sparsesolverreport& operator=(const sparsesolverreport &rhs);
    virtual ~sparsesolverreport();
    ae_int_t terminationtype;
    ae_int_t nmv;
    ae_int_t iterationscount;
    double r2;
};

`sparsesolverstate` class

/*************************************************************************
This object stores state of the sparse linear solver object.

You should use ALGLIB functions to work with this object.
Never try to access its fields directly!
*************************************************************************/
class sparsesolverstate
{
public:
    sparsesolverstate();
    sparsesolverstate(const sparsesolverstate &rhs);
    sparsesolverstate& operator=(const sparsesolverstate &rhs);
    virtual ~sparsesolverstate();
};

`sparsesolvegmres` function

/*************************************************************************
Solving sparse linear system A*x=b using GMRES(k) method.

This function provides convenience API for an 'expert' interface  provided
by SparseSolverState class. Use SparseSolver  API  if  you  need  advanced
functions like providing initial point, using out-of-core API and so on.

INPUT PARAMETERS:
    A       -   sparse NxN matrix in any sparse storage format. Using  CRS
                format   is   recommended   because   it  avoids  internal
                conversion.
                An exception will be generated if  A  is  not  NxN  matrix
                (where  N  is  a  size   specified  during  solver  object
                creation).
    B       -   right part, array[N]
    K       -   k parameter for  GMRES(k), k>=0.  Zero  value  means  that
                algorithm will choose it automatically.
    EpsF    -   stopping condition, EpsF>=0. The algorithm will stop  when
                residual will decrease below EpsF*|B|. Having EpsF=0 means
                that this stopping condition is ignored.
    MaxIts  -   stopping condition, MaxIts>=0.  The  algorithm  will  stop
                after performing MaxIts iterations. Zero  value  means  no
                limit.

NOTE: having both EpsF=0 and MaxIts=0 means that stopping criteria will be
      chosen automatically.

OUTPUT PARAMETERS:
    X       -   array[N], the solution
    Rep     -   solution report:
                * Rep.TerminationType completion code:
                    * -5    CG method was used for a matrix which  is  not
                            positive definite
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                    *  8    the  algorithm  was  terminated   early  with
                            SparseSolverRequestTermination() being called
                            from other thread.
                * Rep.IterationsCount contains iterations count
                * Rep.NMV contains number of matrix-vector calculations
                * Rep.R2 contains squared residual

  -- ALGLIB --
     Copyright 25.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolvegmres(const sparsematrix &a, const real_1d_array &b, const ae_int_t k, const double epsf, const ae_int_t maxits, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

`sparsesolvercreate` function

/*************************************************************************
This function initializes sparse linear iterative solver object.

This solver can be used  to  solve  nonsymmetric  and  symmetric  positive
definite NxN (square) linear systems.

The solver provides  'expert'  API  which  allows  advanced  control  over
algorithms being used, including ability to get progress report, terminate
long-running solver from other thread, out-of-core solution and so on.

NOTE: there are also convenience  functions  that  allows  quick  one-line
      access to the solvers:
      * SparseSolveCG() to solve SPD linear systems
      * SparseSolveGMRES() to solve unsymmetric linear systems.

NOTE: if you want to solve MxN (rectangular) linear problem  you  may  use
      LinLSQR solver provided by ALGLIB.

USAGE (A is given by the SparseMatrix structure):

    1. User initializes algorithm state with SparseSolverCreate() call
    2. User  selects   algorithm  with one of the SparseSolverSetAlgo???()
       functions. By default, GMRES(k) is used with automatically chosen k
    3. Optionally, user tunes solver parameters, sets starting point, etc.
    4. Depending on whether system is symmetric or not, user calls:
       * SparseSolverSolveSymmetric() for a  symmetric system given by its
         lower or upper triangle
       * SparseSolverSolve() for a nonsymmetric system or a symmetric  one
         given by the full matrix
    5. User calls SparseSolverResults() to get the solution

    It is possible to call SparseSolverSolve???() again to  solve  another
    task with same dimensionality but different matrix and/or  right  part
    without reinitializing SparseSolverState structure.

USAGE (out-of-core mode):

    1. User initializes algorithm state with SparseSolverCreate() call
    2. User  selects   algorithm  with one of the SparseSolverSetAlgo???()
       functions. By default, GMRES(k) is used with automatically chosen k
    3. Optionally, user tunes solver parameters, sets starting point, etc.
    4. After that user should work with out-of-core interface  in  a  loop
       like one given below:

        > alglib.sparsesolveroocstart(state)
        > while alglib.sparsesolverooccontinue(state) do
        >     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
        >     alglib.sparsesolveroocgetrequestdata(state, out X)
        >     if RequestType=0 then
        >         [calculate  Y=A*X, with X=R^N]
        >     alglib.sparsesolveroocsendresult(state, in Y)
        > alglib.sparsesolveroocstop(state, out X, out Report)

INPUT PARAMETERS:
    N       -   problem dimensionality (fixed at start-up)

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolvercreate(const ae_int_t n, sparsesolverstate &state, const xparams _xparams = alglib::xdefault);

`sparsesolverooccontinue` function

/*************************************************************************
This function performs iterative solution of  the  linear  system  in  the
out-of-core mode. It should be used in conjunction with other out-of-core-
related functions of this subspackage in a loop like one given below:

> alglib.sparsesolveroocstart(state)
> while alglib.sparsesolverooccontinue(state) do
>     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
>     alglib.sparsesolveroocgetrequestdata(state, out X)
>     if RequestType=0 then
>         [calculate  Y=A*X, with X=R^N]
>     alglib.sparsesolveroocsendresult(state, in Y)
> alglib.sparsesolveroocstop(state, out X, out Report)

  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
bool sparsesolverooccontinue(sparsesolverstate &state, const xparams _xparams = alglib::xdefault);

`sparsesolveroocgetrequestdata` function

/*************************************************************************
This function is used  to  retrieve  vector  associated  with  out-of-core
request sent by the solver to user code. Depending  on  the  request  type
(returned by the SparseSolverOOCGetRequestInfo()) this  vector  should  be
multiplied by A or subjected to another processing.

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like one given below:

> alglib.sparsesolveroocstart(state)
> while alglib.sparsesolverooccontinue(state) do
>     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
>     alglib.sparsesolveroocgetrequestdata(state, out X)
>     if RequestType=0 then
>         [calculate  Y=A*X, with X=R^N]
>     alglib.sparsesolveroocsendresult(state, in Y)
> alglib.sparsesolveroocstop(state, out X, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    X               -   possibly  preallocated   storage;  reallocated  if
                        needed, left unchanged, if large enough  to  store
                        request data.

OUTPUT PARAMETERS:
    X               -   array[N] or larger, leading N elements are  filled
                        with vector X.


  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolveroocgetrequestdata(sparsesolverstate &state, real_1d_array &x, const xparams _xparams = alglib::xdefault);

`sparsesolveroocgetrequestdata1` function

/*************************************************************************
This function is used to retrieve scalar value associated with out-of-core
request sent by the solver to user code. In  the  current  ALGLIB  version
this function is used to retrieve squared residual  norm  during  progress
reports.

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode

OUTPUT PARAMETERS:
    V               -   scalar value associated with the current request


  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolveroocgetrequestdata1(sparsesolverstate &state, double &v, const xparams _xparams = alglib::xdefault);

`sparsesolveroocgetrequestinfo` function

/*************************************************************************
This function is used to retrieve information  about  out-of-core  request
sent by the solver:
* RequestType=0  means that matrix-vector products A*x is requested
* RequestType=-1 means that solver reports its progress; this  request  is
  returned only when reports are activated wit SparseSolverSetXRep().

This function returns just request type; in order  to  get contents of the
trial vector, use sparsesolveroocgetrequestdata().

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like one given below:

> alglib.sparsesolveroocstart(state)
> while alglib.sparsesolverooccontinue(state) do
>     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
>     alglib.sparsesolveroocgetrequestdata(state, out X)
>     if RequestType=0 then
>         [calculate  Y=A*X, with X=R^N]
>     alglib.sparsesolveroocsendresult(state, in Y)
> alglib.sparsesolveroocstop(state, out X, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode

OUTPUT PARAMETERS:
    RequestType     -   type of the request to process:
                        * 0   for matrix-vector product A*x, with A  being
                          NxN system matrix  and X being N-dimensional
                          vector
                        *-1   for location and residual report


  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolveroocgetrequestinfo(sparsesolverstate &state, ae_int_t &requesttype, const xparams _xparams = alglib::xdefault);

`sparsesolveroocsendresult` function

/*************************************************************************
This function is used to send user reply to out-of-core  request  sent  by
the solver. Usually it is product A*x for vector X returned by the solver.

It should be used in conjunction with other out-of-core-related  functions
of this subspackage in a loop like one given below:

> alglib.sparsesolveroocstart(state)
> while alglib.sparsesolverooccontinue(state) do
>     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
>     alglib.sparsesolveroocgetrequestdata(state, out X)
>     if RequestType=0 then
>         [calculate  Y=A*X, with X=R^N]
>     alglib.sparsesolveroocsendresult(state, in Y)
> alglib.sparsesolveroocstop(state, out X, out Report)

INPUT PARAMETERS:
    State           -   solver running in out-of-core mode
    AX              -   array[N] or larger, leading N elements contain A*x


  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolveroocsendresult(sparsesolverstate &state, const real_1d_array &ax, const xparams _xparams = alglib::xdefault);

`sparsesolveroocstart` function

/*************************************************************************
This function initiates out-of-core mode of the sparse solver.  It  should
be used in conjunction with other out-of-core-related  functions  of  this
subspackage in a loop like one given below:

> alglib.sparsesolveroocstart(state)
> while alglib.sparsesolverooccontinue(state) do
>     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
>     alglib.sparsesolveroocgetrequestdata(state, out X)
>     if RequestType=0 then
>         [calculate  Y=A*X, with X=R^N]
>     alglib.sparsesolveroocsendresult(state, in Y)
> alglib.sparsesolveroocstop(state, out X, out Report)

INPUT PARAMETERS:
    State       -   solver object

  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolveroocstart(sparsesolverstate &state, const real_1d_array &b, const xparams _xparams = alglib::xdefault);

`sparsesolveroocstop` function

/*************************************************************************
This  function  finalizes out-of-core mode of the linear solver. It should
be used in conjunction with other out-of-core-related  functions  of  this
subspackage in a loop like one given below:

> alglib.sparsesolveroocstart(state)
> while alglib.sparsesolverooccontinue(state) do
>     alglib.sparsesolveroocgetrequestinfo(state, out RequestType)
>     alglib.sparsesolveroocgetrequestdata(state, out X)
>     if RequestType=0 then
>         [calculate  Y=A*X, with X=R^N]
>     alglib.sparsesolveroocsendresult(state, in Y)
> alglib.sparsesolveroocstop(state, out X, out Report)

INPUT PARAMETERS:
    State       -   solver state

OUTPUT PARAMETERS:
    X       -   array[N], the solution.
                Zero-filled on the failure (Rep.TerminationType<0).
    Rep     -   report with additional info:
                * Rep.TerminationType completion code:
                    * -5    CG method was used for a matrix which  is  not
                            positive definite
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                    *  8    the  algorithm  was  terminated   early  with
                            SparseSolverRequestTermination() being called
                            from other thread.
                * Rep.IterationsCount contains iterations count
                * Rep.NMV contains number of matrix-vector calculations
                * Rep.R2 contains squared residual

  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolveroocstop(sparsesolverstate &state, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

`sparsesolverrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of the running solver.  It
can be called from some other thread which wants the   solver to terminate
or when processing an out-of-core request.

As result, solver  stops  at  point  which  was  "current  accepted"  when
the termination request was submitted and returns error code 8 (successful
termination).  Such   termination   is  a smooth  process  which  properly
deallocates all temporaries.

INPUT PARAMETERS:
    State   -   solver structure

NOTE: calling this function on solver which is NOT running  will  have  no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

NOTE: solver clears termination flag on its start, it means that  if  some
      other thread will request termination too soon, its request will went
      unnoticed.

  -- ALGLIB --
     Copyright 01.10.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolverrequesttermination(sparsesolverstate &state, const xparams _xparams = alglib::xdefault);

`sparsesolverresults` function

/*************************************************************************
Sparse solver results.

This function must be called after calling one of the SparseSolverSolve()
functions.

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   solution report:
                * Rep.TerminationType completion code:
                    * -5    CG method was used for a matrix which  is  not
                            positive definite
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                    *  8    the  algorithm  was  terminated   early  with
                            SparseSolverRequestTermination() being called
                            from other thread.
                * Rep.IterationsCount contains iterations count
                * Rep.NMV contains number of matrix-vector calculations
                * Rep.R2 contains squared residual
s
  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesolverresults(sparsesolverstate &state, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

`sparsesolversetalgogmres` function

/*************************************************************************
This function sets the solver algorithm to GMRES(k).

NOTE: if you do not need advanced functionality of the  SparseSolver  API,
      you   may   use   convenience   functions   SparseSolveGMRES()   and
      SparseSolveSymmetricGMRES().

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    K       -   GMRES parameter, K>=0:
                * recommended values are in 10..100 range
                * larger values up to N are possible but have little sense
                  - the algorithm will be slower than any dense solver.
                * values above N are truncated down to N
                * zero value means that  default  value  is  chosen.  This
                  value is 50 in the current version, but  it  may  change
                  in future ALGLIB releases.

  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolversetalgogmres(sparsesolverstate &state, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`sparsesolversetcond` function

/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsF    -   algorithm will be stopped if norm of residual is less than
                EpsF*||b||.
    MaxIts  -   algorithm will be stopped if number of iterations is  more
                than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
If  both  EpsF  and  MaxIts  are  zero then small EpsF will be set to small
value.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesolversetcond(sparsesolverstate &state, const double epsf, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`sparsesolversetstartingpoint` function

/*************************************************************************
This function sets starting point.
By default, zero starting point is used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    X       -   starting point, array[N]

OUTPUT PARAMETERS:
    State   -   new starting point was set

  -- ALGLIB --
     Copyright 24.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolversetstartingpoint(sparsesolverstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`sparsesolversetxrep` function

/*************************************************************************
This function turns on/off reporting during out-of-core processing.

When the solver works in the out-of-core mode, it  can  be  configured  to
report its progress by returning current location. These location  reports
are implemented as a special kind of the out-of-core request:
* SparseSolverOOCGetRequestInfo() returns -1
* SparseSolverOOCGetRequestData() returns current location
* SparseSolverOOCGetRequestData1() returns squared norm of the residual
* SparseSolverOOCSendResult() shall NOT be called

This function has no effect when SparseSolverSolve() is used because  this
function has no method of reporting its progress.

NOTE: when used with GMRES(k), this function reports progress  every  k-th
      iteration.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

  -- ALGLIB --
     Copyright 01.10.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolversetxrep(sparsesolverstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`sparsesolversolve` function

/*************************************************************************
Procedure for the solution of A*x=b with sparse nonsymmetric A

IMPORTANT: this function will work with any solver algorithm  being  used,
           symmetric solver like CG,  or  not.  However,  using  symmetric
           solvers on nonsymmetric problems is  dangerous.  It  may  solve
           the problem up  to  desired  precision  (sometimes,  rarely) or
           terminate with error code signalling  violation  of  underlying
           assumptions.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse NxN matrix in any sparse storage  format.
                Using CRS format is recommended because it avoids internal
                conversion.
                An exception will be generated if  A  is  not  NxN  matrix
                (where  N  is  a  size   specified  during  solver  object
                creation).
    B       -   right part, array[N]

RESULT:
    This function returns no result.
    You can get the solution by calling SparseSolverResults()

  -- ALGLIB --
     Copyright 25.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolversolve(sparsesolverstate &state, const sparsematrix &a, const real_1d_array &b, const xparams _xparams = alglib::xdefault);

`sparsesolversolvesymmetric` function

/*************************************************************************
Procedure for  the  solution of A*x=b with sparse symmetric A given by its
lower or upper triangle.

This function will work with any solver algorithm  being   used,  SPD  one
(like CG) or not (like GMRES). Using unsymmetric solvers (like  GMRES)  on
SPD problems is suboptimal, but still possible.

NOTE: the  solver  behavior is ill-defined  for  a  situation  when a  SPD
      solver is used on indefinite matrix. It  may solve the problem up to
      desired precision (sometimes, rarely)  or  return  with  error  code
      signalling violation of underlying assumptions.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse symmetric NxN matrix in any sparse storage  format.
                Using CRS format is recommended because it avoids internal
                conversion.
                An exception will be generated if  A  is  not  NxN  matrix
                (where  N  is  a  size   specified  during  solver  object
                creation).
    IsUpper -   whether upper or lower triangle of A is used:
                * IsUpper=True  => only upper triangle is used and lower
                                   triangle is not referenced at all
                * IsUpper=False => only lower triangle is used and upper
                                   triangle is not referenced at all
    B       -   right part, array[N]

RESULT:
    This function returns no result.
    You can get the solution by calling SparseSolverResults()

  -- ALGLIB --
     Copyright 25.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolversolvesymmetric(sparsesolverstate &state, const sparsematrix &a, const bool isupper, const real_1d_array &b, const xparams _xparams = alglib::xdefault);

`sparsesolvesymmetricgmres` function

/*************************************************************************
Solving sparse symmetric linear system A*x=b using GMRES(k) method. Sparse
symmetric A is given by its lower or upper triangle.

NOTE: use SparseSolveGMRES() to solve system with nonsymmetric A.

This function provides convenience API for an 'expert' interface  provided
by SparseSolverState class. Use SparseSolver  API  if  you  need  advanced
functions like providing initial point, using out-of-core API and so on.

INPUT PARAMETERS:
    A       -   sparse symmetric NxN matrix in any sparse storage  format.
                Using CRS format is recommended because it avoids internal
                conversion.
                An exception will be generated if  A  is  not  NxN  matrix
                (where  N  is  a  size   specified  during  solver  object
                creation).
    IsUpper -   whether upper or lower triangle of A is used:
                * IsUpper=True  => only upper triangle is used and lower
                                   triangle is not referenced at all
                * IsUpper=False => only lower triangle is used and upper
                                   triangle is not referenced at all
    B       -   right part, array[N]
    K       -   k parameter for  GMRES(k), k>=0.  Zero  value  means  that
                algorithm will choose it automatically.
    EpsF    -   stopping condition, EpsF>=0. The algorithm will stop  when
                residual will decrease below EpsF*|B|. Having EpsF=0 means
                that this stopping condition is ignored.
    MaxIts  -   stopping condition, MaxIts>=0.  The  algorithm  will  stop
                after performing MaxIts iterations. Zero  value  means  no
                limit.

NOTE: having both EpsF=0 and MaxIts=0 means that stopping criteria will be
      chosen automatically.

OUTPUT PARAMETERS:
    X       -   array[N], the solution
    Rep     -   solution report:
                * Rep.TerminationType completion code:
                    * -5    CG method was used for a matrix which  is  not
                            positive definite
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                    *  8    the  algorithm  was  terminated   early  with
                            SparseSolverRequestTermination() being called
                            from other thread.
                * Rep.IterationsCount contains iterations count
                * Rep.NMV contains number of matrix-vector calculations
                * Rep.R2 contains squared residual

  -- ALGLIB --
     Copyright 25.09.2021 by Bochkanov Sergey
*************************************************************************/
void sparsesolvesymmetricgmres(const sparsematrix &a, const bool isupper, const real_1d_array &b, const ae_int_t k, const double epsf, const ae_int_t maxits, real_1d_array &x, sparsesolverreport &rep, const xparams _xparams = alglib::xdefault);

`jacobianelliptic` subpackage

Functions

jacobianellipticfunctions

Examples

`jacobianellipticfunctions` function

/*************************************************************************
Jacobian Elliptic Functions

Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m),
and dn(u|m) of parameter m between 0 and 1, and real
argument u.

These functions are periodic, with quarter-period on the
real axis equal to the complete elliptic integral
ellpk(1.0-m).

Relation to incomplete elliptic integral:
If u = ellik(phi,m), then sn(u|m) = sin(phi),
and cn(u|m) = cos(phi).  Phi is called the amplitude of u.

Computation is by means of the arithmetic-geometric mean
algorithm, except when m is within 1e-9 of 0 or 1.  In the
latter case with m close to 1, the approximation applies
only for phi < pi/2.

ACCURACY:

Tested at random points with u between 0 and 10, m between
0 and 1.

           Absolute error (* = relative error):
arithmetic   function   # trials      peak         rms
   IEEE      phi         10000       9.2e-16*    1.4e-16*
   IEEE      sn          50000       4.1e-15     4.6e-16
   IEEE      cn          40000       3.6e-15     4.4e-16
   IEEE      dn          10000       1.3e-12     1.8e-14

 Peak error observed in consistency check using addition
theorem for sn(u+v) was 4e-16 (absolute).  Also tested by
the above relation to the incomplete elliptic integral.
Accuracy deteriorates when u is large.

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void jacobianellipticfunctions(const double u, const double m, double &sn, double &cn, double &dn, double &ph, const xparams _xparams = alglib::xdefault);

`jarquebera` subpackage

Functions

jarqueberatest

Examples

`jarqueberatest` function

/*************************************************************************
Jarque-Bera test

This test checks hypotheses about the fact that a  given  sample  X  is  a
sample of normal random variable.

Requirements:
    * the number of elements in the sample is not less than 5.

Input parameters:
    X   -   sample. Array whose index goes from 0 to N-1.
    N   -   size of the sample. N>=5

Output parameters:
    P           -   p-value for the test

Accuracy of the approximation used (5<=N<=1951):

p-value  	    relative error (5<=N<=1951)
[1, 0.1]            < 1%
[0.1, 0.01]         < 2%
[0.01, 0.001]       < 6%
[0.001, 0]          wasn't measured

For N>1951 accuracy wasn't measured but it shouldn't be sharply  different
from table values.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void jarqueberatest(const real_1d_array &x, const ae_int_t n, double &p, const xparams _xparams = alglib::xdefault);

`knn` subpackage

Classes

knnbuffer
knnbuilder
knnmodel
knnreport

Functions

knnallerrors
knnavgce
knnavgerror
knnavgrelerror
knnbuilderbuildknnmodel
knnbuildercreate
knnbuildersetdatasetcls
knnbuildersetdatasetreg
knnbuildersetnorm
knnclassify
knncreatebuffer
knnprocess
knnprocess0
knnprocessi
knnrelclserror
knnrewritekeps
knnrmserror
knnserialize
knntsprocess
knnunserialize

Examples

knn_cls		Simple classification with KNN model
knn_reg		Simple classification with KNN model

`knnbuffer` class

/*************************************************************************
Buffer object which is used to perform  various  requests  (usually  model
inference) in the multithreaded mode (multiple threads working  with  same
KNN object).

This object should be created with KNNCreateBuffer().
*************************************************************************/
class knnbuffer
{
public:
    knnbuffer();
    knnbuffer(const knnbuffer &rhs);
    knnbuffer& operator=(const knnbuffer &rhs);
    virtual ~knnbuffer();
};

`knnbuilder` class

/*************************************************************************
A KNN builder object; this object encapsulates  dataset  and  all  related
settings, it is used to create an actual instance of KNN model.
*************************************************************************/
class knnbuilder
{
public:
    knnbuilder();
    knnbuilder(const knnbuilder &rhs);
    knnbuilder& operator=(const knnbuilder &rhs);
    virtual ~knnbuilder();
};

`knnmodel` class

/*************************************************************************
KNN model, can be used for classification or regression
*************************************************************************/
class knnmodel
{
public:
    knnmodel();
    knnmodel(const knnmodel &rhs);
    knnmodel& operator=(const knnmodel &rhs);
    virtual ~knnmodel();
};

`knnreport` class

/*************************************************************************
KNN training report.

Following fields store training set errors:
* relclserror       -   fraction of misclassified cases, [0,1]
* avgce             -   average cross-entropy in bits per symbol
* rmserror          -   root-mean-square error
* avgerror          -   average error
* avgrelerror       -   average relative error

For classification problems:
* RMS, AVG and AVGREL errors are calculated for posterior probabilities

For regression problems:
* RELCLS and AVGCE errors are zero
*************************************************************************/
class knnreport
{
public:
    knnreport();
    knnreport(const knnreport &rhs);
    knnreport& operator=(const knnreport &rhs);
    virtual ~knnreport();
    double relclserror;
    double avgce;
    double rmserror;
    double avgerror;
    double avgrelerror;
};

`knnallerrors` function

/*************************************************************************
Calculates all kinds of errors for the model in one call.

INPUT PARAMETERS:
    Model   -   KNN model
    XY      -   test set:
                * one row per point
                * first NVars columns store independent variables
                * depending on problem type:
                  * next column stores class number in [0,NClasses) -  for
                    classification problems
                  * next NOut columns  store  dependent  variables  -  for
                    regression problems
    NPoints -   test set size, NPoints>=0

OUTPUT PARAMETERS:
    Rep     -   following fields are loaded with errors for both regression
                and classification models:
                * rep.rmserror - RMS error for the output
                * rep.avgerror - average error
                * rep.avgrelerror - average relative error
                following fields are set only  for classification  models,
                zero for regression ones:
                * relclserror   - relative classification error, in [0,1]
                * avgce - average cross-entropy in bits per dataset entry

NOTE: the cross-entropy metric is too unstable when used to  evaluate  KNN
      models (such models can report exactly  zero probabilities),  so  we
      do not recommend using it.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnallerrors(const knnmodel &model, const real_2d_array &xy, const ae_int_t npoints, knnreport &rep, const xparams _xparams = alglib::xdefault);

`knnavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    Model   -   KNN model
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/NPoints.
    Zero if model solves regression task.

NOTE: the cross-entropy metric is too unstable when used to  evaluate  KNN
      models (such models can report exactly  zero probabilities),  so  we
      do not recommend using it.

NOTE: if  you  need several different kinds of error metrics, it is better
      to use knnallerrors() which computes all error metric  with just one
      pass over dataset.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double knnavgce(const knnmodel &model, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`knnavgerror` function

/*************************************************************************
Average error on the test set

Its meaning for regression task is obvious. As for classification problems,
average error means error when estimating posterior probabilities.

INPUT PARAMETERS:
    Model   -   KNN model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average error

NOTE: if  you  need several different kinds of error metrics, it is better
      to use knnallerrors() which computes all error metric  with just one
      pass over dataset.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double knnavgerror(const knnmodel &model, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`knnavgrelerror` function

/*************************************************************************
Average relative error on the test set

Its meaning for regression task is obvious. As for classification problems,
average relative error means error when estimating posterior probabilities.

INPUT PARAMETERS:
    Model   -   KNN model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average relative error

NOTE: if  you  need several different kinds of error metrics, it is better
      to use knnallerrors() which computes all error metric  with just one
      pass over dataset.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double knnavgrelerror(const knnmodel &model, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`knnbuilderbuildknnmodel` function

/*************************************************************************
This subroutine builds KNN model  according  to  current  settings,  using
dataset internally stored in the builder object.

The model being built performs inference using Eps-approximate  K  nearest
neighbors search algorithm, with:
* K=1,  Eps=0 corresponding to the "nearest neighbor algorithm"
* K>1,  Eps=0 corresponding to the "K nearest neighbors algorithm"
* K>=1, Eps>0 corresponding to "approximate nearest neighbors algorithm"

An approximate KNN is a good option for high-dimensional  datasets  (exact
KNN works slowly when dimensions count grows).

An ALGLIB implementation of kd-trees is used to perform k-nn searches.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S       -   KNN builder object
    K       -   number of neighbors to search for, K>=1
    Eps     -   approximation factor:
                * Eps=0 means that exact kNN search is performed
                * Eps>0 means that (1+Eps)-approximate search is performed

OUTPUT PARAMETERS:
    Model       -   KNN model
    Rep         -   report

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnbuilderbuildknnmodel(knnbuilder &s, const ae_int_t k, const double eps, knnmodel &model, knnreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`knnbuildercreate` function

/*************************************************************************
This subroutine creates KNNBuilder object which is used to train KNN models.

By default, new builder stores empty dataset and some  reasonable  default
settings. At the very least, you should specify dataset prior to  building
KNN model. You can also tweak settings of the model construction algorithm
(recommended, although default settings should work well).

Following actions are mandatory:
* calling knnbuildersetdataset() to specify dataset
* calling knnbuilderbuildknnmodel() to build KNN model using current
  dataset and default settings

Additionally, you may call:
* knnbuildersetnorm() to change norm being used

INPUT PARAMETERS:
    none

OUTPUT PARAMETERS:
    S           -   KNN builder

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnbuildercreate(knnbuilder &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`knnbuildersetdatasetcls` function

/*************************************************************************
Specifies classification problem (two  or  more  classes  are  predicted).
There also exists "regression" version of this function.

This subroutine adds dense dataset to the internal storage of the  builder
object. Specifying your dataset in the dense format means that  the  dense
version of the KNN construction algorithm will be invoked.

INPUT PARAMETERS:
    S           -   KNN builder object
    XY          -   array[NPoints,NVars+1] (note:   actual   size  can  be
                    larger, only leading part is used anyway), dataset:
                    * first NVars elements of each row store values of the
                      independent variables
                    * next element stores class index, in [0,NClasses)
    NPoints     -   number of rows in the dataset, NPoints>=1
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2

OUTPUT PARAMETERS:
    S           -   KNN builder

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnbuildersetdatasetcls(knnbuilder &s, const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, const xparams _xparams = alglib::xdefault);

Examples: [1]

`knnbuildersetdatasetreg` function

/*************************************************************************
Specifies regression problem (one or more continuous  output variables are
predicted). There also exists "classification" version of this function.

This subroutine adds dense dataset to the internal storage of the  builder
object. Specifying your dataset in the dense format means that  the  dense
version of the KNN construction algorithm will be invoked.

INPUT PARAMETERS:
    S           -   KNN builder object
    XY          -   array[NPoints,NVars+NOut] (note: actual  size  can  be
                    larger, only leading part is used anyway), dataset:
                    * first NVars elements of each row store values of the
                      independent variables
                    * next NOut elements store  values  of  the  dependent
                      variables
    NPoints     -   number of rows in the dataset, NPoints>=1
    NVars       -   number of independent variables, NVars>=1
    NOut        -   number of dependent variables, NOut>=1

OUTPUT PARAMETERS:
    S           -   KNN builder

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnbuildersetdatasetreg(knnbuilder &s, const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nout, const xparams _xparams = alglib::xdefault);

Examples: [1]

`knnbuildersetnorm` function

/*************************************************************************
This function sets norm type used for neighbor search.

INPUT PARAMETERS:
    S           -   decision forest builder object
    NormType    -   norm type:
                    * 0      inf-norm
                    * 1      1-norm
                    * 2      Euclidean norm (default)

OUTPUT PARAMETERS:
    S           -   decision forest builder

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnbuildersetnorm(knnbuilder &s, const ae_int_t nrmtype, const xparams _xparams = alglib::xdefault);

`knnclassify` function

/*************************************************************************
This function returns most probable class number for an  input  X.  It  is
same as calling knnprocess(model,x,y), then determining i=argmax(y[i]) and
returning i.

A class number in [0,NOut) range in returned for classification  problems,
-1 is returned when this function is called for regression problems.

IMPORTANT: this function is thread-unsafe and modifies internal structures
           of the model! You can not use same model  object  for  parallel
           evaluation from several threads.

           Use knntsprocess() with independent  thread-local  buffers,  if
           you need thread-safe evaluation.

INPUT PARAMETERS:
    Model   -   KNN model
    X       -   input vector,  array[0..NVars-1].

RESULT:
    class number, -1 for regression tasks

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
ae_int_t knnclassify(knnmodel &model, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`knncreatebuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel KNN requests.

KNN subpackage provides two sets of computing functions - ones  which  use
internal buffer of KNN model (these  functions are single-threaded because
they use same buffer, which can not  shared  between  threads),  and  ones
which use external buffer.

This function is used to initialize external buffer.

INPUT PARAMETERS
    Model       -   KNN model which is associated with newly created buffer

OUTPUT PARAMETERS
    Buf         -   external buffer.


IMPORTANT: buffer object should be used only with model which was used  to
           initialize buffer. Any attempt to  use  buffer  with  different
           object is dangerous - you  may   get  integrity  check  failure
           (exception) because sizes of internal  arrays  do  not  fit  to
           dimensions of the model structure.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knncreatebuffer(const knnmodel &model, knnbuffer &buf, const xparams _xparams = alglib::xdefault);

`knnprocess` function

/*************************************************************************
Inference using KNN model.

See also knnprocess0(), knnprocessi() and knnclassify() for options with a
bit more convenient interface.

IMPORTANT: this function is thread-unsafe and modifies internal structures
           of the model! You can not use same model  object  for  parallel
           evaluation from several threads.

           Use knntsprocess() with independent  thread-local  buffers,  if
           you need thread-safe evaluation.

INPUT PARAMETERS:
    Model   -   KNN model
    X       -   input vector,  array[0..NVars-1].
    Y       -   possible preallocated buffer. Reused if long enough.

OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnprocess(knnmodel &model, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`knnprocess0` function

/*************************************************************************
This function returns first component of the  inferred  vector  (i.e.  one
with index #0).

It is a convenience wrapper for knnprocess() intended for either:
* 1-dimensional regression problems
* 2-class classification problems

In the former case this function returns inference result as scalar, which
is definitely more convenient that wrapping it as vector.  In  the  latter
case it returns probability of object belonging to class #0.

If you call it for anything different from two cases above, it  will  work
as defined, i.e. return y[0], although it is of less use in such cases.

IMPORTANT: this function is thread-unsafe and modifies internal structures
           of the model! You can not use same model  object  for  parallel
           evaluation from several threads.

           Use knntsprocess() with independent  thread-local  buffers,  if
           you need thread-safe evaluation.

INPUT PARAMETERS:
    Model   -   KNN model
    X       -   input vector,  array[0..NVars-1].

RESULT:
    Y[0]

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double knnprocess0(knnmodel &model, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`knnprocessi` function

/*************************************************************************
'interactive' variant of knnprocess()  for  languages  like  Python  which
support constructs like "y = knnprocessi(model,x)" and interactive mode of
the interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

IMPORTANT: this  function  is  thread-unsafe  and  may   modify   internal
           structures of the model! You can not use same model  object for
           parallel evaluation from several threads.

           Use knntsprocess()  with  independent  thread-local  buffers if
           you need thread-safe evaluation.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnprocessi(knnmodel &model, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`knnrelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    Model   -   KNN model
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.
    Zero if model solves regression task.

NOTE: if  you  need several different kinds of error metrics, it is better
      to use knnallerrors() which computes all error metric  with just one
      pass over dataset.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double knnrelclserror(const knnmodel &model, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`knnrewritekeps` function

/*************************************************************************
Changing search settings of KNN model.

K and EPS parameters of KNN  (AKNN)  search  are  specified  during  model
construction. However, plain KNN algorithm with Euclidean distance  allows
you to change them at any moment.

NOTE: future versions of KNN model may support advanced versions  of  KNN,
      such as NCA or LMNN. It is possible that such algorithms won't allow
      you to change search settings on the fly. If you call this  function
      for an algorithm which does not support on-the-fly changes, it  will
      throw an exception.

INPUT PARAMETERS:
    Model   -   KNN model
    K       -   K>=1, neighbors count
    EPS     -   accuracy of the EPS-approximate NN search. Set to 0.0,  if
                you want to perform "classic" KNN search.  Specify  larger
                values  if  you  need  to  speed-up  high-dimensional  KNN
                queries.

OUTPUT PARAMETERS:
    nothing on success, exception on failure

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knnrewritekeps(knnmodel &model, const ae_int_t k, const double eps, const xparams _xparams = alglib::xdefault);

`knnrmserror` function

/*************************************************************************
RMS error on the test set.

Its meaning for regression task is obvious. As for classification problems,
RMS error means error when estimating posterior probabilities.

INPUT PARAMETERS:
    Model   -   KNN model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.

NOTE: if  you  need several different kinds of error metrics, it is better
      to use knnallerrors() which computes all error metric  with just one
      pass over dataset.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
double knnrmserror(const knnmodel &model, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`knnserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void knnserialize(const knnmodel &obj, std::string &s_out);
void knnserialize(const knnmodel &obj, std::ostream &s_out);

`knntsprocess` function

/*************************************************************************
Thread-safe procesing using external buffer for temporaries.

This function is thread-safe (i.e .  you  can  use  same  KNN  model  from
multiple threads) as long as you use different buffer objects for different
threads.

INPUT PARAMETERS:
    Model   -   KNN model
    Buf     -   buffer object, must be  allocated  specifically  for  this
                model with knncreatebuffer().
    X       -   input vector,  array[NVars]

OUTPUT PARAMETERS:
    Y       -   result, array[NOut].   Regression  estimate  when  solving
                regression task,  vector  of  posterior  probabilities for
                a classification task.

  -- ALGLIB --
     Copyright 15.02.2019 by Bochkanov Sergey
*************************************************************************/
void knntsprocess(const knnmodel &model, knnbuffer &buf, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`knnunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void knnunserialize(const std::string &s_in, knnmodel &obj);
void knnunserialize(const std::istream &s_in, knnmodel &obj);

knn_cls example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple classification example: classify points (x,y) in 2D space
        // as ones with x>=0 and ones with x<0 (y is ignored, but our classifier
        // has to find out it).
        //
        // First, we have to create KNN builder object, load dataset and specify
        // training settings. Our dataset is specified as matrix, which has following
        // format:
        //
        //     x0 y0 class0
        //     x1 y1 class1
        //     x2 y2 class2
        //     ....
        //
        // Here xi and yi can be any values (and in fact you can have any number of
        // independent variables), and classi MUST be integer number in [0,NClasses)
        // range. In our example we denote points with x>=0 as class #0, and
        // ones with negative xi as class #1.
        //
        // NOTE: if you want to solve regression problem, specify dataset in similar
        //       format, but with dependent variable(s) instead of class labels. You
        //       can have dataset with multiple dependent variables, by the way!
        //
        // For the sake of simplicity, our example includes only 4-point dataset and
        // really simple K=1 nearest neighbor search. Industrial problems typically
        // need larger values of K.
        //
        knnbuilder builder;
        ae_int_t nvars = 2;
        ae_int_t nclasses = 2;
        ae_int_t npoints = 4;
        real_2d_array xy = "[[1,1,0],[1,-1,0],[-1,1,1],[-1,-1,1]]";

        knnbuildercreate(builder);
        knnbuildersetdatasetcls(builder, xy, npoints, nvars, nclasses);

        // we build KNN model with k=1 and eps=0 (exact k-nn search is performed)
        ae_int_t k = 1;
        double eps = 0;
        knnmodel model;
        knnreport rep;
        knnbuilderbuildknnmodel(builder, k, eps, model, rep);

        // with such settings (k=1 is used) you can expect zero classification
        // error on training set. Beautiful results, but remember - in real life
        // you do not need zero TRAINING SET error, you need good generalization.

        printf("%.4f\n", double(rep.relclserror)); // EXPECTED: 0.0000

        // now, let's perform some simple processing with knnprocess()
        real_1d_array x = "[+1,0]";
        real_1d_array y = "[]";
        knnprocess(model, x, y);
        printf("%s\n", y.tostring(3).c_str()); // EXPECTED: [+1,0]

        // another option is to use knnprocess0() which returns just first component
        // of the output vector y. ideal for regression problems and binary classifiers.
        double y0;
        y0 = knnprocess0(model, x);
        printf("%.3f\n", double(y0)); // EXPECTED: 1.000

        // finally, you can use knnclassify() which returns most probable class index (i.e. argmax y[i]).
        ae_int_t i;
        i = knnclassify(model, x);
        printf("%d\n", int(i)); // EXPECTED: 0
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

knn_reg example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple regression example: model f(x,y)=x+y
        //
        // First, we have to create KNN builder object, load dataset and specify
        // training settings. Our dataset is specified as matrix, which has following
        // format:
        //
        //     x0 y0 f0
        //     x1 y1 f1
        //     x2 y2 f2
        //     ....
        //
        // Here xi and yi can be any values, and fi is a dependent function value.
        // By the way, with KNN algorithm you can even model functions with multiple
        // dependent variables!
        //
        // NOTE: you can also solve classification problems with KNN models, see
        //       another example for this unit.
        //
        // For the sake of simplicity, our example includes only 4-point dataset and
        // really simple K=1 nearest neighbor search. Industrial problems typically
        // need larger values of K.
        //
        knnbuilder builder;
        ae_int_t nvars = 2;
        ae_int_t nout = 1;
        ae_int_t npoints = 4;
        real_2d_array xy = "[[1,1,+2],[1,-1,0],[-1,1,0],[-1,-1,-2]]";

        knnbuildercreate(builder);
        knnbuildersetdatasetreg(builder, xy, npoints, nvars, nout);

        // we build KNN model with k=1 and eps=0 (exact k-nn search is performed)
        ae_int_t k = 1;
        double eps = 0;
        knnmodel model;
        knnreport rep;
        knnbuilderbuildknnmodel(builder, k, eps, model, rep);

        // with such settings (k=1 is used) you can expect zero RMS error on the
        // training set. Beautiful results, but remember - in real life you do not
        // need zero TRAINING SET error, you need good generalization.

        printf("%.4f\n", double(rep.rmserror)); // EXPECTED: 0.0000

        // now, let's perform some simple processing with knnprocess()
        real_1d_array x = "[+1,+1]";
        real_1d_array y = "[]";
        knnprocess(model, x, y);
        printf("%s\n", y.tostring(3).c_str()); // EXPECTED: [+2]

        // another option is to use knnprocess0() which returns just first component
        // of the output vector y. ideal for regression problems and binary classifiers.
        double y0;
        y0 = knnprocess0(model, x);
        printf("%.3f\n", double(y0)); // EXPECTED: 2.000

        // there also exist another convenience function, knnclassify(),
        // but it does not work for regression problems - it always returns -1.
        ae_int_t i;
        i = knnclassify(model, x);
        printf("%d\n", int(i)); // EXPECTED: -1
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`laguerre` subpackage

Functions

laguerrecalculate
laguerrecoefficients
laguerresum

Examples

`laguerrecalculate` function

/*************************************************************************
Calculation of the value of the Laguerre polynomial.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial Ln at x
*************************************************************************/
double laguerrecalculate(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`laguerrecoefficients` function

/*************************************************************************
Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void laguerrecoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`laguerresum` function

/*************************************************************************
Summation of Laguerre polynomials using Clenshaw's recurrence formula.

This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Laguerre polynomial at x
*************************************************************************/
double laguerresum(const real_1d_array &c, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`lda` subpackage

Functions

fisherlda
fisherldan

Examples

`fisherlda` function

/*************************************************************************
Multiclass Fisher LDA

The function finds coefficients of a linear  combination  which  optimally
separates training set. Most suited for 2-class problems, see fisherldan()
for an variant that returns N-dimensional basis.

INPUT PARAMETERS:
    XY          -   training set, array[NPoints,NVars+1].
                    First NVars columns store values of independent
                    variables, the next column stores class index (from 0
                    to NClasses-1) which dataset element belongs to.
                    Fractional values are rounded to the nearest integer.
                    The class index must be in the [0,NClasses-1] range,
                    an exception is generated otherwise.
    NPoints     -   training set size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2


OUTPUT PARAMETERS:
    W           -   linear combination coefficients, array[NVars]

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 31.05.2008 by Bochkanov Sergey
*************************************************************************/
void fisherlda(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, real_1d_array &w, const xparams _xparams = alglib::xdefault);
void fisherlda(const real_2d_array &xy, const ae_int_t nclasses, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`fisherldan` function

/*************************************************************************
N-dimensional multiclass Fisher LDA

Subroutine finds coefficients of linear combinations which optimally separates
training set on classes. It returns N-dimensional basis whose vector are sorted
by quality of training set separation (in descending order).

INPUT PARAMETERS:
    XY          -   training set, array[NPoints,NVars+1].
                    First NVars columns store values of independent
                    variables, the next column stores class index (from 0
                    to NClasses-1) which dataset element belongs to.
                    Fractional values are rounded to the nearest integer.
                    The class index must be in the [0,NClasses-1] range,
                    an exception is generated otherwise.
    NPoints     -   training set size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2


OUTPUT PARAMETERS:
    W           -   basis, array[NVars,NVars]
                    columns of matrix stores basis vectors, sorted by
                    quality of training set separation (in descending order)

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 31.05.2008 by Bochkanov Sergey
*************************************************************************/
void fisherldan(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, real_2d_array &w, const xparams _xparams = alglib::xdefault);
void fisherldan(const real_2d_array &xy, const ae_int_t nclasses, real_2d_array &w, const xparams _xparams = alglib::xdefault);

`legendre` subpackage

Functions

legendrecalculate
legendrecoefficients
legendresum

Examples

`legendrecalculate` function

/*************************************************************************
Calculation of the value of the Legendre polynomial Pn.

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial Pn at x
*************************************************************************/
double legendrecalculate(const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`legendrecoefficients` function

/*************************************************************************
Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N

Input parameters:
    N   -   polynomial degree, n>=0

Output parameters:
    C   -   coefficients
*************************************************************************/
void legendrecoefficients(const ae_int_t n, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`legendresum` function

/*************************************************************************
Summation of Legendre polynomials using Clenshaw's recurrence formula.

This routine calculates
    c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x)

Parameters:
    n   -   degree, n>=0
    x   -   argument

Result:
    the value of the Legendre polynomial at x
*************************************************************************/
double legendresum(const real_1d_array &c, const ae_int_t n, const double x, const xparams _xparams = alglib::xdefault);

`lincg` subpackage

Classes

lincgreport
lincgstate

Functions

lincgcreate
lincgresults
lincgsetcond
lincgsetprecdiag
lincgsetprecunit
lincgsetrestartfreq
lincgsetrupdatefreq
lincgsetstartingpoint
lincgsetxrep
lincgsolvesparse

Examples

lincg_d_1

Solution of sparse linear systems with CG

`lincgreport` class

/*************************************************************************

*************************************************************************/
class lincgreport
{
public:
    lincgreport();
    lincgreport(const lincgreport &rhs);
    lincgreport& operator=(const lincgreport &rhs);
    virtual ~lincgreport();
    ae_int_t iterationscount;
    ae_int_t nmv;
    ae_int_t terminationtype;
    double r2;
};

`lincgstate` class

/*************************************************************************
This object stores state of the linear CG method.

You should use ALGLIB functions to work with this object.
Never try to access its fields directly!
*************************************************************************/
class lincgstate
{
public:
    lincgstate();
    lincgstate(const lincgstate &rhs);
    lincgstate& operator=(const lincgstate &rhs);
    virtual ~lincgstate();
};

`lincgcreate` function

/*************************************************************************
This function initializes linear CG Solver. This solver is used  to  solve
symmetric positive definite problems. If you want  to  solve  nonsymmetric
(or non-positive definite) problem you may use LinLSQR solver provided  by
ALGLIB.

USAGE:
1. User initializes algorithm state with LinCGCreate() call
2. User tunes solver parameters with  LinCGSetCond() and other functions
3. Optionally, user sets starting point with LinCGSetStartingPoint()
4. User  calls LinCGSolveSparse() function which takes algorithm state and
   SparseMatrix object.
5. User calls LinCGResults() to get solution
6. Optionally, user may call LinCGSolveSparse()  again  to  solve  another
   problem  with different matrix and/or right part without reinitializing
   LinCGState structure.

INPUT PARAMETERS:
    N       -   problem dimension, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgcreate(const ae_int_t n, lincgstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lincgresults` function

/*************************************************************************
CG-solver: results.

This function must be called after LinCGSolve

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -5    input matrix is either not positive definite,
                            too large or too small
                    * -4    overflow/underflow during solution
                            (ill conditioned problem)
                    *  1    ||residual||<=EpsF*||b||
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            best point found is returned
                * Rep.IterationsCount contains iterations count
                * NMV countains number of matrix-vector calculations

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgresults(const lincgstate &state, real_1d_array &x, lincgreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lincgsetcond` function

/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsF    -   algorithm will be stopped if norm of residual is less than
                EpsF*||b||.
    MaxIts  -   algorithm will be stopped if number of iterations is  more
                than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
If  both  EpsF  and  MaxIts  are  zero then small EpsF will be set to small
value.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetcond(lincgstate &state, const double epsf, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`lincgsetprecdiag` function

/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function.  LinCGSolveSparse() will use diagonal of the  system  matrix  as
preconditioner. This preconditioning mode is active by default.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void lincgsetprecdiag(lincgstate &state, const xparams _xparams = alglib::xdefault);

`lincgsetprecunit` function

/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function. By default, SolveSparse() uses diagonal preconditioner,  but  if
you want to use solver without preconditioning, you can call this function
which forces solver to use unit matrix for preconditioning.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void lincgsetprecunit(lincgstate &state, const xparams _xparams = alglib::xdefault);

`lincgsetrestartfreq` function

/*************************************************************************
This function sets restart frequency. By default, algorithm  is  restarted
after N subsequent iterations.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetrestartfreq(lincgstate &state, const ae_int_t srf, const xparams _xparams = alglib::xdefault);

`lincgsetrupdatefreq` function

/*************************************************************************
This function sets frequency of residual recalculations.

Algorithm updates residual r_k using iterative formula,  but  recalculates
it from scratch after each 10 iterations. It is done to avoid accumulation
of numerical errors and to stop algorithm when r_k starts to grow.

Such low update frequence (1/10) gives very  little  overhead,  but  makes
algorithm a bit more robust against numerical errors. However, you may
change it

INPUT PARAMETERS:
    Freq    -   desired update frequency, Freq>=0.
                Zero value means that no updates will be done.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetrupdatefreq(lincgstate &state, const ae_int_t freq, const xparams _xparams = alglib::xdefault);

`lincgsetstartingpoint` function

/*************************************************************************
This function sets starting point.
By default, zero starting point is used.

INPUT PARAMETERS:
    X       -   starting point, array[N]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetstartingpoint(lincgstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`lincgsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinCGOptimize().

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsetxrep(lincgstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`lincgsolvesparse` function

/*************************************************************************
Procedure for solution of A*x=b with sparse A.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse matrix in the CRS format (you MUST contvert  it  to
                CRS format by calling SparseConvertToCRS() function).
    IsUpper -   whether upper or lower triangle of A is used:
                * IsUpper=True  => only upper triangle is used and lower
                                   triangle is not referenced at all
                * IsUpper=False => only lower triangle is used and upper
                                   triangle is not referenced at all
    B       -   right part, array[N]

RESULT:
    This function returns no result.
    You can get solution by calling LinCGResults()

NOTE: this function uses lightweight preconditioning -  multiplication  by
      inverse of diag(A). If you want, you can turn preconditioning off by
      calling LinCGSetPrecUnit(). However, preconditioning cost is low and
      preconditioner  is  very  important  for  solution  of  badly scaled
      problems.

  -- ALGLIB --
     Copyright 14.11.2011 by Bochkanov Sergey
*************************************************************************/
void lincgsolvesparse(lincgstate &state, const sparsematrix &a, const bool isupper, const real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

lincg_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example illustrates solution of sparse linear systems with
        // conjugate gradient method.
        // 
        // Suppose that we have linear system A*x=b with sparse symmetric
        // positive definite A (represented by sparsematrix object)
        //         [ 5 1       ]
        //         [ 1 7 2     ]
        //     A = [   2 8 1   ]
        //         [     1 4 1 ]
        //         [       1 4 ]
        // and right part b
        //     [  7 ]
        //     [ 17 ]
        // b = [ 14 ]
        //     [ 10 ]
        //     [  6 ]
        // and we want to solve this system using sparse linear CG. In order
        // to do so, we have to create left part (sparsematrix object) and
        // right part (dense array).
        //
        // Initially, sparse matrix is created in the Hash-Table format,
        // which allows easy initialization, but do not allow matrix to be
        // used in the linear solvers. So after construction you should convert
        // sparse matrix to CRS format (one suited for linear operations).
        //
        // It is important to note that in our example we initialize full
        // matrix A, both lower and upper triangles. However, it is symmetric
        // and sparse solver needs just one half of the matrix. So you may
        // save about half of the space by filling only one of the triangles.
        //
        sparsematrix a;
        sparsecreate(5, 5, a);
        sparseset(a, 0, 0, 5.0);
        sparseset(a, 0, 1, 1.0);
        sparseset(a, 1, 0, 1.0);
        sparseset(a, 1, 1, 7.0);
        sparseset(a, 1, 2, 2.0);
        sparseset(a, 2, 1, 2.0);
        sparseset(a, 2, 2, 8.0);
        sparseset(a, 2, 3, 1.0);
        sparseset(a, 3, 2, 1.0);
        sparseset(a, 3, 3, 4.0);
        sparseset(a, 3, 4, 1.0);
        sparseset(a, 4, 3, 1.0);
        sparseset(a, 4, 4, 4.0);

        //
        // Now our matrix is fully initialized, but we have to do one more
        // step - convert it from Hash-Table format to CRS format (see
        // documentation on sparse matrices for more information about these
        // formats).
        //
        // If you omit this call, ALGLIB will generate exception on the first
        // attempt to use A in linear operations. 
        //
        sparseconverttocrs(a);

        //
        // Initialization of the right part
        //
        real_1d_array b = "[7,17,14,10,6]";

        //
        // Now we have to create linear solver object and to use it for the
        // solution of the linear system.
        //
        // NOTE: lincgsolvesparse() accepts additional parameter which tells
        //       what triangle of the symmetric matrix should be used - upper
        //       or lower. Because we've filled both parts of the matrix, we
        //       can use any part - upper or lower.
        //
        lincgstate s;
        lincgreport rep;
        real_1d_array x;
        lincgcreate(5, s);
        lincgsolvesparse(s, a, true, b);
        lincgresults(s, x, rep);

        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.000,2.000,1.000,2.000,1.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`linlsqr` subpackage

Classes

linlsqrreport
linlsqrstate

Functions

linlsqrcreate
linlsqrcreatebuf
linlsqrpeekiterationscount
linlsqrrequesttermination
linlsqrresults
linlsqrsetcond
linlsqrsetlambdai
linlsqrsetprecdiag
linlsqrsetprecunit
linlsqrsetxrep
linlsqrsolvesparse

Examples

linlsqr_d_1

Solution of sparse linear systems with CG

`linlsqrreport` class

/*************************************************************************

*************************************************************************/
class linlsqrreport
{
public:
    linlsqrreport();
    linlsqrreport(const linlsqrreport &rhs);
    linlsqrreport& operator=(const linlsqrreport &rhs);
    virtual ~linlsqrreport();
    ae_int_t iterationscount;
    ae_int_t nmv;
    ae_int_t terminationtype;
};

`linlsqrstate` class

/*************************************************************************
This object stores state of the LinLSQR method.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class linlsqrstate
{
public:
    linlsqrstate();
    linlsqrstate(const linlsqrstate &rhs);
    linlsqrstate& operator=(const linlsqrstate &rhs);
    virtual ~linlsqrstate();
};

`linlsqrcreate` function

/*************************************************************************
This function initializes linear LSQR Solver. This solver is used to solve
non-symmetric (and, possibly, non-square) problems. Least squares solution
is returned for non-compatible systems.

USAGE:
1. User initializes algorithm state with LinLSQRCreate() call
2. User tunes solver parameters with  LinLSQRSetCond() and other functions
3. User  calls  LinLSQRSolveSparse()  function which takes algorithm state
   and SparseMatrix object.
4. User calls LinLSQRResults() to get solution
5. Optionally, user may call LinLSQRSolveSparse() again to  solve  another
   problem  with different matrix and/or right part without reinitializing
   LinLSQRState structure.

INPUT PARAMETERS:
    M       -   number of rows in A
    N       -   number of variables, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: see also linlsqrcreatebuf()  for  version  which  reuses  previously
      allocated place as much as possible.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrcreate(const ae_int_t m, const ae_int_t n, linlsqrstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`linlsqrcreatebuf` function

/*************************************************************************
This function initializes linear LSQR Solver.  It  provides  exactly  same
functionality as linlsqrcreate(), but reuses  previously  allocated  space
as much as possible.

INPUT PARAMETERS:
    M       -   number of rows in A
    N       -   number of variables, N>0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 14.11.2018 by Bochkanov Sergey
*************************************************************************/
void linlsqrcreatebuf(const ae_int_t m, const ae_int_t n, linlsqrstate &state, const xparams _xparams = alglib::xdefault);

`linlsqrpeekiterationscount` function

/*************************************************************************
This function is used to peek into LSQR solver and get  current  iteration
counter. You can safely "peek" into the solver from another thread.

INPUT PARAMETERS:
    S           -   solver object

RESULT:
    iteration counter, in [0,INF)

  -- ALGLIB --
     Copyright 21.05.2018 by Bochkanov Sergey
*************************************************************************/
ae_int_t linlsqrpeekiterationscount(const linlsqrstate &s, const xparams _xparams = alglib::xdefault);

`linlsqrrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of the running solver.  It
can be called from some other thread which wants LSQR solver to  terminate
(obviously, the  thread  running  LSQR  solver can not request termination
because it is already busy working on LSQR).

As result, solver  stops  at  point  which  was  "current  accepted"  when
termination  request  was  submitted  and returns error code 8 (successful
termination).  Such   termination   is  a smooth  process  which  properly
deallocates all temporaries.

INPUT PARAMETERS:
    State   -   solver structure

NOTE: calling this function on solver which is NOT running  will  have  no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

NOTE: solver clears termination flag on its start, it means that  if  some
      other thread will request termination too soon, its request will went
      unnoticed.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void linlsqrrequesttermination(linlsqrstate &state, const xparams _xparams = alglib::xdefault);

`linlsqrresults` function

/*************************************************************************
LSQR solver: results.

This function must be called after LinLSQRSolve

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    *  1    ||Rk||<=EpsB*||B||
                    *  4    ||A^T*Rk||/(||A||*||Rk||)<=EpsA
                    *  5    MaxIts steps was taken
                    *  7    rounding errors prevent further progress,
                            X contains best point found so far.
                            (sometimes returned on singular systems)
                    *  8    user requested termination via calling
                            linlsqrrequesttermination()
                * Rep.IterationsCount contains iterations count
                * NMV countains number of matrix-vector calculations

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrresults(const linlsqrstate &state, real_1d_array &x, linlsqrreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`linlsqrsetcond` function

/*************************************************************************
This function sets stopping criteria.

INPUT PARAMETERS:
    EpsA    -   algorithm will be stopped if ||A^T*Rk||/(||A||*||Rk||)<=EpsA.
    EpsB    -   algorithm will be stopped if ||Rk||<=EpsB*||B||
    MaxIts  -   algorithm will be stopped if number of iterations
                more than MaxIts.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE: if EpsA,EpsB,EpsC and MaxIts are zero then these variables will
be setted as default values.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetcond(linlsqrstate &state, const double epsa, const double epsb, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`linlsqrsetlambdai` function

/*************************************************************************
This function sets optional Tikhonov regularization coefficient.
It is zero by default.

INPUT PARAMETERS:
    LambdaI -   regularization factor, LambdaI>=0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetlambdai(linlsqrstate &state, const double lambdai, const xparams _xparams = alglib::xdefault);

`linlsqrsetprecdiag` function

/*************************************************************************
This  function  changes  preconditioning  settings  of  LinCGSolveSparse()
function.  LinCGSolveSparse() will use diagonal of the  system  matrix  as
preconditioner. This preconditioning mode is active by default.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetprecdiag(linlsqrstate &state, const xparams _xparams = alglib::xdefault);

`linlsqrsetprecunit` function

/*************************************************************************
This  function  changes  preconditioning  settings of LinLSQQSolveSparse()
function. By default, SolveSparse() uses diagonal preconditioner,  but  if
you want to use solver without preconditioning, you can call this function
which forces solver to use unit matrix for preconditioning.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 19.11.2012 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetprecunit(linlsqrstate &state, const xparams _xparams = alglib::xdefault);

`linlsqrsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinCGOptimize().

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsetxrep(linlsqrstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`linlsqrsolvesparse` function

/*************************************************************************
Procedure for solution of A*x=b with sparse A.

INPUT PARAMETERS:
    State   -   algorithm state
    A       -   sparse M*N matrix in the CRS format (you MUST contvert  it
                to CRS format  by  calling  SparseConvertToCRS()  function
                BEFORE you pass it to this function).
    B       -   right part, array[M]

RESULT:
    This function returns no result.
    You can get solution by calling LinCGResults()

NOTE: this function uses lightweight preconditioning -  multiplication  by
      inverse of diag(A). If you want, you can turn preconditioning off by
      calling LinLSQRSetPrecUnit(). However, preconditioning cost is   low
      and preconditioner is very important for solution  of  badly  scaled
      problems.

  -- ALGLIB --
     Copyright 30.11.2011 by Bochkanov Sergey
*************************************************************************/
void linlsqrsolvesparse(linlsqrstate &state, const sparsematrix &a, const real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1]

linlsqr_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example illustrates solution of sparse linear least squares problem
        // with LSQR algorithm.
        // 
        // Suppose that we have least squares problem min|A*x-b| with sparse A
        // represented by sparsematrix object
        //         [ 1 1 ]
        //         [ 1 1 ]
        //     A = [ 2 1 ]
        //         [ 1   ]
        //         [   1 ]
        // and right part b
        //     [ 4 ]
        //     [ 2 ]
        // b = [ 4 ]
        //     [ 1 ]
        //     [ 2 ]
        // and we want to solve this system in the least squares sense using
        // LSQR algorithm. In order to do so, we have to create left part
        // (sparsematrix object) and right part (dense array).
        //
        // Initially, sparse matrix is created in the Hash-Table format,
        // which allows easy initialization, but do not allow matrix to be
        // used in the linear solvers. So after construction you should convert
        // sparse matrix to CRS format (one suited for linear operations).
        //
        sparsematrix a;
        sparsecreate(5, 2, a);
        sparseset(a, 0, 0, 1.0);
        sparseset(a, 0, 1, 1.0);
        sparseset(a, 1, 0, 1.0);
        sparseset(a, 1, 1, 1.0);
        sparseset(a, 2, 0, 2.0);
        sparseset(a, 2, 1, 1.0);
        sparseset(a, 3, 0, 1.0);
        sparseset(a, 4, 1, 1.0);

        //
        // Now our matrix is fully initialized, but we have to do one more
        // step - convert it from Hash-Table format to CRS format (see
        // documentation on sparse matrices for more information about these
        // formats).
        //
        // If you omit this call, ALGLIB will generate exception on the first
        // attempt to use A in linear operations. 
        //
        sparseconverttocrs(a);

        //
        // Initialization of the right part
        //
        real_1d_array b = "[4,2,4,1,2]";

        //
        // Now we have to create linear solver object and to use it for the
        // solution of the linear system.
        //
        linlsqrstate s;
        linlsqrreport rep;
        real_1d_array x;
        linlsqrcreate(5, 2, s);
        linlsqrsolvesparse(s, a, b);
        linlsqrresults(s, x, rep);

        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.000,2.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`linreg` subpackage

Classes

linearmodel
lrreport

Functions

lravgerror
lravgrelerror
lrbuild
lrbuilds
lrbuildz
lrbuildzs
lrpack
lrprocess
lrrmserror
lrunpack

Examples

linreg_d_basic

Linear regression used to build the very basic model and unpack coefficients

`linearmodel` class

/*************************************************************************

*************************************************************************/
class linearmodel
{
public:
    linearmodel();
    linearmodel(const linearmodel &rhs);
    linearmodel& operator=(const linearmodel &rhs);
    virtual ~linearmodel();
};

`lrreport` class

/*************************************************************************
LRReport structure contains additional information about linear model:
* C             -   covariation matrix,  array[0..NVars,0..NVars].
                    C[i,j] = Cov(A[i],A[j])
* RMSError      -   root mean square error on a training set
* AvgError      -   average error on a training set
* AvgRelError   -   average relative error on a training set (excluding
                    observations with zero function value).
* CVRMSError    -   leave-one-out cross-validation estimate of
                    generalization error. Calculated using fast algorithm
                    with O(NVars*NPoints) complexity.
* CVAvgError    -   cross-validation estimate of average error
* CVAvgRelError -   cross-validation estimate of average relative error

All other fields of the structure are intended for internal use and should
not be used outside ALGLIB.
*************************************************************************/
class lrreport
{
public:
    lrreport();
    lrreport(const lrreport &rhs);
    lrreport& operator=(const lrreport &rhs);
    virtual ~lrreport();
    real_2d_array c;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double cvrmserror;
    double cvavgerror;
    double cvavgrelerror;
    ae_int_t ncvdefects;
    integer_1d_array cvdefects;
};

`lravgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    LM      -   linear model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average error.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double lravgerror(const linearmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`lravgrelerror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    LM      -   linear model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average relative error.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double lravgrelerror(const linearmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`lrbuild` function

/*************************************************************************
Linear regression

Subroutine builds model:

    Y = A(0)*X[0] + ... + A(N-1)*X[N-1] + A(N)

and model found in ALGLIB format, covariation matrix, training set  errors
(rms,  average,  average  relative)   and  leave-one-out  cross-validation
estimate of the generalization error. CV  estimate calculated  using  fast
algorithm with O(NPoints*NVars) complexity.

When  covariation  matrix  is  calculated  standard deviations of function
values are assumed to be equal to RMS error on the training set.

INPUT PARAMETERS:
    XY          -   training set, array [0..NPoints-1,0..NVars]:
                    * NVars columns - independent variables
                    * last column - dependent variable
    NPoints     -   training set size, NPoints>NVars+1. An exception is
                    generated otherwise.
    NVars       -   number of independent variables

OUTPUT PARAMETERS:
    LM          -   linear model in the ALGLIB format. Use subroutines of
                    this unit to work with the model.
    Rep         -   additional results, see comments on LRReport structure.

  -- ALGLIB --
     Copyright 02.08.2008 by Bochkanov Sergey
*************************************************************************/
void lrbuild(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);
void lrbuild(const real_2d_array &xy, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lrbuilds` function

/*************************************************************************
Linear regression

Variant of LRBuild which uses vector of standatd deviations (errors in
function values).

INPUT PARAMETERS:
    XY          -   training set, array [0..NPoints-1,0..NVars]:
                    * NVars columns - independent variables
                    * last column - dependent variable
    S           -   standard deviations (errors in function values)
                    array[NPoints], S[i]>0.
    NPoints     -   training set size, NPoints>NVars+1
    NVars       -   number of independent variables

OUTPUT PARAMETERS:
    LM          -   linear model in the ALGLIB format. Use subroutines of
                    this unit to work with the model.
    Rep         -   additional results, see comments on LRReport structure.

  -- ALGLIB --
     Copyright 02.08.2008 by Bochkanov Sergey
*************************************************************************/
void lrbuilds(const real_2d_array &xy, const real_1d_array &s, const ae_int_t npoints, const ae_int_t nvars, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);
void lrbuilds(const real_2d_array &xy, const real_1d_array &s, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);

`lrbuildz` function

/*************************************************************************
Like LRBuild but builds model

    Y = A(0)*X[0] + ... + A(N-1)*X[N-1]

i.e. with zero constant term.

  -- ALGLIB --
     Copyright 30.10.2008 by Bochkanov Sergey
*************************************************************************/
void lrbuildz(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);
void lrbuildz(const real_2d_array &xy, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);

`lrbuildzs` function

/*************************************************************************
Like LRBuildS, but builds model

    Y = A(0)*X[0] + ... + A(N-1)*X[N-1]

i.e. with zero constant term.

  -- ALGLIB --
     Copyright 30.10.2008 by Bochkanov Sergey
*************************************************************************/
void lrbuildzs(const real_2d_array &xy, const real_1d_array &s, const ae_int_t npoints, const ae_int_t nvars, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);
void lrbuildzs(const real_2d_array &xy, const real_1d_array &s, linearmodel &lm, lrreport &rep, const xparams _xparams = alglib::xdefault);

`lrpack` function

/*************************************************************************
"Packs" coefficients and creates linear model in ALGLIB format (LRUnpack
reversed).

INPUT PARAMETERS:
    V           -   coefficients, array[0..NVars]
    NVars       -   number of independent variables

OUTPUT PAREMETERS:
    LM          -   linear model.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
void lrpack(const real_1d_array &v, const ae_int_t nvars, linearmodel &lm, const xparams _xparams = alglib::xdefault);
void lrpack(const real_1d_array &v, linearmodel &lm, const xparams _xparams = alglib::xdefault);

`lrprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    LM      -   linear model
    X       -   input vector,  array[0..NVars-1].

Result:
    value of linear model regression estimate

  -- ALGLIB --
     Copyright 03.09.2008 by Bochkanov Sergey
*************************************************************************/
double lrprocess(const linearmodel &lm, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`lrrmserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    LM      -   linear model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double lrrmserror(const linearmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`lrunpack` function

/*************************************************************************
Unpacks coefficients of linear model.

INPUT PARAMETERS:
    LM          -   linear model in ALGLIB format

OUTPUT PARAMETERS:
    V           -   coefficients, array[0..NVars]
                    constant term (intercept) is stored in the V[NVars].
    NVars       -   number of independent variables (one less than number
                    of coefficients)

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
void lrunpack(const linearmodel &lm, real_1d_array &v, ae_int_t &nvars, const xparams _xparams = alglib::xdefault);

Examples: [1]

linreg_d_basic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
        //
        // We have:
        // * xy - matrix of basic function values (exp(0.5*x)) and expected values
        //
        real_2d_array xy = "[[0.606531,1.133719],[0.670320,1.306522],[0.740818,1.504604],[0.818731,1.554663],[0.904837,1.884638],[1.000000,2.072436],[1.105171,2.257285],[1.221403,2.534068],[1.349859,2.622017],[1.491825,2.897713],[1.648721,3.219371]]";
        ae_int_t nvars;
        linearmodel model;
        lrreport rep;
        real_1d_array c;

        lrbuildz(xy, 11, 1, model, rep);
        lrunpack(model, c, nvars);
        printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.98650,0.00000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`logit` subpackage

Classes

logitmodel
mnlreport

Functions

mnlavgce
mnlavgerror
mnlavgrelerror
mnlclserror
mnlpack
mnlprocess
mnlprocessi
mnlrelclserror
mnlrmserror
mnltrainh
mnlunpack

Examples

`logitmodel` class

/*************************************************************************

*************************************************************************/
class logitmodel
{
public:
    logitmodel();
    logitmodel(const logitmodel &rhs);
    logitmodel& operator=(const logitmodel &rhs);
    virtual ~logitmodel();
};

`mnlreport` class

/*************************************************************************
MNLReport structure contains information about training process:
* NGrad     -   number of gradient calculations
* NHess     -   number of Hessian calculations
*************************************************************************/
class mnlreport
{
public:
    mnlreport();
    mnlreport(const mnlreport &rhs);
    mnlreport& operator=(const mnlreport &rhs);
    virtual ~mnlreport();
    ae_int_t ngrad;
    ae_int_t nhess;
};

`mnlavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/(NPoints*ln(2)).

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
double mnlavgce(logitmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mnlavgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average error (error when estimating posterior probabilities).

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double mnlavgerror(logitmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mnlavgrelerror` function

/*************************************************************************
Average relative error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    average relative error (error when estimating posterior probabilities).

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double mnlavgrelerror(logitmodel &lm, const real_2d_array &xy, const ae_int_t ssize, const xparams _xparams = alglib::xdefault);

`mnlclserror` function

/*************************************************************************
Classification error on test set = MNLRelClsError*NPoints

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
ae_int_t mnlclserror(logitmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mnlpack` function

/*************************************************************************
"Packs" coefficients and creates logit model in ALGLIB format (MNLUnpack
reversed).

INPUT PARAMETERS:
    A           -   model (see MNLUnpack)
    NVars       -   number of independent variables
    NClasses    -   number of classes

OUTPUT PARAMETERS:
    LM          -   logit model.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void mnlpack(const real_2d_array &a, const ae_int_t nvars, const ae_int_t nclasses, logitmodel &lm, const xparams _xparams = alglib::xdefault);

`mnlprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    LM      -   logit model, passed by non-constant reference
                (some fields of structure are used as temporaries
                when calculating model output).
    X       -   input vector,  array[0..NVars-1].
    Y       -   (possibly) preallocated buffer; if size of Y is less than
                NClasses, it will be reallocated.If it is large enough, it
                is NOT reallocated, so we can save some time on reallocation.

OUTPUT PARAMETERS:
    Y       -   result, array[0..NClasses-1]
                Vector of posterior probabilities for classification task.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void mnlprocess(logitmodel &lm, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`mnlprocessi` function

/*************************************************************************
'interactive'  variant  of  MNLProcess  for  languages  like  Python which
support constructs like "Y = MNLProcess(LM,X)" and interactive mode of the
interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void mnlprocessi(logitmodel &lm, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`mnlrelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
double mnlrelclserror(logitmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mnlrmserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    LM      -   logit model
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error (error when estimating posterior probabilities).

  -- ALGLIB --
     Copyright 30.08.2008 by Bochkanov Sergey
*************************************************************************/
double mnlrmserror(logitmodel &lm, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mnltrainh` function

/*************************************************************************
This subroutine trains logit model.

INPUT PARAMETERS:
    XY          -   training set, array[0..NPoints-1,0..NVars]
                    First NVars columns store values of independent
                    variables, next column stores number of class (from 0
                    to NClasses-1) which dataset element belongs to. Fractional
                    values are rounded to nearest integer.
    NPoints     -   training set size, NPoints>=1
    NVars       -   number of independent variables, NVars>=1
    NClasses    -   number of classes, NClasses>=2

OUTPUT PARAMETERS:
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<NVars+2, NVars<1, NClasses<2).
                    *  1, if task has been solved
    LM          -   model built
    Rep         -   training report

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void mnltrainh(const real_2d_array &xy, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nclasses, ae_int_t &info, logitmodel &lm, mnlreport &rep, const xparams _xparams = alglib::xdefault);

`mnlunpack` function

/*************************************************************************
Unpacks coefficients of logit model. Logit model have form:

    P(class=i) = S(i) / (S(0) + S(1) + ... +S(M-1))
          S(i) = Exp(A[i,0]*X[0] + ... + A[i,N-1]*X[N-1] + A[i,N]), when i<M-1
        S(M-1) = 1

INPUT PARAMETERS:
    LM          -   logit model in ALGLIB format

OUTPUT PARAMETERS:
    V           -   coefficients, array[0..NClasses-2,0..NVars]
    NVars       -   number of independent variables
    NClasses    -   number of classes

  -- ALGLIB --
     Copyright 10.09.2008 by Bochkanov Sergey
*************************************************************************/
void mnlunpack(const logitmodel &lm, real_2d_array &a, ae_int_t &nvars, ae_int_t &nclasses, const xparams _xparams = alglib::xdefault);

`lsfit` subpackage

Classes

barycentricfitreport
lsfitreport
lsfitstate
polynomialfitreport

Functions

barycentricfitfloaterhormann
barycentricfitfloaterhormannwc
logisticcalc4
logisticcalc5
logisticfit4
logisticfit45x
logisticfit4ec
logisticfit5
logisticfit5ec
lsfitcreatef
lsfitcreatefg
lsfitcreatewf
lsfitcreatewfg
lsfitfit
lsfititeration
lsfitlinear
lsfitlinearc
lsfitlinearw
lsfitlinearwc
lsfitresults
lsfitsetbc
lsfitsetcond
lsfitsetgradientcheck
lsfitsetlc
lsfitsetnonmonotonicsteps
lsfitsetnumdiff
lsfitsetscale
lsfitsetstpmax
lsfitsetxrep
lstfitpiecewiselinearrdp
lstfitpiecewiselinearrdpfixed
polynomialfit
polynomialfitwc
spline1dfitcubicwc
spline1dfithermitedeprecated
spline1dfithermitewc

Examples

lsfit_d_lin		Unconstrained (general) linear least squares fitting with and without weights
lsfit_d_linc		Constrained (general) linear least squares fitting with and without weights
lsfit_d_nlf		Nonlinear fitting using function value only
lsfit_d_nlfb		Bound contstrained nonlinear fitting using function value only
lsfit_d_nlfg		Nonlinear fitting using gradient
lsfit_d_nlscale		Nonlinear fitting with custom scaling and bound constraints
lsfit_d_pol		Unconstrained polynomial fitting
lsfit_d_polc		Constrained polynomial fitting
lsfit_d_spline		Unconstrained fitting by penalized regression spline
lsfit_t_4pl		4-parameter logistic fitting
lsfit_t_5pl		5-parameter logistic fitting

`barycentricfitreport` class

/*************************************************************************
Barycentric fitting report:
    TerminationType completion code: >0 for success, <0 for failure
    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error
    TaskRCond       reciprocal of task's condition number
*************************************************************************/
class barycentricfitreport
{
public:
    barycentricfitreport();
    barycentricfitreport(const barycentricfitreport &rhs);
    barycentricfitreport& operator=(const barycentricfitreport &rhs);
    virtual ~barycentricfitreport();
    ae_int_t terminationtype;
    double taskrcond;
    ae_int_t dbest;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
};

`lsfitreport` class

/*************************************************************************
Least squares fitting report. This structure contains informational fields
which are set by fitting functions provided by this unit.

Different functions initialize different sets of  fields,  so  you  should
read documentation on specific function you used in order  to  know  which
fields are initialized.

    TerminationType filled by all solvers:
                    * positive values, usually 1, denote success
                    * negative values denote various failure scenarios

    TaskRCond       reciprocal of task's condition number
    IterationsCount number of internal iterations

    VarIdx          if user-supplied gradient contains errors  which  were
                    detected by nonlinear fitter, this  field  is  set  to
                    index  of  the  first  component  of gradient which is
                    suspected to be spoiled by bugs.

    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error

    WRMSError       weighted RMS error

    CovPar          covariance matrix for parameters, filled by some solvers
    ErrPar          vector of errors in parameters, filled by some solvers
    ErrCurve        vector of fit errors -  variability  of  the  best-fit
                    curve, filled by some solvers.
    Noise           vector of per-point noise estimates, filled by
                    some solvers.
    R2              coefficient of determination (non-weighted, non-adjusted),
                    filled by some solvers.
*************************************************************************/
class lsfitreport
{
public:
    lsfitreport();
    lsfitreport(const lsfitreport &rhs);
    lsfitreport& operator=(const lsfitreport &rhs);
    virtual ~lsfitreport();
    ae_int_t terminationtype;
    double taskrcond;
    ae_int_t iterationscount;
    ae_int_t varidx;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
    double wrmserror;
    real_2d_array covpar;
    real_1d_array errpar;
    real_1d_array errcurve;
    real_1d_array noise;
    double r2;
};

`lsfitstate` class

/*************************************************************************
Nonlinear fitter.

You should use ALGLIB functions to work with fitter.
Never try to access its fields directly!
*************************************************************************/
class lsfitstate
{
public:
    lsfitstate();
    lsfitstate(const lsfitstate &rhs);
    lsfitstate& operator=(const lsfitstate &rhs);
    virtual ~lsfitstate();
};

`polynomialfitreport` class

/*************************************************************************
Polynomial fitting report:
    TerminationType completion code: >0 for success, <0 for failure
    TaskRCond       reciprocal of task's condition number
    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error
*************************************************************************/
class polynomialfitreport
{
public:
    polynomialfitreport();
    polynomialfitreport(const polynomialfitreport &rhs);
    polynomialfitreport& operator=(const polynomialfitreport &rhs);
    virtual ~polynomialfitreport();
    ae_int_t terminationtype;
    double taskrcond;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
};

`barycentricfitfloaterhormann` function

/*************************************************************************
Rational least squares fitting using  Floater-Hormann  rational  functions
with optimal D chosen from [0,9].

Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
functions. Different values of D are tried, optimal  D  (least  root  mean
square error) is chosen.  Task  is  linear, so linear least squares solver
is used. Complexity  of  this  computational  scheme is  O(N*M^2)  (mostly
dominated by the least squares solver).

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of points, N>0.
    M   -   number of basis functions ( = number_of_nodes), M>=2.

OUTPUT PARAMETERS:
    B   -   barycentric interpolant.
    Rep -   fitting report. The following fields are set:
            * Rep.TerminationType is a completion code, always set to 1
            * DBest         best value of the D parameter
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricfitfloaterhormann(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, barycentricinterpolant &b, barycentricfitreport &rep, const xparams _xparams = alglib::xdefault);

`barycentricfitfloaterhormannwc` function

/*************************************************************************
Weghted rational least  squares  fitting  using  Floater-Hormann  rational
functions  with  optimal  D  chosen  from  [0,9],  with  constraints   and
individual weights.

Equidistant  grid  with M node on [min(x),max(x)]  is  used to build basis
functions. Different values of D are tried, optimal D (least WEIGHTED root
mean square error) is chosen.  Task  is  linear,  so  linear least squares
solver  is  used.  Complexity  of  this  computational  scheme is O(N*M^2)
(mostly dominated by the least squares solver).

SEE ALSO
* BarycentricFitFloaterHormann(), "lightweight" fitting without invididual
  weights and constraints.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points, N>0.
    XC  -   points where function values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that S(XC[i])=YC[i]
            * DC[i]=1   means that S'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints, 0<=K<M.
            K=0 means no constraints (XC/YC/DC are not used in such cases)
    M   -   number of basis functions ( = number_of_nodes), M>=2.

OUTPUT PARAMETERS:
    B   -   barycentric interpolant. Undefined for rep.terminationtype<0.
    Rep -   fitting report. The following fields are set:
            * Rep.TerminationType is a completion code:
              * set to  1 on success
              * set to -3 on failure due to  problematic  constraints:
                either too many  constraints,  degenerate  constraints
                or inconsistent constraints were passed
            * DBest         best value of the D parameter
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error
                            NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroutine doesn't calculate task's condition number for K<>0.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained barycentric interpolants:
* excessive  constraints  can  be  inconsistent.   Floater-Hormann   basis
  functions aren't as flexible as splines (although they are very smooth).
* the more evenly constraints are spread across [min(x),max(x)],  the more
  chances that they will be consistent
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints on the function  VALUES at the interval
  boundaries. Note that consustency of the  constraints  on  the  function
  DERIVATIVES is NOT guaranteed (you can use in such cases  cubic  splines
  which are more flexible).
* another  special  case  is ONE constraint on the function value (OR, but
  not AND, derivative) anywhere in the interval

Our final recommendation is to use constraints  WHEN  AND  ONLY  WHEN  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricfitfloaterhormannwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, barycentricinterpolant &b, barycentricfitreport &rep, const xparams _xparams = alglib::xdefault);

`logisticcalc4` function

/*************************************************************************
This function calculates value of four-parameter logistic (4PL)  model  at
specified point X. 4PL model has following form:

    F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B))

INPUT PARAMETERS:
    X       -   current point, X>=0:
                * zero X is correctly handled even for B<=0
                * negative X results in exception.
    A, B, C, D- parameters of 4PL model:
                * A is unconstrained
                * B is unconstrained; zero or negative values are handled
                  correctly.
                * C>0, non-positive value results in exception
                * D is unconstrained

RESULT:
    model value at X

NOTE: if B=0, denominator is assumed to be equal to 2.0 even  for  zero  X
      (strictly speaking, 0^0 is undefined).

NOTE: this function also throws exception  if  all  input  parameters  are
      correct, but overflow was detected during calculations.

NOTE: this function performs a lot of checks;  if  you  need  really  high
      performance, consider evaluating model  yourself,  without  checking
      for degenerate cases.


  -- ALGLIB PROJECT --
     Copyright 14.05.2014 by Bochkanov Sergey
*************************************************************************/
double logisticcalc4(const double x, const double a, const double b, const double c, const double d, const xparams _xparams = alglib::xdefault);

Examples: [1]

`logisticcalc5` function

/*************************************************************************
This function calculates value of five-parameter logistic (5PL)  model  at
specified point X. 5PL model has following form:

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)

INPUT PARAMETERS:
    X       -   current point, X>=0:
                * zero X is correctly handled even for B<=0
                * negative X results in exception.
    A, B, C, D, G- parameters of 5PL model:
                * A is unconstrained
                * B is unconstrained; zero or negative values are handled
                  correctly.
                * C>0, non-positive value results in exception
                * D is unconstrained
                * G>0, non-positive value results in exception

RESULT:
    model value at X

NOTE: if B=0, denominator is assumed to be equal to Power(2.0,G) even  for
      zero X (strictly speaking, 0^0 is undefined).

NOTE: this function also throws exception  if  all  input  parameters  are
      correct, but overflow was detected during calculations.

NOTE: this function performs a lot of checks;  if  you  need  really  high
      performance, consider evaluating model  yourself,  without  checking
      for degenerate cases.


  -- ALGLIB PROJECT --
     Copyright 14.05.2014 by Bochkanov Sergey
*************************************************************************/
double logisticcalc5(const double x, const double a, const double b, const double c, const double d, const double g, const xparams _xparams = alglib::xdefault);

Examples: [1]

`logisticfit4` function

/*************************************************************************
This function fits four-parameter logistic (4PL) model  to  data  provided
by user. 4PL model has following form:

    F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B))

Here:
    * A, D - unconstrained (see LogisticFit4EC() for constrained 4PL)
    * B>=0
    * C>0

IMPORTANT: output of this function is constrained in  such  way that  B>0.
           Because 4PL model is symmetric with respect to B, there  is  no
           need to explore  B<0.  Constraining  B  makes  algorithm easier
           to stabilize and debug.
           Users  who  for  some  reason  prefer to work with negative B's
           should transform output themselves (swap A and D, replace B  by
           -B).

4PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints. Results from the previous round are used as initial guess.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.

OUTPUT PARAMETERS:
    A, B, C, D- parameters of 4PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: for stability reasons the B parameter is restricted by [1/1000,1000]
      range. It prevents  algorithm from making trial steps  deep into the
      area of bad parameters.

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc4() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void logisticfit4(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, double &a, double &b, double &c, double &d, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`logisticfit45x` function

/*************************************************************************
This is "expert" 4PL/5PL fitting function, which can be used if  you  need
better control over fitting process than provided  by  LogisticFit4()  or
LogisticFit5().

This function fits model of the form

    F(x|A,B,C,D)   = D+(A-D)/(1+Power(x/C,B))           (4PL model)

or

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)    (5PL model)

Here:
    * A, D - unconstrained
    * B>=0 for 4PL, unconstrained for 5PL
    * C>0
    * G>0 (if present)

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.
    CnstrLeft-  optional equality constraint for model value at the   left
                boundary (at X=0). Specify NAN (Not-a-Number)  if  you  do
                not need constraint on the model value at X=0 (in C++  you
                can pass alglib::fp_nan as parameter, in  C#  it  will  be
                Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    CnstrRight- optional equality constraint for model value at X=infinity.
                Specify NAN (Not-a-Number) if you do not  need  constraint
                on the model value (in C++  you can pass alglib::fp_nan as
                parameter, in  C# it will  be Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    Is4PL   -   whether 4PL or 5PL models are fitted
    LambdaV -   regularization coefficient, LambdaV>=0.
                Set it to zero unless you know what you are doing.
    EpsX    -   stopping condition (step size), EpsX>=0.
                Zero value means that small step is automatically chosen.
                See notes below for more information.
    RsCnt   -   number of repeated restarts from  random  points.  4PL/5PL
                models are prone to problem of bad local extrema. Utilizing
                multiple random restarts allows  us  to  improve algorithm
                convergence.
                RsCnt>=0.
                Zero value means that function automatically choose  small
                amount of restarts (recommended).

OUTPUT PARAMETERS:
    A, B, C, D- parameters of 4PL model
    G       -   parameter of 5PL model; for Is4PL=True, G=1 is returned.
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: for better stability B  parameter is restricted by [+-1/1000,+-1000]
      range, and G is restricted by [1/10,10] range. It prevents algorithm
      from making trial steps deep into the area of bad parameters.

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc5() function.

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.

EQUALITY CONSTRAINTS ON PARAMETERS

4PL/5PL solver supports equality constraints on model values at  the  left
boundary (X=0) and right  boundary  (X=infinity).  These  constraints  are
completely optional and you can specify both of them, only  one  -  or  no
constraints at all.

Parameter  CnstrLeft  contains  left  constraint (or NAN for unconstrained
fitting), and CnstrRight contains right  one.  For  4PL,  left  constraint
ALWAYS corresponds to parameter A, and right one is ALWAYS  constraint  on
D. That's because 4PL model is normalized in such way that B>=0.

For 5PL model things are different. Unlike  4PL  one,  5PL  model  is  NOT
symmetric with respect to  change  in  sign  of  B. Thus, negative B's are
possible, and left constraint may constrain parameter A (for positive B's)
- or parameter D (for negative B's). Similarly changes  meaning  of  right
constraint.

You do not have to decide what parameter to  constrain  -  algorithm  will
automatically determine correct parameters as fitting progresses. However,
question highlighted above is important when you interpret fitting results.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void logisticfit45x(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const double cnstrleft, const double cnstrright, const bool is4pl, const double lambdav, const double epsx, const ae_int_t rscnt, double &a, double &b, double &c, double &d, double &g, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

`logisticfit4ec` function

/*************************************************************************
This function fits four-parameter logistic (4PL) model  to  data  provided
by user, with optional constraints on parameters A and D.  4PL  model  has
following form:

    F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B))

Here:
    * A, D - with optional equality constraints
    * B>=0
    * C>0

IMPORTANT: output of this function is constrained in  such  way that  B>0.
           Because 4PL model is symmetric with respect to B, there  is  no
           need to explore  B<0.  Constraining  B  makes  algorithm easier
           to stabilize and debug.
           Users  who  for  some  reason  prefer to work with negative B's
           should transform output themselves (swap A and D, replace B  by
           -B).

4PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints. Results from the previous round are used as initial guess.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.
    CnstrLeft-  optional equality constraint for model value at the   left
                boundary (at X=0). Specify NAN (Not-a-Number)  if  you  do
                not need constraint on the model value at X=0 (in C++  you
                can pass alglib::fp_nan as parameter, in  C#  it  will  be
                Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    CnstrRight- optional equality constraint for model value at X=infinity.
                Specify NAN (Not-a-Number) if you do not  need  constraint
                on the model value (in C++  you can pass alglib::fp_nan as
                parameter, in  C# it will  be Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.

OUTPUT PARAMETERS:
    A, B, C, D- parameters of 4PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: for stability reasons the B parameter is restricted by [1/1000,1000]
      range. It prevents  algorithm from making trial steps  deep into the
      area of bad parameters.

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc4() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.

EQUALITY CONSTRAINTS ON PARAMETERS

4PL/5PL solver supports equality constraints on model values at  the  left
boundary (X=0) and right  boundary  (X=infinity).  These  constraints  are
completely optional and you can specify both of them, only  one  -  or  no
constraints at all.

Parameter  CnstrLeft  contains  left  constraint (or NAN for unconstrained
fitting), and CnstrRight contains right  one.  For  4PL,  left  constraint
ALWAYS corresponds to parameter A, and right one is ALWAYS  constraint  on
D. That's because 4PL model is normalized in such way that B>=0.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void logisticfit4ec(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const double cnstrleft, const double cnstrright, double &a, double &b, double &c, double &d, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

`logisticfit5` function

/*************************************************************************
This function fits five-parameter logistic (5PL) model  to  data  provided
by user. 5PL model has following form:

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)

Here:
    * A, D - unconstrained
    * B - unconstrained
    * C>0
    * G>0

IMPORTANT: unlike in  4PL  fitting,  output  of  this  function   is   NOT
           constrained in  such  way that B is guaranteed to be  positive.
           Furthermore,  unlike  4PL,  5PL  model  is  NOT  symmetric with
           respect to B, so you can NOT transform model to equivalent one,
           with B having desired sign (>0 or <0).

5PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".  Parameter  G  is
  fixed at G=1.
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints on B and C, but with G still equal to 1.  Results  from  the
  previous round are used as initial guess.
* third Levenberg-Marquardt round relaxes constraints on G  and  tries  two
  different models - one with B>0 and one with B<0.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.

OUTPUT PARAMETERS:
    A,B,C,D,G-  parameters of 5PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: for better stability B  parameter is restricted by [+-1/1000,+-1000]
      range, and G is restricted by [1/10,10] range. It prevents algorithm
      from making trial steps deep into the area of bad parameters.

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc5() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void logisticfit5(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, double &a, double &b, double &c, double &d, double &g, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`logisticfit5ec` function

/*************************************************************************
This function fits five-parameter logistic (5PL) model  to  data  provided
by user, subject to optional equality constraints on parameters A  and  D.
5PL model has following form:

    F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G)

Here:
    * A, D - with optional equality constraints
    * B - unconstrained
    * C>0
    * G>0

IMPORTANT: unlike in  4PL  fitting,  output  of  this  function   is   NOT
           constrained in  such  way that B is guaranteed to be  positive.
           Furthermore,  unlike  4PL,  5PL  model  is  NOT  symmetric with
           respect to B, so you can NOT transform model to equivalent one,
           with B having desired sign (>0 or <0).

5PL fitting is implemented as follows:
* we perform small number of restarts from random locations which helps to
  solve problem of bad local extrema. Locations are only partially  random
  - we use input data to determine good  initial  guess,  but  we  include
  controlled amount of randomness.
* we perform Levenberg-Marquardt fitting with very  tight  constraints  on
  parameters B and C - it allows us to find good  initial  guess  for  the
  second stage without risk of running into "flat spot".  Parameter  G  is
  fixed at G=1.
* second  Levenberg-Marquardt  round  is   performed   without   excessive
  constraints on B and C, but with G still equal to 1.  Results  from  the
  previous round are used as initial guess.
* third Levenberg-Marquardt round relaxes constraints on G  and  tries  two
  different models - one with B>0 and one with B<0.
* after fitting is done, we compare results with best values found so far,
  rewrite "best solution" if needed, and move to next random location.

Overall algorithm is very stable and is not prone to  bad  local  extrema.
Furthermore, it automatically scales when input data have  very  large  or
very small range.

INPUT PARAMETERS:
    X       -   array[N], stores X-values.
                MUST include only non-negative numbers  (but  may  include
                zero values). Can be unsorted.
    Y       -   array[N], values to fit.
    N       -   number of points. If N is less than  length  of  X/Y, only
                leading N elements are used.
    CnstrLeft-  optional equality constraint for model value at the   left
                boundary (at X=0). Specify NAN (Not-a-Number)  if  you  do
                not need constraint on the model value at X=0 (in C++  you
                can pass alglib::fp_nan as parameter, in  C#  it  will  be
                Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.
    CnstrRight- optional equality constraint for model value at X=infinity.
                Specify NAN (Not-a-Number) if you do not  need  constraint
                on the model value (in C++  you can pass alglib::fp_nan as
                parameter, in  C# it will  be Double.NaN).
                See  below,  section  "EQUALITY  CONSTRAINTS"   for   more
                information about constraints.

OUTPUT PARAMETERS:
    A,B,C,D,G-  parameters of 5PL model
    Rep     -   fitting report. This structure has many fields,  but  ONLY
                ONES LISTED BELOW ARE SET:
                * Rep.IterationsCount - number of iterations performed
                * Rep.RMSError - root-mean-square error
                * Rep.AvgError - average absolute error
                * Rep.AvgRelError - average relative error (calculated for
                  non-zero Y-values)
                * Rep.MaxError - maximum absolute error
                * Rep.R2 - coefficient of determination,  R-squared.  This
                  coefficient   is  calculated  as  R2=1-RSS/TSS  (in case
                  of nonlinear  regression  there  are  multiple  ways  to
                  define R2, each of them giving different results).

NOTE: for better stability B  parameter is restricted by [+-1/1000,+-1000]
      range, and G is restricted by [1/10,10] range. It prevents algorithm
      from making trial steps deep into the area of bad parameters.

NOTE: after  you  obtained  coefficients,  you  can  evaluate  model  with
      LogisticCalc5() function.

NOTE: if you need better control over fitting process than provided by this
      function, you may use LogisticFit45X().

NOTE: step is automatically scaled according to scale of parameters  being
      fitted before we compare its length with EpsX. Thus,  this  function
      can be used to fit data with very small or very large values without
      changing EpsX.

EQUALITY CONSTRAINTS ON PARAMETERS

5PL solver supports equality constraints on model  values  at   the   left
boundary (X=0) and right  boundary  (X=infinity).  These  constraints  are
completely optional and you can specify both of them, only  one  -  or  no
constraints at all.

Parameter  CnstrLeft  contains  left  constraint (or NAN for unconstrained
fitting), and CnstrRight contains right  one.

Unlike 4PL one, 5PL model is NOT symmetric with respect to  change in sign
of B. Thus, negative B's are possible, and left constraint  may  constrain
parameter A (for positive B's)  -  or  parameter  D  (for  negative  B's).
Similarly changes meaning of right constraint.

You do not have to decide what parameter to  constrain  -  algorithm  will
automatically determine correct parameters as fitting progresses. However,
question highlighted above is important when you interpret fitting results.


  -- ALGLIB PROJECT --
     Copyright 14.02.2014 by Bochkanov Sergey
*************************************************************************/
void logisticfit5ec(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const double cnstrleft, const double cnstrright, double &a, double &b, double &c, double &d, double &g, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

`lsfitcreatef` function

/*************************************************************************
Nonlinear least squares fitting using function values only.

Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.

Nonlinear task min(F(c)) is solved, where

    F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]).

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted
    DiffStep-   numerical differentiation step, >0; Obviously,  step  size
                should not be too large in order to get a  good  numerical
                derivative. However, it also  should  not  be   too  small
                because numerical errors are greatly amplified by numerical
                differentiation.
                By default, symmetric 3-point formula which provides  good
                accuracy is used. It can be changed to a  faster  but less
                precise 2-point one with minlmsetnumdiff() function.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

IMPORTANT: the LSFIT optimizer  supports  parallel  model  evaluation  and
           parallel numerical  differentiation  ('callback  parallelism').
           This feature, which is present in commercial  ALGLIB  editions,
           greatly accelerates fits with large datasets  and/or  expensive
           target functions.

           Callback parallelism is usually beneficial when a  single  pass
           over  the   entire   dataset   requires   more   than   several
           milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial  version'
           section,  and  comments  on  lsfitfit()   function   for   more
           information.

  -- ALGLIB --
     Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatef(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, const double diffstep, lsfitstate &state, const xparams _xparams = alglib::xdefault);
void lsfitcreatef(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const double diffstep, lsfitstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`lsfitcreatefg` function

/*************************************************************************
Nonlinear least squares fitting using gradient only, without individual
weights.

Nonlinear task min(F(c)) is solved, where

    F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]) and its gradient.

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

IMPORTANT: the LSFIT optimizer  supports  parallel  model  evaluation  and
           parallel numerical  differentiation  ('callback  parallelism').
           This feature, which is present in commercial  ALGLIB  editions,
           greatly accelerates fits with large datasets  and/or  expensive
           target functions.

           Callback parallelism is usually beneficial when a  single  pass
           over  the   entire   dataset   requires   more   than   several
           milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial  version'
           section,  and  comments  on  lsfitfit()   function   for   more
           information.

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatefg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, lsfitstate &state, const xparams _xparams = alglib::xdefault);
void lsfitcreatefg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &c, lsfitstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitcreatewf` function

/*************************************************************************
Weighted nonlinear least squares fitting using function values only.

Combination of numerical differentiation and secant updates is used to
obtain function Jacobian.

Nonlinear task min(F(c)) is solved, where

    F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]).

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    W       -   weights, array[0..N-1]
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted
    DiffStep-   numerical differentiation step, >0; Obviously,  step  size
                should not be too large in order to get a  good  numerical
                derivative. However, it also  should  not  be   too  small
                because numerical errors are greatly amplified by numerical
                differentiation.
                By default, symmetric 3-point formula which provides  good
                accuracy is used. It can be changed to a  faster  but less
                precise 2-point one with minlmsetnumdiff() function.


OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

IMPORTANT: the LSFIT optimizer  supports  parallel  model  evaluation  and
           parallel numerical  differentiation  ('callback  parallelism').
           This feature, which is present in commercial  ALGLIB  editions,
           greatly accelerates fits with large datasets  and/or  expensive
           target functions.

           Callback parallelism is usually beneficial when a  single  pass
           over  the   entire   dataset   requires   more   than   several
           milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial  version'
           section,  and  comments  on  lsfitfit()   function   for   more
           information.

  -- ALGLIB --
     Copyright 18.10.2008 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewf(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, const double diffstep, lsfitstate &state, const xparams _xparams = alglib::xdefault);
void lsfitcreatewf(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const double diffstep, lsfitstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`lsfitcreatewfg` function

/*************************************************************************
Weighted nonlinear least squares fitting using gradient only.

Nonlinear task min(F(c)) is solved, where

    F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2,

    * N is a number of points,
    * M is a dimension of a space points belong to,
    * K is a dimension of a space of parameters being fitted,
    * w is an N-dimensional vector of weight coefficients,
    * x is a set of N points, each of them is an M-dimensional vector,
    * c is a K-dimensional vector of parameters being fitted

This subroutine uses only f(c,x[i]) and its gradient.

INPUT PARAMETERS:
    X       -   array[0..N-1,0..M-1], points (one row = one point)
    Y       -   array[0..N-1], function values.
    W       -   weights, array[0..N-1]
    C       -   array[0..K-1], initial approximation to the solution,
    N       -   number of points, N>1
    M       -   dimension of space
    K       -   number of parameters being fitted

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

See also:
    LSFitResults
    LSFitCreateFG (fitting without weights)
    LSFitCreateWFGH (fitting using Hessian)

IMPORTANT: the LSFIT optimizer  supports  parallel  model  evaluation  and
           parallel numerical  differentiation  ('callback  parallelism').
           This feature, which is present in commercial  ALGLIB  editions,
           greatly accelerates fits with large datasets  and/or  expensive
           target functions.

           Callback parallelism is usually beneficial when a  single  pass
           over  the   entire   dataset   requires   more   than   several
           milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial  version'
           section,  and  comments  on  lsfitfit()   function   for   more
           information.

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitcreatewfg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, const ae_int_t n, const ae_int_t m, const ae_int_t k, lsfitstate &state, const xparams _xparams = alglib::xdefault);
void lsfitcreatewfg(const real_2d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &c, lsfitstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitfit` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear fitter

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


CALLBACK PARALLELISM:

The  LSFIT  optimizer  supports  parallel  model  evaluation  and parallel
numerical differentiation ('callback parallelism'). This feature, which is
present in commercial ALGLIB editions, greatly accelerates fits with large
datasets and/or expensive target functions.

Callback parallelism is usually beneficial when a  single  pass  over  the
entire dataset requires more than several milliseconds. In this  case  the
job of computing model values at  dataset  points  can  be  split  between
multiple threads.

If you employ a numerical differentiation scheme, you can also parallelize
computation of different components of a numerical gradient. Generally, the
mode computationally demanding your problem is (many points, numerical differentiation,
expensive model), the more you can get for multithreading.

ALGLIB Reference Manual, 'Working with commercial  version' section,
describes how to activate callback parallelism for your programming language.

CALLBACK ARGUMENTS

This algorithm is somewhat unusual because  it  works  with  parameterized
function f(C,X), where X is  a  function  argument (we  have  many  points
which  are  characterized  by different  argument  values),  and  C  is  a
parameter to fit.

For example, if we want to do linear  fit  by  f(c0,c1,x) = c0*x+c1,  then
x will be argument, and {c0,c1} will be parameters.

It is important to understand that this algorithm finds   minimum  in  the
space of function PARAMETERS (not  arguments),  so  it  needs  derivatives
of f() with respect to C, not X.

In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1}
instead of {df/dx} = {c0}.

  -- ALGLIB --
     Copyright 17.12.2023 by Bochkanov Sergey
*************************************************************************/
void lsfitfit(lsfitstate &state,
    void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &c, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void lsfitfit(lsfitstate &state,
    void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr),
    void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &c, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`lsfititeration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool lsfititeration(lsfitstate &state, const xparams _xparams = alglib::xdefault);

`lsfitlinear` function

/*************************************************************************
Linear least squares fitting.

QR decomposition is used to reduce task to MxM, then triangular solver  or
SVD-based solver is used depending on condition number of the  system.  It
allows to maximize speed and retain decent accuracy.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I, J] - value of J-th basis function in I-th point.
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.

OUTPUT PARAMETERS:
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * Rep.TerminationType is a completion code, always set  to
                  1 which denotes success
                * Rep.TaskRCond     reciprocal of condition number
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinear(const real_1d_array &y, const real_2d_array &fmatrix, const ae_int_t n, const ae_int_t m, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);
void lsfitlinear(const real_1d_array &y, const real_2d_array &fmatrix, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitlinearc` function

/*************************************************************************
Constained linear least squares fitting.

This  is  variation  of LSFitLinear(),  which searchs for min|A*x=b| given
that  K  additional  constaints  C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints,  then  LSFitLinear()
is called.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I,J] - value of J-th basis function in I-th point.
    CMatrix -   a table of constaints, array[0..K-1,0..M].
                I-th row of CMatrix corresponds to I-th linear constraint:
                CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.
    K       -   number of constraints, 0 <= K < M
                K=0 corresponds to absence of constraints.

OUTPUT PARAMETERS:
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * Rep.TerminationType is a completion code:
                  * set to  1 on success
                  * set to -3 on failure due to  problematic  constraints:
                    either too many  constraints (M or  more),  degenerate
                    constraints (some constraints are repetead  twice)  or
                    inconsistent constraints are specified
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

IMPORTANT:  errors  in  parameters  are  calculated  without  taking  into
            account boundary/linear constraints! Presence  of  constraints
            changes distribution of errors, but there is no  easy  way  to
            account for constraints when you calculate covariance matrix.

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearc(const real_1d_array &y, const real_2d_array &fmatrix, const real_2d_array &cmatrix, const ae_int_t n, const ae_int_t m, const ae_int_t k, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);
void lsfitlinearc(const real_1d_array &y, const real_2d_array &fmatrix, const real_2d_array &cmatrix, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitlinearw` function

/*************************************************************************
Weighted linear least squares fitting.

QR decomposition is used to reduce task to MxM, then triangular solver  or
SVD-based solver is used depending on condition number of the  system.  It
allows to maximize speed and retain decent accuracy.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    W       -   array[0..N-1]  Weights  corresponding to function  values.
                Each summand in square  sum  of  approximation  deviations
                from  given  values  is  multiplied  by  the   square   of
                corresponding weight.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I, J] - value of J-th basis function in I-th point.
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.

OUTPUT PARAMETERS:
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. Following fields are set:
                * Rep.TerminationType always set to 1 (success)
                * Rep.TaskRCond     reciprocal of condition number
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearw(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, const ae_int_t n, const ae_int_t m, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);
void lsfitlinearw(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitlinearwc` function

/*************************************************************************
Weighted constained linear least squares fitting.

This  is  variation  of LSFitLinearW(), which searchs for min|A*x=b| given
that  K  additional  constaints  C*x=bc are satisfied. It reduces original
task to modified one: min|B*y-d| WITHOUT constraints,  then LSFitLinearW()
is called.

IMPORTANT: if you want to perform  polynomial  fitting,  it  may  be  more
           convenient to use PolynomialFit() function. This function gives
           best  results  on  polynomial  problems  and  solves  numerical
           stability  issues  which  arise  when   you   fit   high-degree
           polynomials to your data.

INPUT PARAMETERS:
    Y       -   array[0..N-1] Function values in  N  points.
    W       -   array[0..N-1]  Weights  corresponding to function  values.
                Each summand in square  sum  of  approximation  deviations
                from  given  values  is  multiplied  by  the   square   of
                corresponding weight.
    FMatrix -   a table of basis functions values, array[0..N-1, 0..M-1].
                FMatrix[I,J] - value of J-th basis function in I-th point.
    CMatrix -   a table of constaints, array[0..K-1,0..M].
                I-th row of CMatrix corresponds to I-th linear constraint:
                CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M]
    N       -   number of points used. N>=1.
    M       -   number of basis functions, M>=1.
    K       -   number of constraints, 0 <= K < M
                K=0 corresponds to absence of constraints.

OUTPUT PARAMETERS:
    C       -   decomposition coefficients, array[0..M-1]
    Rep     -   fitting report. The following fields are set:
                * Rep.TerminationType is a completion code:
                  * set to  1 on success
                  * set to -3 on failure due to  problematic  constraints:
                    either too many  constraints (M or  more),  degenerate
                    constraints (some constraints are repetead  twice)  or
                    inconsistent constraints are specified
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(F*CovPar*F')),
                    where F is functions matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

IMPORTANT:  errors  in  parameters  are  calculated  without  taking  into
            account boundary/linear constraints! Presence  of  constraints
            changes distribution of errors, but there is no  easy  way  to
            account for constraints when you calculate covariance matrix.

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 07.09.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitlinearwc(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, const real_2d_array &cmatrix, const ae_int_t n, const ae_int_t m, const ae_int_t k, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);
void lsfitlinearwc(const real_1d_array &y, const real_1d_array &w, const real_2d_array &fmatrix, const real_2d_array &cmatrix, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitresults` function

/*************************************************************************
Nonlinear least squares fitting results.

Called after return from LSFitFit().

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    C       -   array[K], solution
    Rep     -   optimization report. On success following fields are set:
                * TerminationType   completion code:
                    * -8    optimizer   detected  NAN/INF  in  the  target
                            function and/or gradient
                    * -7    gradient verification failed.
                            See LSFitSetGradientCheck() for more information.
                    * -3    inconsistent constraints
                    *  2    relative step is no more than EpsX.
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible
                * R2                non-adjusted coefficient of determination
                                    (non-weighted)
                * RMSError          rms error on the (X,Y).
                * AvgError          average error on the (X,Y).
                * AvgRelError       average relative error on the non-zero Y
                * MaxError          maximum error
                                    NON-WEIGHTED ERRORS ARE CALCULATED
                * WRMSError         weighted rms error on the (X,Y).

ERRORS IN PARAMETERS

This  solver  also  calculates different kinds of errors in parameters and
fills corresponding fields of report:
* Rep.CovPar        covariance matrix for parameters, array[K,K].
* Rep.ErrPar        errors in parameters, array[K],
                    errpar = sqrt(diag(CovPar))
* Rep.ErrCurve      vector of fit errors - standard deviations of empirical
                    best-fit curve from "ideal" best-fit curve built  with
                    infinite number of samples, array[N].
                    errcurve = sqrt(diag(J*CovPar*J')),
                    where J is Jacobian matrix.
* Rep.Noise         vector of per-point estimates of noise, array[N]

IMPORTANT:  errors  in  parameters  are  calculated  without  taking  into
            account boundary/linear constraints! Presence  of  constraints
            changes distribution of errors, but there is no  easy  way  to
            account for constraints when you calculate covariance matrix.

NOTE:       noise in the data is estimated as follows:
            * for fitting without user-supplied  weights  all  points  are
              assumed to have same level of noise, which is estimated from
              the data
            * for fitting with user-supplied weights we assume that  noise
              level in I-th point is inversely proportional to Ith weight.
              Coefficient of proportionality is estimated from the data.

NOTE:       we apply small amount of regularization when we invert squared
            Jacobian and calculate covariance matrix. It  guarantees  that
            algorithm won't divide by zero  during  inversion,  but  skews
            error estimates a bit (fractional error is about 10^-9).

            However, we believe that this difference is insignificant  for
            all practical purposes except for the situation when you  want
            to compare ALGLIB results with "reference"  implementation  up
            to the last significant digit.

NOTE:       covariance matrix is estimated using  correction  for  degrees
            of freedom (covariances are divided by N-M instead of dividing
            by N).

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitresults(const lsfitstate &state, real_1d_array &c, lsfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`lsfitsetbc` function

/*************************************************************************
This function sets boundary constraints for underlying optimizer

Boundary constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetBC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[K].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (latter is recommended because
                it will allow solver to use better algorithm).
    BndU    -   upper bounds, array[K].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (latter is recommended because
                it will allow solver to use better algorithm).

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: unlike other constrained optimization algorithms, this solver  has
following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by bound constraints

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void lsfitsetbc(lsfitstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`lsfitsetcond` function

/*************************************************************************
Stopping conditions for nonlinear least squares fitting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by LSFitSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations   is    unlimited.   Only   Levenberg-Marquardt
                iterations  are  counted  (L-BFGS/CG  iterations  are  NOT
                counted because their cost is very low compared to that of
                LM).

NOTE

Passing EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to  automatic
stopping criterion selection (according to the scheme used by MINLM unit).


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void lsfitsetcond(lsfitstate &state, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`lsfitsetgradientcheck` function

/*************************************************************************
This  subroutine  turns  on  verification  of  the  user-supplied analytic
gradient:
* user calls this subroutine before fitting begins
* LSFitFit() is called
* prior to actual fitting, for  each  point  in  data  set  X_i  and  each
  component  of  parameters  being  fited C_j algorithm performs following
  steps:
  * two trial steps are made to C_j-TestStep*S[j] and C_j+TestStep*S[j],
    where C_j is j-th parameter and S[j] is a scale of j-th parameter
  * if needed, steps are bounded with respect to constraints on C[]
  * F(X_i|C) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point
  * in case difference between prediction and actual value is higher  than
    some predetermined threshold, algorithm stops with completion code -7;
    Rep.VarIdx is set to index of the parameter with incorrect derivative.
* after verification is over, algorithm proceeds to the actual optimization.

NOTE 1: verification needs N*K (points count * parameters count)  gradient
        evaluations. It is very costly and you should use it only for  low
        dimensional  problems,  when  you  want  to  be  sure  that you've
        correctly calculated analytic derivatives. You should not  use  it
        in the production code  (unless  you  want  to  check  derivatives
        provided by some third party).

NOTE 2: you  should  carefully  choose  TestStep. Value which is too large
        (so large that function behaviour is significantly non-cubic) will
        lead to false alarms. You may use  different  step  for  different
        parameters by means of setting scale with LSFitSetScale().

NOTE 3: this function may lead to false positives. In case it reports that
        I-th  derivative was calculated incorrectly, you may decrease test
        step  and  try  one  more  time  - maybe your function changes too
        sharply  and  your  step  is  too  large for such rapidly chanding
        function.

NOTE 4: this function works only for optimizers created with LSFitCreateWFG()
        or LSFitCreateFG() constructors.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification

  -- ALGLIB --
     Copyright 15.06.2012 by Bochkanov Sergey
*************************************************************************/
void lsfitsetgradientcheck(lsfitstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

`lsfitsetlc` function

/*************************************************************************
This function sets linear constraints for underlying optimizer

Linear constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetLC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

IMPORTANT: if you have linear constraints, it is strongly  recommended  to
           set scale of variables with lsfitsetscale(). QP solver which is
           used to calculate linearly constrained steps heavily relies  on
           good scaling of input problems.

NOTE: linear  (non-box)  constraints  are  satisfied only approximately  -
      there  always  exists some violation due  to  numerical  errors  and
      algorithmic limitations.

NOTE: general linear constraints  add  significant  overhead  to  solution
      process. Although solver performs roughly same amount of  iterations
      (when compared  with  similar  box-only  constrained  problem), each
      iteration   now    involves  solution  of  linearly  constrained  QP
      subproblem, which requires ~3-5 times more Cholesky  decompositions.
      Thus, if you can reformulate your problem in such way  this  it  has
      only box constraints, it may be beneficial to do so.

  -- ALGLIB --
     Copyright 29.04.2017 by Bochkanov Sergey
*************************************************************************/
void lsfitsetlc(lsfitstate &state, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void lsfitsetlc(lsfitstate &state, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

`lsfitsetnonmonotonicsteps` function

/*************************************************************************
This function is used  to  activate/deactivate  nonmonotonic  steps.  Such
steps may improve  convergence  on  noisy  problems  or  ones  with  minor
smoothness defects.

In  its  standard  mode,  LSFIT solver compares squared errors f[1] at the
trial point with the value at the current  point  f[0].  Only  steps  that
decrease f() are accepted.

When the nonmonotonic mode is activated, f[1]  is  compared  with  maximum
over several previous  locations:  max(f[0],f[-1],...,f[-CNT]).  We  still
accept only steps that decrease  f(),  however  our  reference  value  has
changed. The net results is that f[1]>f[0] are now allowed.

Nonmonotonic steps can help to handle minor defects in the objective (e.g.
small  noise,  discontinuous  jumps  or  nonsmoothness).  However,  it  is
important  that  the  overall  shape  of  the  problem  is  still  smooth.
It  may  also  help  to  minimize  perfectly  smooth  targets with complex
geometries by allowing to jump through curved valleys.

However, sometimes nonmonotonic steps degrade convergence by  allowing  an
optimizer to wander too far away from the solution, so this feature should
be used only after careful testing.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    Cnt     -   nonmonotonic memory length, Cnt>=0:
                * 0 for traditional monotonic steps
                * 2..3 is recommended for the nonmonotonic optimization

  -- ALGLIB --
     Copyright 07.04.2024 by Bochkanov Sergey
*************************************************************************/
void lsfitsetnonmonotonicsteps(lsfitstate &state, const ae_int_t cnt, const xparams _xparams = alglib::xdefault);

`lsfitsetnumdiff` function

/*************************************************************************
This function sets specific finite  difference  formula  to  be  used  for
numerical differentiation.

It works only for optimizers configured to use numerical  differentiation;
in other cases it has no effect.

INPUT PARAMETERS:
    State       -   optimizer
    FormulaType -   formula type:
                    * 3 for a 3-point formula, which is also  known  as  a
                      symmetric difference quotient (the formula  actually
                      uses only two function values per variable:  at  x+h
                      and x-h). A good choice for medium-accuracy  setups,
                      a default option.
                    * 2 for a forward (or backward, depending  on variable
                      bounds)  finite   difference  (f(x+h)-f(x))/h.  This
                      formula has the lowest accuracy. However, it  is  4x
                      faster than the 5-point formula and 2x  faster  than
                      the 3-point one because, in addition to the  central
                      value f(x), it needs only  one  additional  function
                      evaluation per variable.


  -- ALGLIB --
     Copyright 03.12.2024 by Bochkanov Sergey
*************************************************************************/
void lsfitsetnumdiff(lsfitstate &state, const ae_int_t formulatype, const xparams _xparams = alglib::xdefault);

`lsfitsetscale` function

/*************************************************************************
This function sets scaling coefficients for underlying optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Generally, scale is NOT considered to be a form of preconditioner.  But LM
optimizer is unique in that it uses scaling matrix both  in  the  stopping
condition tests and as Marquardt damping factor.

Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it  is
easier  to  converge  when  variables  are  properly  scaled, so premature
stopping is possible when very badly scalled variables are  combined  with
relaxed stopping conditions).

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void lsfitsetscale(lsfitstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`lsfitsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

NOTE: non-zero StpMax leads to moderate  performance  degradation  because
intermediate  step  of  preconditioned L-BFGS optimization is incompatible
with limits on step size.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void lsfitsetstpmax(lsfitstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`lsfitsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

When reports are needed, State.C (current parameters) and State.F (current
value of fitting function) are reported.


  -- ALGLIB --
     Copyright 15.08.2010 by Bochkanov Sergey
*************************************************************************/
void lsfitsetxrep(lsfitstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`lstfitpiecewiselinearrdp` function

/*************************************************************************
This  subroutine fits piecewise linear curve to points with Ramer-Douglas-
Peucker algorithm, which stops after achieving desired precision.

IMPORTANT:
* it performs non-least-squares fitting; it builds curve, but  this  curve
  does not minimize some least squares  metric.  See  description  of  RDP
  algorithm (say, in Wikipedia) for more details on WHAT is performed.
* this function does NOT work with parametric curves  (i.e.  curves  which
  can be represented as {X(t),Y(t)}. It works with curves   which  can  be
  represented as Y(X). Thus, it is impossible to model figures like circles
  with this functions.
  If  you  want  to  work  with  parametric   curves,   you   should   use
  ParametricRDPFixed() function provided  by  "Parametric"  subpackage  of
  "Interpolation" package.

INPUT PARAMETERS:
    X       -   array of X-coordinates:
                * at least N elements
                * can be unordered (points are automatically sorted)
                * this function may accept non-distinct X (see below for
                  more information on handling of such inputs)
    Y       -   array of Y-coordinates:
                * at least N elements
    N       -   number of elements in X/Y
    Eps     -   positive number, desired precision.


OUTPUT PARAMETERS:
    X2      -   X-values of corner points for piecewise approximation,
                has length NSections+1 or zero (for NSections=0).
    Y2      -   Y-values of corner points,
                has length NSections+1 or zero (for NSections=0).
    NSections-  number of sections found by algorithm,
                NSections can be zero for degenerate datasets
                (N<=1 or all X[] are non-distinct).

NOTE: X2/Y2 are ordered arrays, i.e. (X2[0],Y2[0]) is  a  first  point  of
      curve, (X2[NSection-1],Y2[NSection-1]) is the last point.

  -- ALGLIB --
     Copyright 02.10.2014 by Bochkanov Sergey
*************************************************************************/
void lstfitpiecewiselinearrdp(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const double eps, real_1d_array &x2, real_1d_array &y2, ae_int_t &nsections, const xparams _xparams = alglib::xdefault);

`lstfitpiecewiselinearrdpfixed` function

/*************************************************************************
This  subroutine fits piecewise linear curve to points with Ramer-Douglas-
Peucker algorithm, which stops after generating specified number of linear
sections.

IMPORTANT:
* it does NOT perform least-squares fitting; it  builds  curve,  but  this
  curve does not minimize some least squares metric.  See  description  of
  RDP algorithm (say, in Wikipedia) for more details on WHAT is performed.
* this function does NOT work with parametric curves  (i.e.  curves  which
  can be represented as {X(t),Y(t)}. It works with curves   which  can  be
  represented as Y(X). Thus,  it  is  impossible  to  model  figures  like
  circles  with  this  functions.
  If  you  want  to  work  with  parametric   curves,   you   should   use
  ParametricRDPFixed() function provided  by  "Parametric"  subpackage  of
  "Interpolation" package.

INPUT PARAMETERS:
    X       -   array of X-coordinates:
                * at least N elements
                * can be unordered (points are automatically sorted)
                * this function may accept non-distinct X (see below for
                  more information on handling of such inputs)
    Y       -   array of Y-coordinates:
                * at least N elements
    N       -   number of elements in X/Y
    M       -   desired number of sections:
                * at most M sections are generated by this function
                * less than M sections can be generated if we have N<M
                  (or some X are non-distinct).

OUTPUT PARAMETERS:
    X2      -   X-values of corner points for piecewise approximation,
                has length NSections+1 or zero (for NSections=0).
    Y2      -   Y-values of corner points,
                has length NSections+1 or zero (for NSections=0).
    NSections-  number of sections found by algorithm, NSections<=M,
                NSections can be zero for degenerate datasets
                (N<=1 or all X[] are non-distinct).

NOTE: X2/Y2 are ordered arrays, i.e. (X2[0],Y2[0]) is  a  first  point  of
      curve, (X2[NSection-1],Y2[NSection-1]) is the last point.

  -- ALGLIB --
     Copyright 02.10.2014 by Bochkanov Sergey
*************************************************************************/
void lstfitpiecewiselinearrdpfixed(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, real_1d_array &x2, real_1d_array &y2, ae_int_t &nsections, const xparams _xparams = alglib::xdefault);

`polynomialfit` function

/*************************************************************************
Fitting by polynomials in barycentric form. This function provides  simple
unterface for unconstrained unweighted fitting. See  PolynomialFitWC()  if
you need constrained fitting.

The task is linear, thus the linear least  squares  solver  is  used.  The
complexity of this computational scheme is O(N*M^2), mostly  dominated  by
the least squares solver

SEE ALSO:
    PolynomialFitWC()

NOTES:
    you can convert P from barycentric form  to  the  power  or  Chebyshev
    basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions  from
    POLINT subpackage.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of points, N>0
            * if given, only leading N elements of X/Y are used
            * if not given, automatically determined from sizes of X/Y
    M   -   number of basis functions (= polynomial_degree + 1), M>=1

OUTPUT PARAMETERS:
    P   -   interpolant in barycentric form for Rep.TerminationType>0.
            undefined for Rep.TerminationType<0.
    Rep -   fitting report. The following fields are set:
                * Rep.TerminationType is a completion code which is always
                  set to 1 (success)
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfit(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, barycentricinterpolant &p, polynomialfitreport &rep, const xparams _xparams = alglib::xdefault);
void polynomialfit(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, barycentricinterpolant &p, polynomialfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialfitwc` function

/*************************************************************************
Weighted  fitting by polynomials in barycentric form, with constraints  on
function values or first derivatives.

Small regularizing term is used when solving constrained tasks (to improve
stability).

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

SEE ALSO:
    PolynomialFit()

NOTES:
    you can convert P from barycentric form  to  the  power  or  Chebyshev
    basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions  from
    the POLINT subpackage.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points, N>0.
            * if given, only leading N elements of X/Y/W are used
            * if not given, automatically determined from sizes of X/Y/W
    XC  -   points where polynomial values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that P(XC[i])=YC[i]
            * DC[i]=1   means that P'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints, 0<=K<M.
            K=0 means no constraints (XC/YC/DC are not used in such cases)
    M   -   number of basis functions (= polynomial_degree + 1), M>=1

OUTPUT PARAMETERS:
    P   -   interpolant in barycentric form for Rep.TerminationType>0.
            undefined for Rep.TerminationType<0.
    Rep -   fitting report. The following fields are set:
                * Rep.TerminationType is a completion code:
                  * set to  1 on success
                  * set to -3 on failure due to  problematic  constraints:
                    either too many  constraints,  degenerate  constraints
                    or inconsistent constraints were passed
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained regression splines:
* even simple constraints can be inconsistent, see  Wikipedia  article  on
  this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the one special cases, however, we can  guarantee  consistency.  This
  case  is:  M>1  and constraints on the function values (NOT DERIVATIVES)

Our final recommendation is to use constraints  WHEN  AND  ONLY  when  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 10.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialfitwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, barycentricinterpolant &p, polynomialfitreport &rep, const xparams _xparams = alglib::xdefault);
void polynomialfitwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t m, barycentricinterpolant &p, polynomialfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dfitcubicwc` function

/*************************************************************************
Weighted fitting by cubic  spline,  with constraints on function values or
derivatives.

Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is  used to build
basis functions. Basis functions are cubic splines with continuous  second
derivatives  and  non-fixed first  derivatives  at  interval  ends.  Small
regularizing term is used  when  solving  constrained  tasks  (to  improve
stability).

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

IMPORTANT: ALGLIB has a much faster version  of  the  cubic spline fitting
           function - spline1dfit(). This function performs least  squares
           fit in O(max(M,N)) time/memory. However, it  does  not  support
           constraints.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y/W are processed
            * if not given, automatically determined from X/Y/W sizes
    XC  -   points where spline values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that S(XC[i])=YC[i]
            * DC[i]=1   means that S'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints (optional):
            * 0<=K<M.
            * K=0 means no constraints (XC/YC/DC are not used)
            * if given, only first K elements of XC/YC/DC are used
            * if not given, automatically determined from XC/YC/DC
    M   -   number of basis functions ( = number_of_nodes+2), M>=4.

OUTPUT PARAMETERS:
    S   -   spline interpolant.
    Rep     -   fitting report. The following fields are set:
                * Rep.TerminationType is a completion code:
                  * set to  1 on success
                  * set to -3 on failure due to  problematic  constraints:
                    either too many  constraints,  degenerate  constraints
                    or inconsistent constraints were passed
                * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are  piecewise  cubic
  functions, and it is easy to create an example, where  large  number  of
  constraints  concentrated  in  small  area will result in inconsistency.
  Just because spline is not flexible enough to satisfy all of  them.  And
  same constraints spread across the  [min(x),max(x)]  will  be  perfectly
  consistent.
* the more evenly constraints are spread across [min(x),max(x)],  the more
  chances that they will be consistent
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints IS NOT GUARANTEED.
* in the several special cases, however, we CAN guarantee consistency.
* one of this cases is constraints  on  the  function  values  AND/OR  its
  derivatives at the interval boundaries.
* another  special  case  is ONE constraint on the function value (OR, but
  not AND, derivative) anywhere in the interval

Our final recommendation is to use constraints  WHEN  AND  ONLY  WHEN  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfitcubicwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfitcubicwc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

`spline1dfithermitedeprecated` function

/*************************************************************************
Deprecated fitting function with O(N*M^2+M^3) running time. Superseded  by
spline1dfit().

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermitedeprecated(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfithermitedeprecated(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

`spline1dfithermitewc` function

/*************************************************************************
Weighted  fitting  by Hermite spline,  with constraints on function values
or first derivatives.

Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is  used  to  build
basis functions. Basis functions are Hermite splines.  Small  regularizing
term is used when solving constrained tasks (to improve stability).

Task is linear, so linear least squares solver is used. Complexity of this
computational scheme is O(N*M^2), mostly dominated by least squares solver

IMPORTANT: ALGLIB has a much faster version  of  the  cubic spline fitting
           function - spline1dfit(). This function performs least  squares
           fit in O(max(M,N)) time/memory. However, it  does  not  support
           constraints.

INPUT PARAMETERS:
    X   -   points, array[0..N-1].
    Y   -   function values, array[0..N-1].
    W   -   weights, array[0..N-1]
            Each summand in square  sum  of  approximation deviations from
            given  values  is  multiplied  by  the square of corresponding
            weight. Fill it by 1's if you don't  want  to  solve  weighted
            task.
    N   -   number of points (optional):
            * N>0
            * if given, only first N elements of X/Y/W are processed
            * if not given, automatically determined from X/Y/W sizes
    XC  -   points where spline values/derivatives are constrained,
            array[0..K-1].
    YC  -   values of constraints, array[0..K-1]
    DC  -   array[0..K-1], types of constraints:
            * DC[i]=0   means that S(XC[i])=YC[i]
            * DC[i]=1   means that S'(XC[i])=YC[i]
            SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS
    K   -   number of constraints (optional):
            * 0<=K<M.
            * K=0 means no constraints (XC/YC/DC are not used)
            * if given, only first K elements of XC/YC/DC are used
            * if not given, automatically determined from XC/YC/DC
    M   -   number of basis functions (= 2 * number of nodes),
            M>=4,
            M IS EVEN!

OUTPUT PARAMETERS:
    S   -   spline interpolant.
    Rep     -   fitting report. The following fields are set:
                * Rep.TerminationType is a completion code:
                  * set to  1 on success
                  * set to -3 on failure due to  problematic  constraints:
                    either too many  constraints,  degenerate  constraints
                    or inconsistent constraints were passed
                  * RMSError      rms error on the (X,Y).
                * AvgError      average error on the (X,Y).
                * AvgRelError   average relative error on the non-zero Y
                * MaxError      maximum error
                                NON-WEIGHTED ERRORS ARE CALCULATED

IMPORTANT:
    this subroitine doesn't calculate task's condition number for K<>0.

IMPORTANT:
    this subroitine supports only even M's


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES:

Setting constraints can lead  to undesired  results,  like ill-conditioned
behavior, or inconsistency being detected. From the other side,  it allows
us to improve quality of the fit. Here we summarize  our  experience  with
constrained regression splines:
* excessive constraints can be inconsistent. Splines are  piecewise  cubic
  functions, and it is easy to create an example, where  large  number  of
  constraints  concentrated  in  small  area will result in inconsistency.
  Just because spline is not flexible enough to satisfy all of  them.  And
  same constraints spread across the  [min(x),max(x)]  will  be  perfectly
  consistent.
* the more evenly constraints are spread across [min(x),max(x)],  the more
  chances that they will be consistent
* the  greater  is  M (given  fixed  constraints),  the  more chances that
  constraints will be consistent
* in the general case, consistency of constraints is NOT GUARANTEED.
* in the several special cases, however, we can guarantee consistency.
* one of this cases is  M>=4  and   constraints  on   the  function  value
  (AND/OR its derivative) at the interval boundaries.
* another special case is M>=4  and  ONE  constraint on the function value
  (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)]

Our final recommendation is to use constraints  WHEN  AND  ONLY  when  you
can't solve your task without them. Anything beyond  special  cases  given
above is not guaranteed and may result in inconsistency.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB PROJECT --
     Copyright 18.08.2009 by Bochkanov Sergey
*************************************************************************/
void spline1dfithermitewc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t k, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfithermitewc(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const real_1d_array &xc, const real_1d_array &yc, const integer_1d_array &dc, const ae_int_t m, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

lsfit_d_lin example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
        //
        // We have:
        // * y - vector of experimental data
        // * fmatrix -  matrix of basis functions calculated at sample points
        //              Actually, we have only one basis function F0 = exp(0.5*x).
        //
        real_2d_array fmatrix = "[[0.606531],[0.670320],[0.740818],[0.818731],[0.904837],[1.000000],[1.105171],[1.221403],[1.349859],[1.491825],[1.648721]]";
        real_1d_array y = "[1.133719, 1.306522, 1.504604, 1.554663, 1.884638, 2.072436, 2.257285, 2.534068, 2.622017, 2.897713, 3.219371]";
        real_1d_array c;
        lsfitreport rep;

        //
        // Linear fitting without weights
        //
        lsfitlinear(y, fmatrix, c, rep);
        printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.98650]

        //
        // Linear fitting with individual weights.
        // Slightly different result is returned.
        //
        real_1d_array w = "[1.414213, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]";
        lsfitlinearw(y, w, fmatrix, c, rep);
        printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.983354]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_linc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate linear fitting by f(x|a,b) = a*x+b
        // with simple constraint f(0)=0.
        //
        // We have:
        // * y - vector of experimental data
        // * fmatrix -  matrix of basis functions sampled at [0,1] with step 0.2:
        //                  [ 1.0   0.0 ]
        //                  [ 1.0   0.2 ]
        //                  [ 1.0   0.4 ]
        //                  [ 1.0   0.6 ]
        //                  [ 1.0   0.8 ]
        //                  [ 1.0   1.0 ]
        //              first column contains value of first basis function (constant term)
        //              second column contains second basis function (linear term)
        // * cmatrix -  matrix of linear constraints:
        //                  [ 1.0  0.0  0.0 ]
        //              first two columns contain coefficients before basis functions,
        //              last column contains desired value of their sum.
        //              So [1,0,0] means "1*constant_term + 0*linear_term = 0" 
        //
        real_1d_array y = "[0.072436,0.246944,0.491263,0.522300,0.714064,0.921929]";
        real_2d_array fmatrix = "[[1,0.0],[1,0.2],[1,0.4],[1,0.6],[1,0.8],[1,1.0]]";
        real_2d_array cmatrix = "[[1,0,0]]";
        real_1d_array c;
        lsfitreport rep;

        //
        // Constrained fitting without weights
        //
        lsfitlinearc(y, fmatrix, cmatrix, c, rep);
        printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [0,0.932933]

        //
        // Constrained fitting with individual weights
        //
        real_1d_array w = "[1, 1.414213, 1, 1, 1, 1]";
        lsfitlinearwc(y, w, fmatrix, cmatrix, c, rep);
        printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [0,0.938322]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_nlf example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}
int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate exponential fitting by
        //
        //     f(x) = exp(-c*x^2)
        //
        // using numerical differentiation.
        //
        // IMPORTANT: the LSFIT optimizer supports parallel model  evaluation  and
        //            parallel numerical differentiation ('callback parallelism').
        //            This feature, which is present in commercial ALGLIB editions
        //            greatly  accelerates  fits  with   large   datasets   and/or
        //            expensive target functions.
        //
        //            Callback parallelism is usually  beneficial  when  a  single
        //            pass over the entire  dataset  requires  more  than  several
        //            milliseconds. This particular example,  of  course,  is  not
        //            suited for callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  lsfitfit()  function  for  more
        //            information.
        //
        real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
        real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
        real_1d_array c = "[0.3]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        lsfitstate state;
        lsfitreport rep;
        double diffstep = 0.0001;

        //
        // Fitting without weights
        //
        lsfitcreatef(x, y, c, diffstep, state);
        lsfitsetcond(state, epsx, maxits);
        alglib::lsfitfit(state, function_cx_1_func);
        lsfitresults(state, c, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 2
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]

        //
        // Fitting with weights
        // (you can change weights and see how it changes result)
        //
        real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
        lsfitcreatewf(x, y, w, c, diffstep, state);
        lsfitsetcond(state, epsx, maxits);
        alglib::lsfitfit(state, function_cx_1_func);
        lsfitresults(state, c, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 2
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_nlfb example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}
int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate exponential fitting by
        //
        //     f(x) = exp(-c*x^2)
        //
        // subject to box constraints
        //
        //     0.0 <= c <= 1.0
        //
        // using function value only. An unconstrained solution is c=1.5, but because of
        // constraints we should get c=1.0 (at the boundary).
        //
        // IMPORTANT: the LSFIT optimizer supports parallel model  evaluation  and
        //            parallel numerical differentiation ('callback parallelism').
        //            This feature, which is present in commercial ALGLIB editions
        //            greatly  accelerates  fits  with   large   datasets   and/or
        //            expensive target functions.
        //
        //            Callback parallelism is usually  beneficial  when  a  single
        //            pass over the entire  dataset  requires  more  than  several
        //            milliseconds. This particular example,  of  course,  is  not
        //            suited for callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  lsfitfit()  function  for  more
        //            information.
        //
        real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
        real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
        real_1d_array c = "[0.3]";
        real_1d_array bndl = "[0.0]";
        real_1d_array bndu = "[1.0]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        lsfitstate state;
        lsfitreport rep;
        double diffstep = 0.0001;

        lsfitcreatef(x, y, c, diffstep, state);
        lsfitsetbc(state, bndl, bndu);
        lsfitsetcond(state, epsx, maxits);
        alglib::lsfitfit(state, function_cx_1_func);
        lsfitresults(state, c, rep);
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.0]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_nlfg example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0))
    // where x is a position on X-axis and c is adjustable parameter
    func = exp(-c[0]*pow(x[0],2));
}
void function_cx_1_grad(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    // this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
    // where x is a position on X-axis and c is adjustable parameter.
    // IMPORTANT: gradient is calculated with respect to C, not to X
    func = exp(-c[0]*pow(x[0],2));
    grad[0] = -pow(x[0],2)*func;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate exponential fitting by
        //
        //     f(x) = exp(-c*x^2)
        //
        // using function value and gradient (with respect to c).
        //
        // IMPORTANT: the LSFIT optimizer supports parallel model  evaluation  and
        //            parallel numerical differentiation ('callback parallelism').
        //            This feature, which is present in commercial ALGLIB editions
        //            greatly  accelerates  fits  with   large   datasets   and/or
        //            expensive target functions.
        //
        //            Callback parallelism is usually  beneficial  when  a  single
        //            pass over the entire  dataset  requires  more  than  several
        //            milliseconds. This particular example,  of  course,  is  not
        //            suited for callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  lsfitfit()  function  for  more
        //            information.
        //
        real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
        real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
        real_1d_array c = "[0.3]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        lsfitstate state;
        lsfitreport rep;

        //
        // Fitting without weights
        //
        lsfitcreatefg(x, y, c, state);
        lsfitsetcond(state, epsx, maxits);
        alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad);
        lsfitresults(state, c, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 2
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]

        //
        // Fitting with weights
        // (you can change weights and see how it changes result)
        //
        real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
        lsfitcreatewfg(x, y, w, c, state);
        lsfitsetcond(state, epsx, maxits);
        alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad);
        lsfitresults(state, c, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 2
        printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_nlscale example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;
void function_debt_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr) 
{
    //
    // this callback calculates f(c,x)=c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1))
    //
    func = c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1));
}
int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate fitting by
        //
        //     f(x) = c[0]*(1+c[1]*((x-1999)^c[2]-1))
        //
        // subject to box constraints
        //
        //     -INF  < c[0] < +INF
        //      -10 <= c[1] <= +10
        //      0.1 <= c[2] <= 2.0
        //
        // The data we want to fit are time series of Japan national debt
        // collected from 2000 to 2008 measured in USD (dollars, not
        // millions of dollars).
        //
        // Our variables are:
        //     c[0] - debt value at initial moment (2000),
        //     c[1] - direction coefficient (growth or decrease),
        //     c[2] - curvature coefficient.
        // You may see that our variables are badly scaled - first one 
        // is order of 10^12, and next two are somewhere about 1 in 
        // magnitude. Such problem is difficult to solve without some
        // kind of scaling.
        // That is exactly where lsfitsetscale() function can be used.
        // We set scale of our variables to [1.0E12, 1, 1], which allows
        // us to easily solve this problem.
        //
        // You can try commenting out lsfitsetscale() call - and you will 
        // see that algorithm will fail to converge.
        //
        // IMPORTANT: the LSFIT optimizer supports parallel model  evaluation  and
        //            parallel numerical differentiation ('callback parallelism').
        //            This feature, which is present in commercial ALGLIB editions
        //            greatly  accelerates  fits  with   large   datasets   and/or
        //            expensive target functions.
        //
        //            Callback parallelism is usually  beneficial  when  a  single
        //            pass over the entire  dataset  requires  more  than  several
        //            milliseconds. This particular example,  of  course,  is  not
        //            suited for callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  lsfitfit()  function  for  more
        //            information.
        //
        real_2d_array x = "[[2000],[2001],[2002],[2003],[2004],[2005],[2006],[2007],[2008]]";
        real_1d_array y = "[4323239600000.0, 4560913100000.0, 5564091500000.0, 6743189300000.0, 7284064600000.0, 7050129600000.0, 7092221500000.0, 8483907600000.0, 8625804400000.0]";
        real_1d_array c = "[1.0e+13, 1, 1]";
        double epsx = 1.0e-5;
        real_1d_array bndl = "[-inf, -10, 0.1]";
        real_1d_array bndu = "[+inf, +10, 2.0]";
        real_1d_array s = "[1.0e+12, 1, 1]";
        ae_int_t maxits = 0;
        lsfitstate state;
        lsfitreport rep;
        double diffstep = 1.0e-5;

        lsfitcreatef(x, y, c, diffstep, state);
        lsfitsetcond(state, epsx, maxits);
        lsfitsetbc(state, bndl, bndu);
        lsfitsetscale(state, s);
        alglib::lsfitfit(state, function_debt_func);
        lsfitresults(state, c, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 2
        printf("%s\n", c.tostring(-2).c_str()); // EXPECTED: [4.142560E+12, 0.434240, 0.565376]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_pol example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates polynomial fitting.
        //
        // Fitting is done by two (M=2) functions from polynomial basis:
        //     f0 = 1
        //     f1 = x
        // Basically, it just a linear fit; more complex polynomials may be used
        // (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
        // us to demonstrate polynomialfit() function in action.
        //
        // We have:
        // * x      set of abscissas
        // * y      experimental data
        //
        // Additionally we demonstrate weighted fitting, where second point has
        // more weight than other ones.
        //
        real_1d_array x = "[0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]";
        real_1d_array y = "[0.00,0.05,0.26,0.32,0.33,0.43,0.60,0.60,0.77,0.98,1.02]";
        ae_int_t m = 2;
        double t = 2;
        barycentricinterpolant p;
        polynomialfitreport rep;
        double v;

        //
        // Fitting without individual weights
        //
        // NOTE: result is returned as barycentricinterpolant structure.
        //       if you want to get representation in the power basis,
        //       you can use barycentricbar2pow() function to convert
        //       from barycentric to power representation (see docs for 
        //       POLINT subpackage for more info).
        //
        polynomialfit(x, y, m, p, rep);
        v = barycentriccalc(p, t);
        printf("%.2f\n", double(v)); // EXPECTED: 2.011

        //
        // Fitting with individual weights
        //
        // NOTE: slightly different result is returned
        //
        real_1d_array w = "[1,1.414213562,1,1,1,1,1,1,1,1,1]";
        real_1d_array xc = "[]";
        real_1d_array yc = "[]";
        integer_1d_array dc = "[]";
        polynomialfitwc(x, y, w, xc, yc, dc, m, p, rep);
        v = barycentriccalc(p, t);
        printf("%.2f\n", double(v)); // EXPECTED: 2.023
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_polc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates polynomial fitting.
        //
        // Fitting is done by two (M=2) functions from polynomial basis:
        //     f0 = 1
        //     f1 = x
        // with simple constraint on function value
        //     f(0) = 0
        // Basically, it just a linear fit; more complex polynomials may be used
        // (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
        // us to demonstrate polynomialfit() function in action.
        //
        // We have:
        // * x      set of abscissas
        // * y      experimental data
        // * xc     points where constraints are placed
        // * yc     constraints on derivatives
        // * dc     derivative indices
        //          (0 means function itself, 1 means first derivative)
        //
        real_1d_array x = "[1.0,1.0]";
        real_1d_array y = "[0.9,1.1]";
        real_1d_array w = "[1,1]";
        real_1d_array xc = "[0]";
        real_1d_array yc = "[0]";
        integer_1d_array dc = "[0]";
        double t = 2;
        ae_int_t m = 2;
        barycentricinterpolant p;
        polynomialfitreport rep;
        double v;

        polynomialfitwc(x, y, w, xc, yc, dc, m, p, rep);
        v = barycentriccalc(p, t);
        printf("%.2f\n", double(v)); // EXPECTED: 2.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_d_spline example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // In this example we demonstrate penalized spline fitting of noisy data
        //
        // We have:
        // * x - abscissas
        // * y - vector of experimental data, straight line with small noise
        //
        real_1d_array x = "[0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90]";
        real_1d_array y = "[0.10,0.00,0.30,0.40,0.30,0.40,0.62,0.68,0.75,0.95]";
        double v;
        spline1dinterpolant s;
        spline1dfitreport rep;

        //
        // Fit with VERY small amount of smoothing (eps = 1.0E-9)
        // and large number of basis functions (M=50).
        //
        // With such small regularization penalized spline almost fully reproduces function values
        //
        spline1dfit(x, y, 50, 0.000000001, s, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        v = spline1dcalc(s, 0.0);
        printf("%.1f\n", double(v)); // EXPECTED: 0.10

        //
        // Fit with VERY large amount of smoothing eps=1000000
        // and large number of basis functions (M=50).
        //
        // With such regularization our spline should become close to the straight line fit.
        // We will compare its value in x=1.0 with results obtained from such fit.
        //
        spline1dfit(x, y, 50, 1000000, s, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        v = spline1dcalc(s, 1.0);
        printf("%.2f\n", double(v)); // EXPECTED: 0.969

        //
        // In real life applications you may need some moderate degree of fitting,
        // so we try to fit once more with eps=0.1.
        //
        spline1dfit(x, y, 50, 0.1, s, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_t_4pl example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_1d_array x = "[1,2,3,4,5,6,7,8]";
        real_1d_array y = "[0.06313223,0.44552624,0.61838364,0.71385108,0.77345838,0.81383140,0.84280033,0.86449822]";
        ae_int_t n = 8;
        double a;
        double b;
        double c;
        double d;
        lsfitreport rep;

        //
        // Test logisticfit4() on carefully designed data with a priori known answer.
        //
        logisticfit4(x, y, n, a, b, c, d, rep);
        printf("%.1f\n", double(a)); // EXPECTED: -1.000
        printf("%.1f\n", double(b)); // EXPECTED: 1.200
        printf("%.1f\n", double(c)); // EXPECTED: 0.900
        printf("%.1f\n", double(d)); // EXPECTED: 1.000

        //
        // Evaluate model at point x=0.5
        //
        double v;
        v = logisticcalc4(0.5, a, b, c, d);
        printf("%.2f\n", double(v)); // EXPECTED: -0.33874308
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

lsfit_t_5pl example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_1d_array x = "[1,2,3,4,5,6,7,8]";
        real_1d_array y = "[0.1949776139,0.5710060208,0.726002637,0.8060434158,0.8534547965,0.8842071579,0.9054773317,0.9209088299]";
        ae_int_t n = 8;
        double a;
        double b;
        double c;
        double d;
        double g;
        lsfitreport rep;

        //
        // Test logisticfit5() on carefully designed data with a priori known answer.
        //
        logisticfit5(x, y, n, a, b, c, d, g, rep);
        printf("%.1f\n", double(a)); // EXPECTED: -1.000
        printf("%.1f\n", double(b)); // EXPECTED: 1.200
        printf("%.1f\n", double(c)); // EXPECTED: 0.900
        printf("%.1f\n", double(d)); // EXPECTED: 1.000
        printf("%.1f\n", double(g)); // EXPECTED: 1.200

        //
        // Evaluate model at point x=0.5
        //
        double v;
        v = logisticcalc5(0.5, a, b, c, d, g);
        printf("%.2f\n", double(v)); // EXPECTED: -0.2354656824
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`mannwhitneyu` subpackage

Functions

mannwhitneyutest

Examples

`mannwhitneyutest` function

/*************************************************************************
Mann-Whitney U-test

This test checks hypotheses about whether X  and  Y  are  samples  of  two
continuous distributions of the same shape  and  same  median  or  whether
their medians are different.

The following tests are performed:
    * two-tailed test (null hypothesis - the medians are equal)
    * left-tailed test (null hypothesis - the median of the  first  sample
      is greater than or equal to the median of the second sample)
    * right-tailed test (null hypothesis - the median of the first  sample
      is less than or equal to the median of the second sample).

Requirements:
    * the samples are independent
    * X and Y are continuous distributions (or discrete distributions well-
      approximating continuous distributions)
    * distributions of X and Y have the  same  shape.  The  only  possible
      difference is their position (i.e. the value of the median)
    * the number of elements in each sample is not less than 5
    * the scale of measurement should be ordinal, interval or ratio  (i.e.
      the test could not be applied to nominal variables).

The test is non-parametric and doesn't require distributions to be normal.

Input parameters:
    X   -   sample 1. Array whose index goes from 0 to N-1.
    N   -   size of the sample. N>=5
    Y   -   sample 2. Array whose index goes from 0 to M-1.
    M   -   size of the sample. M>=5

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

To calculate p-values, special approximation is used. This method lets  us
calculate p-values with satisfactory  accuracy  in  interval  [0.0001, 1].
There is no approximation outside the [0.0001, 1] interval. Therefore,  if
the significance level outlies this interval, the test returns 0.0001.

Relative precision of approximation of p-value:

N          M          Max.err.   Rms.err.
5..10      N..10      1.4e-02    6.0e-04
5..10      N..100     2.2e-02    5.3e-06
10..15     N..15      1.0e-02    3.2e-04
10..15     N..100     1.0e-02    2.2e-05
15..100    N..100     6.1e-03    2.7e-06

For N,M>100 accuracy checks weren't put into  practice,  but  taking  into
account characteristics of asymptotic approximation used, precision should
not be sharply different from the values for interval [5, 100].

NOTE: P-value approximation was  optimized  for  0.0001<=p<=0.2500.  Thus,
      P's outside of this interval are enforced to these bounds. Say,  you
      may quite often get P equal to exactly 0.25 or 0.0001.

  -- ALGLIB --
     Copyright 09.04.2007 by Bochkanov Sergey
*************************************************************************/
void mannwhitneyutest(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`matdet` subpackage

Functions

cmatrixdet
cmatrixludet
rmatrixdet
rmatrixludet
spdmatrixcholeskydet
spdmatrixdet

Examples

matdet_d_1		Determinant calculation, real matrix, short form
matdet_d_2		Determinant calculation, real matrix, full form
matdet_d_3		Determinant calculation, complex matrix, short form
matdet_d_4		Determinant calculation, complex matrix, full form
matdet_d_5		Determinant calculation, complex matrix with zero imaginary part, short form

`cmatrixdet` function

/*************************************************************************
Calculation of the determinant of a general matrix

Input parameters:
    A       -   matrix, array[0..N-1, 0..N-1]
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: determinant of matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
alglib::complex cmatrixdet(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
alglib::complex cmatrixdet(const complex_2d_array &a, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`cmatrixludet` function

/*************************************************************************
Determinant calculation of the matrix given by its LU decomposition.

Input parameters:
    A       -   LU decomposition of the matrix (output of
                RMatrixLU subroutine).
    Pivots  -   table of permutations which were made during
                the LU decomposition.
                Output of RMatrixLU subroutine.
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: matrix determinant.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, const xparams _xparams = alglib::xdefault);
alglib::complex cmatrixludet(const complex_2d_array &a, const integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);

`rmatrixdet` function

/*************************************************************************
Calculation of the determinant of a general matrix

Input parameters:
    A       -   matrix, array[0..N-1, 0..N-1]
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: determinant of matrix A.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
double rmatrixdet(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double rmatrixdet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`rmatrixludet` function

/*************************************************************************
Determinant calculation of the matrix given by its LU decomposition.

Input parameters:
    A       -   LU decomposition of the matrix (output of
                RMatrixLU subroutine).
    Pivots  -   table of permutations which were made during
                the LU decomposition.
                Output of RMatrixLU subroutine.
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

Result: matrix determinant.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double rmatrixludet(const real_2d_array &a, const integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskydet` function

/*************************************************************************
Determinant calculation of the matrix given by the Cholesky decomposition.

Input parameters:
    A       -   Cholesky decomposition,
                output of SMatrixCholesky subroutine.
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)

As the determinant is equal to the product of squares of diagonal elements,
it's not necessary to specify which triangle - lower or upper - the matrix
is stored in.

Result:
    matrix determinant.

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double spdmatrixcholeskydet(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);
double spdmatrixcholeskydet(const real_2d_array &a, const xparams _xparams = alglib::xdefault);

`spdmatrixdet` function

/*************************************************************************
Determinant calculation of the symmetric positive definite matrix.

Input parameters:
    A       -   matrix, array[N,N]
    N       -   (optional) size of matrix A:
                * if given, only principal NxN submatrix is processed and
                  overwritten. other elements are unchanged.
                * if not given, automatically determined from matrix size
                  (A must be square matrix)
    IsUpper -   storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used/changed  by
                  function
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used/changed  by
                  function

Result:
    determinant of matrix A.
    If matrix A is not positive definite, an exception is generated.

  -- ALGLIB --
     Copyright 2005-2008 by Bochkanov Sergey
*************************************************************************/
double spdmatrixdet(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
double spdmatrixdet(const real_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);

matdet_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array b = "[[1,2],[2,1]]";
        double a;
        a = rmatrixdet(b);
        printf("%.3f\n", double(a)); // EXPECTED: -3
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matdet_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array b = "[[5,4],[4,5]]";
        double a;
        a = rmatrixdet(b, 2);
        printf("%.3f\n", double(a)); // EXPECTED: 9
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matdet_d_3 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        complex_2d_array b = "[[1+1i,2],[2,1-1i]]";
        alglib::complex a;
        a = cmatrixdet(b);
        printf("%s\n", a.tostring(3).c_str()); // EXPECTED: -2
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matdet_d_4 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        alglib::complex a;
        complex_2d_array b = "[[5i,4],[4i,5]]";
        a = cmatrixdet(b, 2);
        printf("%s\n", a.tostring(3).c_str()); // EXPECTED: 9i
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matdet_d_5 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        alglib::complex a;
        complex_2d_array b = "[[9,1],[2,1]]";
        a = cmatrixdet(b);
        printf("%s\n", a.tostring(3).c_str()); // EXPECTED: 7
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`matgen` subpackage

Functions

cmatrixrndcond
cmatrixrndorthogonal
cmatrixrndorthogonalfromtheleft
cmatrixrndorthogonalfromtheright
hmatrixrndcond
hmatrixrndmultiply
hpdmatrixrndcond
rmatrixrndcond
rmatrixrndorthogonal
rmatrixrndorthogonalfromtheleft
rmatrixrndorthogonalfromtheright
smatrixrndcond
smatrixrndmultiply
spdmatrixrndcond

Examples

`cmatrixrndcond` function

/*************************************************************************
Generation of random NxN complex matrix with given condition number C and
norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`cmatrixrndorthogonal` function

/*************************************************************************
Generation of a random Haar distributed orthogonal complex matrix

INPUT PARAMETERS:
    N   -   matrix size, N>=1

OUTPUT PARAMETERS:
    A   -   orthogonal NxN matrix, array[0..N-1,0..N-1]

NOTE: this function uses algorithm  described  in  Stewart, G. W.  (1980),
      "The Efficient Generation of  Random  Orthogonal  Matrices  with  an
      Application to Condition Estimators".

      Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
      * takes an NxN one
      * takes uniformly distributed unit vector of dimension N+1.
      * constructs a Householder reflection from the vector, then applies
        it to the smaller matrix (embedded in the larger size with a 1 at
        the bottom right corner).

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonal(const ae_int_t n, complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`cmatrixrndorthogonalfromtheleft` function

/*************************************************************************
Multiplication of MxN complex matrix by MxM random Haar distributed
complex orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   Q*A, where Q is random MxM orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonalfromtheleft(complex_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`cmatrixrndorthogonalfromtheright` function

/*************************************************************************
Multiplication of MxN complex matrix by NxN random Haar distributed
complex orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void cmatrixrndorthogonalfromtheright(complex_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`hmatrixrndcond` function

/*************************************************************************
Generation of random NxN Hermitian matrix with given condition number  and
norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void hmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`hmatrixrndmultiply` function

/*************************************************************************
Hermitian multiplication of NxN matrix by random Haar distributed
complex orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..N-1, 0..N-1]
    N   -   matrix size

OUTPUT PARAMETERS:
    A   -   Q^H*A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void hmatrixrndmultiply(complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`hpdmatrixrndcond` function

/*************************************************************************
Generation of random NxN Hermitian positive definite matrix with given
condition number and norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random HPD matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixrndcond(const ae_int_t n, const double c, complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`rmatrixrndcond` function

/*************************************************************************
Generation of random NxN matrix with given condition number and norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void rmatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);

`rmatrixrndorthogonal` function

/*************************************************************************
Generation of a random uniformly distributed (Haar) orthogonal matrix

INPUT PARAMETERS:
    N   -   matrix size, N>=1

OUTPUT PARAMETERS:
    A   -   orthogonal NxN matrix, array[0..N-1,0..N-1]

NOTE: this function uses algorithm  described  in  Stewart, G. W.  (1980),
      "The Efficient Generation of  Random  Orthogonal  Matrices  with  an
      Application to Condition Estimators".

      Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it:
      * takes an NxN one
      * takes uniformly distributed unit vector of dimension N+1.
      * constructs a Householder reflection from the vector, then applies
        it to the smaller matrix (embedded in the larger size with a 1 at
        the bottom right corner).

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonal(const ae_int_t n, real_2d_array &a, const xparams _xparams = alglib::xdefault);

`rmatrixrndorthogonalfromtheleft` function

/*************************************************************************
Multiplication of MxN matrix by MxM random Haar distributed orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   Q*A, where Q is random MxM orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonalfromtheleft(real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixrndorthogonalfromtheright` function

/*************************************************************************
Multiplication of MxN matrix by NxN random Haar distributed orthogonal matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..M-1, 0..N-1]
    M, N-   matrix size

OUTPUT PARAMETERS:
    A   -   A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void rmatrixrndorthogonalfromtheright(real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`smatrixrndcond` function

/*************************************************************************
Generation of random NxN symmetric matrix with given condition number  and
norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void smatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);

`smatrixrndmultiply` function

/*************************************************************************
Symmetric multiplication of NxN matrix by random Haar distributed
orthogonal  matrix

INPUT PARAMETERS:
    A   -   matrix, array[0..N-1, 0..N-1]
    N   -   matrix size

OUTPUT PARAMETERS:
    A   -   Q'*A*Q, where Q is random NxN orthogonal matrix

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void smatrixrndmultiply(real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`spdmatrixrndcond` function

/*************************************************************************
Generation of random NxN symmetric positive definite matrix with given
condition number and norm2(A)=1

INPUT PARAMETERS:
    N   -   matrix size
    C   -   condition number (in 2-norm)

OUTPUT PARAMETERS:
    A   -   random SPD matrix with norm2(A)=1 and cond(A)=C

  -- ALGLIB routine --
     04.12.2009
     Bochkanov Sergey
*************************************************************************/
void spdmatrixrndcond(const ae_int_t n, const double c, real_2d_array &a, const xparams _xparams = alglib::xdefault);

`matinv` subpackage

Classes

matinvreport

Functions

cmatrixinverse
cmatrixluinverse
cmatrixtrinverse
hpdmatrixcholeskyinverse
hpdmatrixinverse
rmatrixinverse
rmatrixluinverse
rmatrixtrinverse
spdmatrixcholeskyinverse
spdmatrixinverse

Examples

matinv_d_c1		Complex matrix inverse
matinv_d_hpd1		HPD matrix inverse
matinv_d_r1		Real matrix inverse
matinv_d_spd1		SPD matrix inverse

`matinvreport` class

/*************************************************************************
Matrix inverse report:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for failure due to the matrix being singular or
                         nearly-singular
* r1                reciprocal of condition number in 1-norm
* rinf              reciprocal of condition number in inf-norm
*************************************************************************/
class matinvreport
{
public:
    matinvreport();
    matinvreport(const matinvreport &rhs);
    matinvreport& operator=(const matinvreport &rhs);
    virtual ~matinvreport();
    ae_int_t terminationtype;
    double r1;
    double rinf;
};

`cmatrixinverse` function

/*************************************************************************
Inversion of a general matrix.

Input parameters:
    A       -   matrix
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

Output parameters:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
void cmatrixinverse(complex_2d_array &a, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixinverse(complex_2d_array &a, matinvreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`cmatrixluinverse` function

/*************************************************************************
Inversion of a matrix given by its LU decomposition.

INPUT PARAMETERS:
    A       -   LU decomposition of the matrix
                (output of CMatrixLU subroutine).
    Pivots  -   table of permutations
                (the output of CMatrixLU subroutine).
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixluinverse(complex_2d_array &a, const integer_1d_array &pivots, matinvreport &rep, const xparams _xparams = alglib::xdefault);

`cmatrixtrinverse` function

/*************************************************************************
Triangular matrix inverse (complex)

The subroutine inverts the following types of matrices:
    * upper triangular
    * upper triangular with unit diagonal
    * lower triangular
    * lower triangular with unit diagonal

In case of an upper (lower) triangular matrix,  the  inverse  matrix  will
also be upper (lower) triangular, and after the end of the algorithm,  the
inverse matrix replaces the source matrix. The elements  below (above) the
main diagonal are not changed by the algorithm.

If  the matrix  has a unit diagonal, the inverse matrix also  has  a  unit
diagonal, and the diagonal elements are not passed to the algorithm.

INPUT PARAMETERS:
    A       -   matrix, array[0..N-1, 0..N-1].
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   diagonal type (optional):
                * if True, matrix has unit diagonal (a[i,i] are NOT used)
                * if False, matrix diagonal is arbitrary
                * if not given, False is assumed

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void cmatrixtrinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void cmatrixtrinverse(complex_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);

`hpdmatrixcholeskyinverse` function

/*************************************************************************
Inversion of a Hermitian positive definite matrix which is given
by Cholesky decomposition.

Input parameters:
    A       -   Cholesky decomposition of the matrix to be inverted:
                A=U'*U or A = L*L'.
                Output of  HPDMatrixCholesky subroutine.
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   storage type:
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixcholeskyinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixcholeskyinverse(complex_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);

`hpdmatrixinverse` function

/*************************************************************************
Inversion of a Hermitian positive definite matrix.

Given an upper or lower triangle of a Hermitian positive definite matrix,
the algorithm generates matrix A^-1 and saves the upper or lower triangle
depending on the input.

INPUT PARAMETERS:
    A       -   matrix to be inverted (upper or lower triangle), array[N,N]
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   storage type:
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void hpdmatrixinverse(complex_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void hpdmatrixinverse(complex_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixinverse` function

/*************************************************************************
Inversion of a general matrix.

INPUT PARAMETERS:
    A       -   matrix.
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixinverse(real_2d_array &a, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixinverse(real_2d_array &a, matinvreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rmatrixluinverse` function

/*************************************************************************
Inversion of a matrix given by its LU decomposition.

INPUT PARAMETERS:
    A       -   LU decomposition of the matrix
                (output of RMatrixLU subroutine).
    Pivots  -   table of permutations
                (the output of RMatrixLU subroutine).
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     05.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, const ae_int_t n, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixluinverse(real_2d_array &a, const integer_1d_array &pivots, matinvreport &rep, const xparams _xparams = alglib::xdefault);

`rmatrixtrinverse` function

/*************************************************************************
Triangular matrix inverse (real)

The subroutine inverts the following types of matrices:
    * upper triangular
    * upper triangular with unit diagonal
    * lower triangular
    * lower triangular with unit diagonal

In case of an upper (lower) triangular matrix,  the  inverse  matrix  will
also be upper (lower) triangular, and after the end of the algorithm,  the
inverse matrix replaces the source matrix. The elements  below (above) the
main diagonal are not changed by the algorithm.

If  the matrix  has a unit diagonal, the inverse matrix also  has  a  unit
diagonal, and the diagonal elements are not passed to the algorithm.

INPUT PARAMETERS:
    A       -   matrix, array[0..N-1, 0..N-1].
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   diagonal type (optional):
                * if True, matrix has unit diagonal (a[i,i] are NOT used)
                * if False, matrix diagonal is arbitrary
                * if not given, False is assumed

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 05.02.2010 by Bochkanov Sergey
*************************************************************************/
void rmatrixtrinverse(real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void rmatrixtrinverse(real_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskyinverse` function

/*************************************************************************
Inversion of a symmetric positive definite matrix which is given
by Cholesky decomposition.

INPUT PARAMETERS:
    A       -   Cholesky decomposition of the matrix to be inverted:
                A=U'*U or A = L*L'.
                Output of  SPDMatrixCholesky subroutine.
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   storage type:
                * if True, the symmetric  matrix  A  is given by its upper
                  triangle, and the lower triangle isn't  used/changed  by
                  the function
                * if False, the symmetric matrix  A  is given by its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  the function

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0,   corresponding      triangle
                  contains inverse matrix,   the  other  triangle  is  not
                  modified.
                * for rep.terminationtype<0,  corresponding  triangle   is
                  zero-filled; the other triangle is not modified.
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void spdmatrixcholeskyinverse(real_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyinverse(real_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);

`spdmatrixinverse` function

/*************************************************************************
Inversion of a symmetric positive definite matrix.

Given an upper or lower triangle of a symmetric positive definite matrix,
the algorithm generates matrix A^-1 and saves the upper or lower triangle
depending on the input.

INPUT PARAMETERS:
    A       -   matrix to be inverted (upper or lower triangle), array[N,N]
    N       -   size of the matrix A (optional):
                * if given, only principal NxN submatrix is processed  and
                  overwritten. Trailing elements are unchanged.
                * if not given, the size  is automatically determined from
                  the matrix size (A must be a square matrix)
    IsUpper -   storage type:
                * if True, symmetric  matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't  used/changed  by
                  function
                * if False,  symmetric matrix  A  is  given  by  its lower
                  triangle, and the  upper triangle isn't used/changed  by
                  function

OUTPUT PARAMETERS:
    A       -   inverse of matrix A, array[N,N]:
                * for rep.terminationtype>0, contains matrix inverse
                * for rep.terminationtype<0, zero-filled
    Rep     -   solver report:
                * rep.terminationtype>0 for success, <0 for failure
                * see below for more info

SOLVER REPORT

Subroutine sets following fields of the Rep structure:
* terminationtype   completion code:
                    *  1 for success
                    * -3 for a singular or extremely ill-conditioned matrix
* r1                reciprocal of condition number: 1/cond(A), 1-norm.
* rinf              reciprocal of condition number: 1/cond(A), inf-norm.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.02.2010
     Bochkanov Sergey
*************************************************************************/
void spdmatrixinverse(real_2d_array &a, const ae_int_t n, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);
void spdmatrixinverse(real_2d_array &a, const bool isupper, matinvreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

matinv_d_c1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        complex_2d_array a = "[[1i,-1],[1i,1]]";
        matinvreport rep;
        cmatrixinverse(a, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[-0.5i,-0.5i],[-0.5,0.5]]
        printf("%.4f\n", double(rep.r1)); // EXPECTED: 0.5
        printf("%.4f\n", double(rep.rinf)); // EXPECTED: 0.5
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matinv_d_hpd1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        complex_2d_array a = "[[2,1],[1,2]]";
        matinvreport rep;

        //
        // The matrix is given by its upper and lower triangles
        //
        //     [ 2 1 ]
        //     [ 1 2 ]
        //
        // However, hpdmatrixinverse() accepts and modifies only one triangle - either
        // the upper or the lower one. The other triangle is left untouched. In this example
        // we modify the lower triangle. Thus, the inverse matrix is
        //
        //     [  2/3 -1/3 ]
        //     [ -1/3  2/3 ]
        //
        // but only lower triangle is returned, and the upper triangle is not modified:
        //
        //     [  2/3   1  ]
        //     [ -1/3  2/3 ]
        //
        //
        bool isupper = false;
        hpdmatrixinverse(a, isupper, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.666666,1],[-0.333333,0.666666]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matinv_d_r1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array a = "[[1,-1],[1,1]]";
        matinvreport rep;
        rmatrixinverse(a, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.5,0.5],[-0.5,0.5]]
        printf("%.4f\n", double(rep.r1)); // EXPECTED: 0.5
        printf("%.4f\n", double(rep.rinf)); // EXPECTED: 0.5
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

matinv_d_spd1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array a = "[[2,1],[1,2]]";
        matinvreport rep;

        //
        // The matrix is given by its upper and lower triangles
        //
        //     [ 2 1 ]
        //     [ 1 2 ]
        //
        // However, spdmatrixinverse() accepts and modifies only one triangle - either
        // the upper or the lower one. The other triangle is left untouched. In this example
        // we modify the lower triangle. Thus, the inverse matrix is
        //
        //     [  2/3 -1/3 ]
        //     [ -1/3  2/3 ]
        //
        // but only lower triangle is returned, and the upper triangle is not modified:
        //
        //     [  2/3   1  ]
        //     [ -1/3  2/3 ]
        //
        //
        bool isupper = false;
        spdmatrixinverse(a, isupper, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.666666,1],[-0.333333,0.666666]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`mcpd` subpackage

Classes

mcpdreport
mcpdstate

Functions

mcpdaddbc
mcpdaddec
mcpdaddtrack
mcpdcreate
mcpdcreateentry
mcpdcreateentryexit
mcpdcreateexit
mcpdresults
mcpdsetbc
mcpdsetec
mcpdsetlc
mcpdsetpredictionweights
mcpdsetprior
mcpdsettikhonovregularizer
mcpdsolve

Examples

mcpd_simple1		Simple unconstrained MCPD model (no entry/exit states)
mcpd_simple2		Simple MCPD model (no entry/exit states) with equality constraints

`mcpdreport` class

/*************************************************************************
This structure is a MCPD training report:
    InnerIterationsCount    -   number of inner iterations of the
                                underlying optimization algorithm
    OuterIterationsCount    -   number of outer iterations of the
                                underlying optimization algorithm
    NFEV                    -   number of merit function evaluations
    TerminationType         -   termination type
                                (same as for MinBLEIC optimizer, positive
                                values denote success, negative ones -
                                failure)

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
class mcpdreport
{
public:
    mcpdreport();
    mcpdreport(const mcpdreport &rhs);
    mcpdreport& operator=(const mcpdreport &rhs);
    virtual ~mcpdreport();
    ae_int_t inneriterationscount;
    ae_int_t outeriterationscount;
    ae_int_t nfev;
    ae_int_t terminationtype;
};

`mcpdstate` class

/*************************************************************************
This structure is a MCPD (Markov Chains for Population Data) solver.

You should use ALGLIB functions in order to work with this object.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
class mcpdstate
{
public:
    mcpdstate();
    mcpdstate(const mcpdstate &rhs);
    mcpdstate& operator=(const mcpdstate &rhs);
    virtual ~mcpdstate();
};

`mcpdaddbc` function

/*************************************************************************
This function is used to add bound constraints  on  the  elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This  function  can  be  used to ADD bound constraint for one element of P
without changing constraints for other elements.

You  can  also  use  MCPDSetBC()  function  which  allows to  place  bound
constraints  on arbitrary subset of elements of P.   Set of constraints is
specified  by  BndL/BndU matrices, which may contain arbitrary combination
of finite numbers or infinities (like -INF<x<=0.5 or 0.1<=x<+INF).

These functions (MCPDSetBC and MCPDAddBC) interact as follows:
* there is internal matrix of bound constraints which is stored in the
  MCPD solver
* MCPDSetBC() replaces this matrix by another one (SET)
* MCPDAddBC() modifies one element of this matrix and  leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddBC()  call  preserves  all  modifications  done by previous
  calls,  while  MCPDSetBC()  completely discards all changes  done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    I       -   row index of element being constrained
    J       -   column index of element being constrained
    BndL    -   lower bound
    BndU    -   upper bound

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdaddbc(mcpdstate &s, const ae_int_t i, const ae_int_t j, const double bndl, const double bndu, const xparams _xparams = alglib::xdefault);

`mcpdaddec` function

/*************************************************************************
This function is used to add equality constraints on the elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This function can be used to ADD equality constraint for one element of  P
without changing constraints for other elements.

You  can  also  use  MCPDSetEC()  function  which  allows  you  to specify
arbitrary set of equality constraints in one call.

These functions (MCPDSetEC and MCPDAddEC) interact as follows:
* there is internal matrix of equality constraints which is stored in the
  MCPD solver
* MCPDSetEC() replaces this matrix by another one (SET)
* MCPDAddEC() modifies one element of this matrix and leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddEC()  call  preserves  all  modifications done by previous
  calls,  while  MCPDSetEC()  completely discards all changes done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    I       -   row index of element being constrained
    J       -   column index of element being constrained
    C       -   value (constraint for P[I,J]).  Can  be  either  NAN  (no
                constraint) or finite value from [0,1].

NOTES:

1. infinite values of C  will lead to exception being thrown. Values  less
than 0.0 or greater than 1.0 will lead to error code being returned  after
call to MCPDSolve().

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdaddec(mcpdstate &s, const ae_int_t i, const ae_int_t j, const double c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mcpdaddtrack` function

/*************************************************************************
This  function  is  used to add a track - sequence of system states at the
different moments of its evolution.

You  may  add  one  or several tracks to the MCPD solver. In case you have
several tracks, they won't overwrite each other. For example,  if you pass
two tracks, A1-A2-A3 (system at t=A+1, t=A+2 and t=A+3) and B1-B2-B3, then
solver will try to model transitions from t=A+1 to t=A+2, t=A+2 to  t=A+3,
t=B+1 to t=B+2, t=B+2 to t=B+3. But it WONT mix these two tracks - i.e. it
wont try to model transition from t=A+3 to t=B+1.

INPUT PARAMETERS:
    S       -   solver
    XY      -   track, array[K,N]:
                * I-th row is a state at t=I
                * elements of XY must be non-negative (exception will be
                  thrown on negative elements)
    K       -   number of points in a track
                * if given, only leading K rows of XY are used
                * if not given, automatically determined from size of XY

NOTES:

1. Track may contain either proportional or population data:
   * with proportional data all rows of XY must sum to 1.0, i.e. we have
     proportions instead of absolute population values
   * with population data rows of XY contain population counts and generally
     do not sum to 1.0 (although they still must be non-negative)

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdaddtrack(mcpdstate &s, const real_2d_array &xy, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void mcpdaddtrack(mcpdstate &s, const real_2d_array &xy, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mcpdcreate` function

/*************************************************************************
DESCRIPTION:

This function creates MCPD (Markov Chains for Population Data) solver.

This  solver  can  be  used  to find transition matrix P for N-dimensional
prediction  problem  where transition from X[i] to X[i+1] is  modelled  as
    X[i+1] = P*X[i]
where X[i] and X[i+1] are N-dimensional population vectors (components  of
each X are non-negative), and P is a N*N transition matrix (elements of  P
are non-negative, each column sums to 1.0).

Such models arise when when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is constant, i.e. there is no new individuals and no one
  leaves population
* you want to model transitions of individuals from one state into another

USAGE:

Here we give very brief outline of the MCPD. We strongly recommend you  to
read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
on data analysis which is available at http://www.alglib.net/dataanalysis/

1. User initializes algorithm state with MCPDCreate() call

2. User  adds  one  or  more  tracks -  sequences of states which describe
   evolution of a system being modelled from different starting conditions

3. User may add optional boundary, equality  and/or  linear constraints on
   the coefficients of P by calling one of the following functions:
   * MCPDSetEC() to set equality constraints
   * MCPDSetBC() to set bound constraints
   * MCPDSetLC() to set linear constraints

4. Optionally,  user  may  set  custom  weights  for prediction errors (by
   default, algorithm assigns non-equal, automatically chosen weights  for
   errors in the prediction of different components of X). It can be  done
   with a call of MCPDSetPredictionWeights() function.

5. User calls MCPDSolve() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.

6. User calls MCPDResults() to get solution

INPUT PARAMETERS:
    N       -   problem dimension, N>=1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdcreate(const ae_int_t n, mcpdstate &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mcpdcreateentry` function

/*************************************************************************
DESCRIPTION:

This function is a specialized version of MCPDCreate()  function,  and  we
recommend  you  to read comments for this function for general information
about MCPD solver.

This  function  creates  MCPD (Markov Chains for Population  Data)  solver
for "Entry-state" model,  i.e. model  where transition from X[i] to X[i+1]
is modelled as
    X[i+1] = P*X[i]
where
    X[i] and X[i+1] are N-dimensional state vectors
    P is a N*N transition matrix
and  one  selected component of X[] is called "entry" state and is treated
in a special way:
    system state always transits from "entry" state to some another state
    system state can not transit from any state into "entry" state
Such conditions basically mean that row of P which corresponds to  "entry"
state is zero.

Such models arise when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is NOT constant -  at every moment of time there is some
  (unpredictable) amount of "new" individuals, which can transit into  one
  of the states at the next turn, but still no one leaves population
* you want to model transitions of individuals from one state into another
* but you do NOT want to predict amount of "new"  individuals  because  it
  does not depends on individuals already present (hence  system  can  not
  transit INTO entry state - it can only transit FROM it).

This model is discussed  in  more  details  in  the ALGLIB User Guide (see
http://www.alglib.net/dataanalysis/ for more data).

INPUT PARAMETERS:
    N       -   problem dimension, N>=2
    EntryState- index of entry state, in 0..N-1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdcreateentry(const ae_int_t n, const ae_int_t entrystate, mcpdstate &s, const xparams _xparams = alglib::xdefault);

`mcpdcreateentryexit` function

/*************************************************************************
DESCRIPTION:

This function is a specialized version of MCPDCreate()  function,  and  we
recommend  you  to read comments for this function for general information
about MCPD solver.

This  function  creates  MCPD (Markov Chains for Population  Data)  solver
for "Entry-Exit-states" model, i.e. model where  transition  from  X[i] to
X[i+1] is modelled as
    X[i+1] = P*X[i]
where
    X[i] and X[i+1] are N-dimensional state vectors
    P is a N*N transition matrix
one selected component of X[] is called "entry" state and is treated in  a
special way:
    system state always transits from "entry" state to some another state
    system state can not transit from any state into "entry" state
and another one component of X[] is called "exit" state and is treated  in
a special way too:
    system state can transit from any state into "exit" state
    system state can not transit from "exit" state into any other state
    transition operator discards "exit" state (makes it zero at each turn)
Such conditions basically mean that:
    row of P which corresponds to "entry" state is zero
    column of P which corresponds to "exit" state is zero
Multiplication by such P may decrease sum of vector components.

Such models arise when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is NOT constant
* at every moment of time there is some (unpredictable)  amount  of  "new"
  individuals, which can transit into one of the states at the next turn
* some  individuals  can  move  (predictably)  into "exit" state and leave
  population at the next turn
* you want to model transitions of individuals from one state into another,
  including transitions from the "entry" state and into the "exit" state.
* but you do NOT want to predict amount of "new"  individuals  because  it
  does not depends on individuals already present (hence  system  can  not
  transit INTO entry state - it can only transit FROM it).

This model is discussed  in  more  details  in  the ALGLIB User Guide (see
http://www.alglib.net/dataanalysis/ for more data).

INPUT PARAMETERS:
    N       -   problem dimension, N>=2
    EntryState- index of entry state, in 0..N-1
    ExitState-  index of exit state, in 0..N-1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdcreateentryexit(const ae_int_t n, const ae_int_t entrystate, const ae_int_t exitstate, mcpdstate &s, const xparams _xparams = alglib::xdefault);

`mcpdcreateexit` function

/*************************************************************************
DESCRIPTION:

This function is a specialized version of MCPDCreate()  function,  and  we
recommend  you  to read comments for this function for general information
about MCPD solver.

This  function  creates  MCPD (Markov Chains for Population  Data)  solver
for "Exit-state" model,  i.e. model  where  transition from X[i] to X[i+1]
is modelled as
    X[i+1] = P*X[i]
where
    X[i] and X[i+1] are N-dimensional state vectors
    P is a N*N transition matrix
and  one  selected component of X[] is called "exit"  state and is treated
in a special way:
    system state can transit from any state into "exit" state
    system state can not transit from "exit" state into any other state
    transition operator discards "exit" state (makes it zero at each turn)
Such  conditions  basically  mean  that  column  of P which corresponds to
"exit" state is zero. Multiplication by such P may decrease sum of  vector
components.

Such models arise when:
* there is some population of individuals
* individuals can have different states
* individuals can transit from one state to another
* population size is NOT constant - individuals can move into "exit" state
  and leave population at the next turn, but there are no new individuals
* amount of individuals which leave population can be predicted
* you want to model transitions of individuals from one state into another
  (including transitions into the "exit" state)

This model is discussed  in  more  details  in  the ALGLIB User Guide (see
http://www.alglib.net/dataanalysis/ for more data).

INPUT PARAMETERS:
    N       -   problem dimension, N>=2
    ExitState-  index of exit state, in 0..N-1

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdcreateexit(const ae_int_t n, const ae_int_t exitstate, mcpdstate &s, const xparams _xparams = alglib::xdefault);

`mcpdresults` function

/*************************************************************************
MCPD results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    P       -   array[N,N], transition matrix
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one. Speaking short, positive values  denote
                success, negative ones are failures.
                More information about fields of this  structure  can  be
                found in the comments on MCPDReport datatype.


  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdresults(const mcpdstate &s, real_2d_array &p, mcpdreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mcpdsetbc` function

/*************************************************************************
This function is used to add bound constraints  on  the  elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This  function  can  be  used  to  place bound   constraints  on arbitrary
subset  of  elements  of  P.  Set of constraints is specified by BndL/BndU
matrices, which may contain arbitrary combination  of  finite  numbers  or
infinities (like -INF<x<=0.5 or 0.1<=x<+INF).

You can also use MCPDAddBC() function which allows to ADD bound constraint
for one element of P without changing constraints for other elements.

These functions (MCPDSetBC and MCPDAddBC) interact as follows:
* there is internal matrix of bound constraints which is stored in the
  MCPD solver
* MCPDSetBC() replaces this matrix by another one (SET)
* MCPDAddBC() modifies one element of this matrix and  leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddBC()  call  preserves  all  modifications  done by previous
  calls,  while  MCPDSetBC()  completely discards all changes  done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    BndL    -   lower bounds constraints, array[N,N]. Elements of BndL can
                be finite numbers or -INF.
    BndU    -   upper bounds constraints, array[N,N]. Elements of BndU can
                be finite numbers or +INF.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsetbc(mcpdstate &s, const real_2d_array &bndl, const real_2d_array &bndu, const xparams _xparams = alglib::xdefault);

`mcpdsetec` function

/*************************************************************************
This function is used to add equality constraints on the elements  of  the
transition matrix P.

MCPD solver has four types of constraints which can be placed on P:
* user-specified equality constraints (optional)
* user-specified bound constraints (optional)
* user-specified general linear constraints (optional)
* basic constraints (always present):
  * non-negativity: P[i,j]>=0
  * consistency: every column of P sums to 1.0

Final  constraints  which  are  passed  to  the  underlying  optimizer are
calculated  as  intersection  of all present constraints. For example, you
may specify boundary constraint on P[0,0] and equality one:
    0.1<=P[0,0]<=0.9
    P[0,0]=0.5
Such  combination  of  constraints  will  be  silently  reduced  to  their
intersection, which is P[0,0]=0.5.

This  function  can  be  used  to  place equality constraints on arbitrary
subset of elements of P. Set of constraints is specified by EC, which  may
contain either NAN's or finite numbers from [0,1]. NAN denotes absence  of
constraint, finite number denotes equality constraint on specific  element
of P.

You can also  use  MCPDAddEC()  function  which  allows  to  ADD  equality
constraint  for  one  element  of P without changing constraints for other
elements.

These functions (MCPDSetEC and MCPDAddEC) interact as follows:
* there is internal matrix of equality constraints which is stored in  the
  MCPD solver
* MCPDSetEC() replaces this matrix by another one (SET)
* MCPDAddEC() modifies one element of this matrix and  leaves  other  ones
  unchanged (ADD)
* thus  MCPDAddEC()  call  preserves  all  modifications  done by previous
  calls,  while  MCPDSetEC()  completely discards all changes  done to the
  equality constraints.

INPUT PARAMETERS:
    S       -   solver
    EC      -   equality constraints, array[N,N]. Elements of  EC  can  be
                either NAN's or finite  numbers from  [0,1].  NAN  denotes
                absence  of  constraints,  while  finite  value    denotes
                equality constraint on the corresponding element of P.

NOTES:

1. infinite values of EC will lead to exception being thrown. Values  less
than 0.0 or greater than 1.0 will lead to error code being returned  after
call to MCPDSolve().

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsetec(mcpdstate &s, const real_2d_array &ec, const xparams _xparams = alglib::xdefault);

`mcpdsetlc` function

/*************************************************************************
This function is used to set linear equality/inequality constraints on the
elements of the transition matrix P.

This function can be used to set one or several general linear constraints
on the elements of P. Two types of constraints are supported:
* equality constraints
* inequality constraints (both less-or-equal and greater-or-equal)

Coefficients  of  constraints  are  specified  by  matrix  C (one  of  the
parameters).  One  row  of  C  corresponds  to  one  constraint.   Because
transition  matrix P has N*N elements,  we  need  N*N columns to store all
coefficients  (they  are  stored row by row), and one more column to store
right part - hence C has N*N+1 columns.  Constraint  kind is stored in the
CT array.

Thus, I-th linear constraint is
    P[0,0]*C[I,0] + P[0,1]*C[I,1] + .. + P[0,N-1]*C[I,N-1] +
        + P[1,0]*C[I,N] + P[1,1]*C[I,N+1] + ... +
        + P[N-1,N-1]*C[I,N*N-1]  ?=?  C[I,N*N]
where ?=? can be either "=" (CT[i]=0), "<=" (CT[i]<0) or ">=" (CT[i]>0).

Your constraint may involve only some subset of P (less than N*N elements).
For example it can be something like
    P[0,0] + P[0,1] = 0.5
In this case you still should pass matrix  with N*N+1 columns, but all its
elements (except for C[0,0], C[0,1] and C[0,N*N-1]) will be zero.

INPUT PARAMETERS:
    S       -   solver
    C       -   array[K,N*N+1] - coefficients of constraints
                (see above for complete description)
    CT      -   array[K] - constraint types
                (see above for complete description)
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsetlc(mcpdstate &s, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void mcpdsetlc(mcpdstate &s, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

`mcpdsetpredictionweights` function

/*************************************************************************
This function is used to change prediction weights

MCPD solver scales prediction errors as follows
    Error(P) = ||W*(y-P*x)||^2
where
    x is a system state at time t
    y is a system state at time t+1
    P is a transition matrix
    W is a diagonal scaling matrix

By default, weights are chosen in order  to  minimize  relative prediction
error instead of absolute one. For example, if one component of  state  is
about 0.5 in magnitude and another one is about 0.05, then algorithm  will
make corresponding weights equal to 2.0 and 20.0.

INPUT PARAMETERS:
    S       -   solver
    PW      -   array[N], weights:
                * must be non-negative values (exception will be thrown otherwise)
                * zero values will be replaced by automatically chosen values

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsetpredictionweights(mcpdstate &s, const real_1d_array &pw, const xparams _xparams = alglib::xdefault);

`mcpdsetprior` function

/*************************************************************************
This  function  allows to set prior values used for regularization of your
problem.

By default, regularizing term is equal to r*||P-prior_P||^2, where r is  a
small non-zero value,  P is transition matrix, prior_P is identity matrix,
||X||^2 is a sum of squared elements of X.

This  function  allows  you to change prior values prior_P. You  can  also
change r with MCPDSetTikhonovRegularizer() function.

INPUT PARAMETERS:
    S       -   solver
    PP      -   array[N,N], matrix of prior values:
                1. elements must be real numbers from [0,1]
                2. columns must sum to 1.0.
                First property is checked (exception is thrown otherwise),
                while second one is not checked/enforced.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsetprior(mcpdstate &s, const real_2d_array &pp, const xparams _xparams = alglib::xdefault);

`mcpdsettikhonovregularizer` function

/*************************************************************************
This function allows to  tune  amount  of  Tikhonov  regularization  being
applied to your problem.

By default, regularizing term is equal to r*||P-prior_P||^2, where r is  a
small non-zero value,  P is transition matrix, prior_P is identity matrix,
||X||^2 is a sum of squared elements of X.

This  function  allows  you to change coefficient r. You can  also  change
prior values with MCPDSetPrior() function.

INPUT PARAMETERS:
    S       -   solver
    V       -   regularization  coefficient, finite non-negative value. It
                is  not  recommended  to specify zero value unless you are
                pretty sure that you want it.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsettikhonovregularizer(mcpdstate &s, const double v, const xparams _xparams = alglib::xdefault);

`mcpdsolve` function

/*************************************************************************
This function is used to start solution of the MCPD problem.

After return from this function, you can use MCPDResults() to get solution
and completion code.

  -- ALGLIB --
     Copyright 23.05.2010 by Bochkanov Sergey
*************************************************************************/
void mcpdsolve(mcpdstate &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

mcpd_simple1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple MCPD example
        //
        // We have a loan portfolio. Our loans can be in one of two states:
        // * normal loans ("good" ones)
        // * past due loans ("bad" ones)
        //
        // We assume that:
        // * loans can transition from any state to any other state. In 
        //   particular, past due loan can become "good" one at any moment 
        //   with same (fixed) probability. Not realistic, but it is toy example :)
        // * portfolio size does not change over time
        //
        // Thus, we have following model
        //     state_new = P*state_old
        // where
        //         ( p00  p01 )
        //     P = (          )
        //         ( p10  p11 )
        //
        // We want to model transitions between these two states using MCPD
        // approach (Markov Chains for Proportional/Population Data), i.e.
        // to restore hidden transition matrix P using actual portfolio data.
        // We have:
        // * poportional data, i.e. proportion of loans in the normal and past 
        //   due states (not portfolio size measured in some currency, although 
        //   it is possible to work with population data too)
        // * two tracks, i.e. two sequences which describe portfolio
        //   evolution from two different starting states: [1,0] (all loans 
        //   are "good") and [0.8,0.2] (only 80% of portfolio is in the "good"
        //   state)
        //
        mcpdstate s;
        mcpdreport rep;
        real_2d_array p;
        real_2d_array track0 = "[[1.00000,0.00000],[0.95000,0.05000],[0.92750,0.07250],[0.91738,0.08263],[0.91282,0.08718]]";
        real_2d_array track1 = "[[0.80000,0.20000],[0.86000,0.14000],[0.88700,0.11300],[0.89915,0.10085]]";

        mcpdcreate(2, s);
        mcpdaddtrack(s, track0);
        mcpdaddtrack(s, track1);
        mcpdsolve(s);
        mcpdresults(s, p, rep);

        //
        // Hidden matrix P is equal to
        //         ( 0.95  0.50 )
        //         (            )
        //         ( 0.05  0.50 )
        // which means that "good" loans can become "bad" with 5% probability, 
        // while "bad" loans will return to good state with 50% probability.
        //
        printf("%s\n", p.tostring(2).c_str()); // EXPECTED: [[0.95,0.50],[0.05,0.50]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

mcpd_simple2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Simple MCPD example
        //
        // We have a loan portfolio. Our loans can be in one of three states:
        // * normal loans
        // * past due loans
        // * charged off loans
        //
        // We assume that:
        // * normal loan can stay normal or become past due (but not charged off)
        // * past due loan can stay past due, become normal or charged off
        // * charged off loan will stay charged off for the rest of eternity
        // * portfolio size does not change over time
        // Not realistic, but it is toy example :)
        //
        // Thus, we have following model
        //     state_new = P*state_old
        // where
        //         ( p00  p01    )
        //     P = ( p10  p11    )
        //         (      p21  1 )
        // i.e. four elements of P are known a priori.
        //
        // Although it is possible (given enough data) to In order to enforce 
        // this property we set equality constraints on these elements.
        //
        // We want to model transitions between these two states using MCPD
        // approach (Markov Chains for Proportional/Population Data), i.e.
        // to restore hidden transition matrix P using actual portfolio data.
        // We have:
        // * poportional data, i.e. proportion of loans in the current and past 
        //   due states (not portfolio size measured in some currency, although 
        //   it is possible to work with population data too)
        // * two tracks, i.e. two sequences which describe portfolio
        //   evolution from two different starting states: [1,0,0] (all loans 
        //   are "good") and [0.8,0.2,0.0] (only 80% of portfolio is in the "good"
        //   state)
        //
        mcpdstate s;
        mcpdreport rep;
        real_2d_array p;
        real_2d_array track0 = "[[1.000000,0.000000,0.000000],[0.950000,0.050000,0.000000],[0.927500,0.060000,0.012500],[0.911125,0.061375,0.027500],[0.896256,0.060900,0.042844]]";
        real_2d_array track1 = "[[0.800000,0.200000,0.000000],[0.860000,0.090000,0.050000],[0.862000,0.065500,0.072500],[0.851650,0.059475,0.088875],[0.838805,0.057451,0.103744]]";

        mcpdcreate(3, s);
        mcpdaddtrack(s, track0);
        mcpdaddtrack(s, track1);
        mcpdaddec(s, 0, 2, 0.0);
        mcpdaddec(s, 1, 2, 0.0);
        mcpdaddec(s, 2, 2, 1.0);
        mcpdaddec(s, 2, 0, 0.0);
        mcpdsolve(s);
        mcpdresults(s, p, rep);

        //
        // Hidden matrix P is equal to
        //         ( 0.95 0.50      )
        //         ( 0.05 0.25      )
        //         (      0.25 1.00 ) 
        // which means that "good" loans can become past due with 5% probability, 
        // while past due loans will become charged off with 25% probability or
        // return back to normal state with 50% probability.
        //
        printf("%s\n", p.tostring(2).c_str()); // EXPECTED: [[0.95,0.50,0.00],[0.05,0.25,0.00],[0.00,0.25,1.00]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minbc` subpackage

Classes

minbcreport
minbcstate

Functions

minbccreate
minbccreatef
minbciteration
minbcoptguardgradient
minbcoptguardnonc1test0results
minbcoptguardnonc1test1results
minbcoptguardresults
minbcoptguardsmoothness
minbcoptimize
minbcrequesttermination
minbcrestartfrom
minbcresults
minbcresultsbuf
minbcsetbc
minbcsetcond
minbcsetprecdefault
minbcsetprecdiag
minbcsetprecscale
minbcsetscale
minbcsetstpmax
minbcsetxrep

Examples

minbc_d_1		Nonlinear optimization with box constraints
minbc_numdiff		Nonlinear optimization with bound constraints and numerical differentiation

`minbcreport` class

/*************************************************************************
This structure stores optimization report:
* iterationscount           number of iterations
* nfev                      number of gradient evaluations
* terminationtype           termination type (see below)

TERMINATION CODES

terminationtype field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -3    inconsistent constraints.
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated by user who called minbcrequesttermination(). X contains
        point which was "current accepted" when  termination  request  was
        submitted.
*************************************************************************/
class minbcreport
{
public:
    minbcreport();
    minbcreport(const minbcreport &rhs);
    minbcreport& operator=(const minbcreport &rhs);
    virtual ~minbcreport();
    ae_int_t iterationscount;
    ae_int_t nfev;
    ae_int_t varidx;
    ae_int_t terminationtype;
};

`minbcstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinBC subpackage to work with this
object
*************************************************************************/
class minbcstate
{
public:
    minbcstate();
    minbcstate(const minbcstate &rhs);
    minbcstate& operator=(const minbcstate &rhs);
    virtual ~minbcstate();
};

`minbccreate` function

/*************************************************************************
                     BOX CONSTRAINED OPTIMIZATION
          WITH FAST ACTIVATION OF MULTIPLE BOX CONSTRAINTS

DESCRIPTION:
The  subroutine  minimizes  function   F(x) of N arguments subject  to box
constraints (with some of box constraints actually being equality ones).

This optimizer uses algorithm similar to that of MinBLEIC (optimizer  with
general linear constraints), but presence of box-only  constraints  allows
us to use faster constraint activation strategies. On large-scale problems,
with multiple constraints active at the solution, this  optimizer  can  be
several times faster than BLEIC.

REQUIREMENTS:
* user must provide function value and gradient
* starting point X0 must be feasible or
  not too far away from the feasible set
* grad(f) must be Lipschitz continuous on a level set:
  L = { x : f(x)<=f(x0) }
* function must be defined everywhere on the feasible set F

USAGE:

Constrained optimization if far more complex than the unconstrained one.
Here we give very brief outline of the BC optimizer. We strongly recommend
you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
on optimization, which is available at http://www.alglib.net/optimization/

1. User initializes algorithm state with MinBCCreate() call

2. USer adds box constraints by calling MinBCSetBC() function.

3. User sets stopping conditions with MinBCSetCond().

4. User calls MinBCOptimize() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.

5. User calls MinBCResults() to get solution

6. Optionally user may call MinBCRestartFrom() to solve another problem
   with same N but another starting point.
   MinBCRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbccreate(const ae_int_t n, const real_1d_array &x, minbcstate &state, const xparams _xparams = alglib::xdefault);
void minbccreate(const real_1d_array &x, minbcstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minbccreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinBCCreate().  It  uses
finite differences in order to differentiate target function.

Description below contains information which is specific to  this function
only. We recommend to read comments on MinBCCreate() in  order  to  get
more information about creation of BC optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinBCSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less  robust and precise. CG needs exact gradient values. Imprecise
   gradient may slow  down  convergence, especially  on  highly  nonlinear
   problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void minbccreatef(const ae_int_t n, const real_1d_array &x, const double diffstep, minbcstate &state, const xparams _xparams = alglib::xdefault);
void minbccreatef(const real_1d_array &x, const double diffstep, minbcstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minbciteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minbciteration(minbcstate &state, const xparams _xparams = alglib::xdefault);

`minbcoptguardgradient` function

/*************************************************************************
This  function  activates/deactivates verification  of  the  user-supplied
analytic gradient.

Upon  activation  of  this  option  OptGuard  integrity  checker  performs
numerical differentiation of your target function  at  the  initial  point
(note: future versions may also perform check  at  the  final  point)  and
compares numerical gradient with analytic one provided by you.

If difference is too large, an error flag is set and optimization  session
continues. After optimization session is over, you can retrieve the report
which  stores  both  gradients  and  specific  components  highlighted  as
suspicious by the OptGuard.

The primary OptGuard report can be retrieved with minbcoptguardresults().

IMPORTANT: gradient check is a high-overhead option which  will  cost  you
           about 3*N additional function evaluations. In many cases it may
           cost as much as the rest of the optimization session.

           YOU SHOULD NOT USE IT IN THE PRODUCTION CODE UNLESS YOU WANT TO
           CHECK DERIVATIVES PROVIDED BY SOME THIRD PARTY.

NOTE: unlike previous incarnation of the gradient checking code,  OptGuard
      does NOT interrupt optimization even if it discovers bad gradient.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step used for numerical differentiation:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification
                    You should carefully choose TestStep. Value  which  is
                    too large (so large that  function  behavior  is  non-
                    cubic at this scale) will lead  to  false  alarms. Too
                    short step will result in rounding  errors  dominating
                    numerical derivative.

                    You may use different step for different parameters by
                    means of setting scale with minbcsetscale().

=== EXPLANATION ==========================================================

In order to verify gradient algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void minbcoptguardgradient(minbcstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

`minbcoptguardnonc1test0results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #0

Nonsmoothness (non-C1) test #0 studies  function  values  (not  gradient!)
obtained during line searches and monitors  behavior  of  the  directional
derivative estimate.

This test is less powerful than test #1, but it does  not  depend  on  the
gradient values and thus it is more robust against artifacts introduced by
numerical differentiation.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #0 "strong" report
    lngrep  -   C1 test #0 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbcoptguardnonc1test0results(const minbcstate &state, optguardnonc1test0report &strrep, optguardnonc1test0report &lngrep, const xparams _xparams = alglib::xdefault);

`minbcoptguardnonc1test1results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #1

Nonsmoothness (non-C1)  test  #1  studies  individual  components  of  the
gradient computed during line search.

When precise analytic gradient is provided this test is more powerful than
test #0  which  works  with  function  values  and  ignores  user-provided
gradient.  However,  test  #0  becomes  more   powerful   when   numerical
differentiation is employed (in such cases test #1 detects  higher  levels
of numerical noise and becomes too conservative).

This test also tells specific components of the gradient which violate  C1
continuity, which makes it more informative than #0, which just tells that
continuity is violated.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* vidx - is an index of the variable in [0,N) with nonsmooth derivative
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], g[] - arrays of length CNT which store step lengths and  gradient
  values at these points; g[i] is evaluated in  x0+stp[i]*d  and  contains
  vidx-th component of the gradient.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #1 "strong" report
    lngrep  -   C1 test #1 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbcoptguardnonc1test1results(minbcstate &state, optguardnonc1test1report &strrep, optguardnonc1test1report &lngrep, const xparams _xparams = alglib::xdefault);

`minbcoptguardresults` function

/*************************************************************************
Results of OptGuard integrity check, should be called  after  optimization
session is over.

=== PRIMARY REPORT =======================================================

OptGuard performs several checks which are intended to catch common errors
in the implementation of nonlinear function/gradient:
* incorrect analytic gradient
* discontinuous (non-C0) target functions (constraints)
* nonsmooth     (non-C1) target functions (constraints)

Each of these checks is activated with appropriate function:
* minbcoptguardgradient() for gradient verification
* minbcoptguardsmoothness() for C0/C1 checks

Following flags are set when these errors are suspected:
* rep.badgradsuspected, and additionally:
  * rep.badgradvidx for specific variable (gradient element) suspected
  * rep.badgradxbase, a point where gradient is tested
  * rep.badgraduser, user-provided gradient  (stored  as  2D  matrix  with
    single row in order to make  report  structure  compatible  with  more
    complex optimizers like MinNLC or MinLM)
  * rep.badgradnum,   reference    gradient    obtained    via   numerical
    differentiation (stored as  2D matrix with single row in order to make
    report structure compatible with more complex optimizers  like  MinNLC
    or MinLM)
* rep.nonc0suspected
* rep.nonc1suspected

=== ADDITIONAL REPORTS/LOGS ==============================================

Several different tests are performed to catch C0/C1 errors, you can  find
out specific test signaled error by looking to:
* rep.nonc0test0positive, for non-C0 test #0
* rep.nonc1test0positive, for non-C1 test #0
* rep.nonc1test1positive, for non-C1 test #1

Additional information (including line search logs)  can  be  obtained  by
means of:
* minbcoptguardnonc1test0results()
* minbcoptguardnonc1test1results()
which return detailed error reports, specific points where discontinuities
were found, and so on.

==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    rep     -   generic OptGuard report;  more  detailed  reports  can  be
                retrieved with other functions.

NOTE: false negatives (nonsmooth problems are not identified as  nonsmooth
      ones) are possible although unlikely.

      The reason  is  that  you  need  to  make several evaluations around
      nonsmoothness  in  order  to  accumulate  enough  information  about
      function curvature. Say, if you start right from the nonsmooth point,
      optimizer simply won't get enough data to understand what  is  going
      wrong before it terminates due to abrupt changes in the  derivative.
      It is also  possible  that  "unlucky"  step  will  move  us  to  the
      termination too quickly.

      Our current approach is to have less than 0.1%  false  negatives  in
      our test examples  (measured  with  multiple  restarts  from  random
      points), and to have exactly 0% false positives.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbcoptguardresults(minbcstate &state, optguardreport &rep, const xparams _xparams = alglib::xdefault);

`minbcoptguardsmoothness` function

/*************************************************************************
This  function  activates/deactivates nonsmoothness monitoring  option  of
the  OptGuard  integrity  checker. Smoothness  monitor  silently  observes
solution process and tries to detect ill-posed problems, i.e. ones with:
a) discontinuous target function (non-C0)
b) nonsmooth     target function (non-C1)

Smoothness monitoring does NOT interrupt optimization  even if it suspects
that your problem is nonsmooth. It just sets corresponding  flags  in  the
OptGuard report which can be retrieved after optimization is over.

Smoothness monitoring is a moderate overhead option which often adds  less
than 1% to the optimizer running time. Thus, you can use it even for large
scale problems.

NOTE: OptGuard does  NOT  guarantee  that  it  will  always  detect  C0/C1
      continuity violations.

      First, minor errors are hard to  catch - say, a 0.0001 difference in
      the model values at two sides of the gap may be due to discontinuity
      of the model - or simply because the model has changed.

      Second, C1-violations  are  especially  difficult  to  detect  in  a
      noninvasive way. The optimizer usually  performs  very  short  steps
      near the nonsmoothness, and differentiation  usually   introduces  a
      lot of numerical noise.  It  is  hard  to  tell  whether  some  tiny
      discontinuity in the slope is due to real nonsmoothness or just  due
      to numerical noise alone.

      Our top priority was to avoid false positives, so in some rare cases
      minor errors may went unnoticed (however, in most cases they can  be
      spotted with restart from different initial point).

INPUT PARAMETERS:
    state   -   algorithm state
    level   -   monitoring level:
                * 0 - monitoring is disabled
                * 1 - noninvasive low-overhead monitoring; function values
                      and/or gradients are recorded, but OptGuard does not
                      try to perform additional evaluations  in  order  to
                      get more information about suspicious locations.

=== EXPLANATION ==========================================================

One major source of headache during optimization  is  the  possibility  of
the coding errors in the target function/constraints (or their gradients).
Such  errors   most   often   manifest   themselves  as  discontinuity  or
nonsmoothness of the target/constraints.

Another frequent situation is when you try to optimize something involving
lots of min() and max() operations, i.e. nonsmooth target. Although not  a
coding error, it is nonsmoothness anyway - and smooth  optimizers  usually
stop right after encountering nonsmoothness, well before reaching solution.

OptGuard integrity checker helps you to catch such situations: it monitors
function values/gradients being passed  to  the  optimizer  and  tries  to
errors. Upon discovering suspicious pair of points it  raises  appropriate
flag (and allows you to continue optimization). When optimization is done,
you can study OptGuard result.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbcoptguardsmoothness(minbcstate &state, const ae_int_t level, const xparams _xparams = alglib::xdefault);
void minbcoptguardsmoothness(minbcstate &state, const xparams _xparams = alglib::xdefault);

`minbcoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied gradient,  and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either  MinBCCreate() for analytical gradient or  MinBCCreateF()
   for numerical differentiation) you should choose appropriate variant of
   MinBCOptimize() - one  which  accepts  function  AND gradient or one
   which accepts function ONLY.

   Be careful to choose variant of MinBCOptimize() which corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinBCOptimize()  and specific
   function used to create optimizer.


                     |         USER PASSED TO MinBCOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinBCCreateF()    |     works               FAILS
   MinBCCreate()     |     FAILS               works

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function  and  MinBCOptimize()  version.   Attemps   to   use   such
   combination (for  example,  to  create optimizer with MinBCCreateF()
   and  to  pass  gradient  information  to  MinCGOptimize()) will lead to
   exception being thrown. Either  you  did  not pass gradient when it WAS
   needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcoptimize(minbcstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minbcoptimize(minbcstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbcrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void minbcrequesttermination(minbcstate &state, const xparams _xparams = alglib::xdefault);

`minbcrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinBCCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcrestartfrom(minbcstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`minbcresults` function

/*************************************************************************
BC results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8    internal integrity control  detected  infinite or
                        NAN   values   in   function/gradient.   Abnormal
                        termination signalled.
                * -3   inconsistent constraints.
                *  1   relative function improvement is no more than EpsF.
                *  2   scaled step is no more than EpsX.
                *  4   scaled gradient norm is no more than EpsG.
                *  5   MaxIts steps was taken
                *  8   terminated by user who called minbcrequesttermination().
                       X contains point which was "current accepted"  when
                       termination request was submitted.
                More information about fields of this  structure  can  be
                found in the comments on MinBCReport datatype.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcresults(const minbcstate &state, real_1d_array &x, minbcreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbcresultsbuf` function

/*************************************************************************
BC results

Buffered implementation of MinBCResults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcresultsbuf(const minbcstate &state, real_1d_array &x, minbcreport &rep, const xparams _xparams = alglib::xdefault);

`minbcsetbc` function

/*************************************************************************
This function sets boundary constraints for BC optimizer.

Boundary constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with MinBCRestartFrom().

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: this solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by  bound  constraints,
  even  when  numerical  differentiation is used (algorithm adjusts  nodes
  according to boundary constraints)

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetbc(minbcstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbcsetcond` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinBCSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinBCSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection.

NOTE: when SetCond() called with non-zero MaxIts, BC solver may perform
      slightly more than MaxIts iterations. I.e., MaxIts  sets  non-strict
      limit on iterations count.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetcond(minbcstate &state, const double epsg, const double epsf, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbcsetprecdefault` function

/*************************************************************************
Modification of the preconditioner: preconditioning is turned off.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetprecdefault(minbcstate &state, const xparams _xparams = alglib::xdefault);

`minbcsetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE 1: D[i] should be positive. Exception will be thrown otherwise.

NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetprecdiag(minbcstate &state, const real_1d_array &d, const xparams _xparams = alglib::xdefault);

`minbcsetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables  with  MinBCSetScale()
call  (before  or after MinBCSetPrecScale() call). Without knowledge of
the scale of your variables scale-based preconditioner will be  just  unit
matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetprecscale(minbcstate &state, const xparams _xparams = alglib::xdefault);

`minbcsetscale` function

/*************************************************************************
This function sets scaling coefficients for BC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

In  most  optimizers  (and  in  the  BC  too)  scaling is NOT a form of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner  by  separate  call  to  one  of  the  MinBCSetPrec...()
functions.

There is a special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minbcsetscale(minbcstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`minbcsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  lead   to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetstpmax(minbcstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`minbcsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinBCOptimize().

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbcsetxrep(minbcstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minbc_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // subject to box constraints
        //
        //     -1<=x<=+1, -1<=y<=+1
        //
        // using MinBC optimizer with:
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see minbcsetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        // * OptGuard integrity check being used to check problem statement
        //   for some common errors like nonsmoothness or bad analytic gradient
        //
        // First, we create optimizer object and tune its properties:
        // * set box constraints
        // * set variable scales
        // * set stopping criteria
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        minbcstate state;
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minbccreate(x, state);
        minbcsetbc(state, bndl, bndu);
        minbcsetscale(state, s);
        minbcsetcond(state, epsg, epsf, epsx, maxits);

        //
        // Then we activate OptGuard integrity checking.
        //
        // OptGuard monitor helps to catch common coding and problem statement
        // issues, like:
        // * discontinuity of the target function (C0 continuity violation)
        // * nonsmoothness of the target function (C1 continuity violation)
        // * erroneous analytic gradient, i.e. one inconsistent with actual
        //   change in the target/constraints
        //
        // OptGuard is essential for early prototyping stages because such
        // problems often result in premature termination of the optimizer
        // which is really hard to distinguish from the correct termination.
        //
        // IMPORTANT: GRADIENT VERIFICATION IS PERFORMED BY MEANS OF NUMERICAL
        //            DIFFERENTIATION. DO NOT USE IT IN PRODUCTION CODE!!!!!!!
        //
        //            Other OptGuard checks add moderate overhead, but anyway
        //            it is better to turn them off when they are not needed.
        //
        minbcoptguardsmoothness(state);
        minbcoptguardgradient(state, 0.001);

        //
        // Optimize and evaluate results
        //
        minbcreport rep;
        alglib::minbcoptimize(state, function1_grad);
        minbcresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]

        //
        // Check that OptGuard did not report errors
        //
        // NOTE: want to test OptGuard? Try breaking the gradient - say, add
        //       1.0 to some of its components.
        //
        optguardreport ogrep;
        minbcoptguardresults(state, ogrep);
        printf("%s\n", ogrep.badgradsuspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minbc_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // subject to box constraints
        //
        //    -1<=x<=+1, -1<=y<=+1
        //
        // using MinBC optimizer with:
        // * numerical differentiation being used
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see minbcsetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        // * OptGuard integrity check being used to check problem statement
        //   for some common errors like nonsmoothness or bad analytic gradient
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        minbcstate state;
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        double diffstep = 1.0e-6;

        //
        // Now we are ready to actually optimize something:
        // * first we create optimizer
        // * we add boundary constraints
        // * we tune stopping conditions
        // * and, finally, optimize and obtain results...
        //
        minbccreatef(x, diffstep, state);
        minbcsetbc(state, bndl, bndu);
        minbcsetscale(state, s);
        minbcsetcond(state, epsg, epsf, epsx, maxits);

        //
        // Then we activate OptGuard integrity checking.
        //
        // Numerical differentiation always produces "correct" gradient
        // (with some truncation error, but unbiased). Thus, we just have
        // to check smoothness properties of the target: C0 and C1 continuity.
        //
        // Sometimes user accidentally tries to solve nonsmooth problems
        // with smooth optimizer. OptGuard helps to detect such situations
        // early, at the prototyping stage.
        //
        minbcoptguardsmoothness(state);

        //
        // Optimize and evaluate results
        //
        minbcreport rep;
        alglib::minbcoptimize(state, function1_func);
        minbcresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]

        //
        // Check that OptGuard did not report errors
        //
        // Want to challenge OptGuard? Try to make your problem
        // nonsmooth by replacing 100*(x+3)^4 by 100*|x+3| and
        // re-run optimizer.
        //
        optguardreport ogrep;
        minbcoptguardresults(state, ogrep);
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minbleic` subpackage

Classes

minbleicreport
minbleicstate

Functions

minbleiccreate
minbleiccreatef
minbleiciteration
minbleicoptguardgradient
minbleicoptguardnonc1test0results
minbleicoptguardnonc1test1results
minbleicoptguardresults
minbleicoptguardsmoothness
minbleicoptimize
minbleicrequesttermination
minbleicrestartfrom
minbleicresults
minbleicresultsbuf
minbleicsetbc
minbleicsetcond
minbleicsetlc
minbleicsetprecdefault
minbleicsetprecdiag
minbleicsetprecscale
minbleicsetscale
minbleicsetstpmax
minbleicsetxrep

Examples

minbleic_d_1		Nonlinear optimization with bound constraints
minbleic_d_2		Nonlinear optimization with linear inequality constraints
minbleic_numdiff		Nonlinear optimization with bound constraints and numerical differentiation

`minbleicreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           number of iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -3    inconsistent constraints. Feasible point is
        either nonexistent or too hard to find. Try to
        restart optimizer with better initial approximation
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated by user who called minbleicrequesttermination(). X contains
        point which was "current accepted" when  termination  request  was
        submitted.

ADDITIONAL FIELDS

There are additional fields which can be used for debugging:
* DebugEqErr                error in the equality constraints (2-norm)
* DebugFS                   f, calculated at projection of initial point
                            to the feasible set
* DebugFF                   f, calculated at the final point
* DebugDX                   |X_start-X_final|
*************************************************************************/
class minbleicreport
{
public:
    minbleicreport();
    minbleicreport(const minbleicreport &rhs);
    minbleicreport& operator=(const minbleicreport &rhs);
    virtual ~minbleicreport();
    ae_int_t iterationscount;
    ae_int_t nfev;
    ae_int_t varidx;
    ae_int_t terminationtype;
    double debugeqerr;
    double debugfs;
    double debugff;
    double debugdx;
    ae_int_t debugfeasqpits;
    ae_int_t debugfeasgpaits;
    ae_int_t inneriterationscount;
    ae_int_t outeriterationscount;
};

`minbleicstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinBLEIC subpackage to work with this
object
*************************************************************************/
class minbleicstate
{
public:
    minbleicstate();
    minbleicstate(const minbleicstate &rhs);
    minbleicstate& operator=(const minbleicstate &rhs);
    virtual ~minbleicstate();
};

`minbleiccreate` function

/*************************************************************************
                     BOUND CONSTRAINED OPTIMIZATION
       WITH ADDITIONAL LINEAR EQUALITY AND INEQUALITY CONSTRAINTS

DESCRIPTION:
The  subroutine  minimizes  function   F(x)  of N arguments subject to any
combination of:
* bound constraints
* linear inequality constraints
* linear equality constraints

REQUIREMENTS:
* user must provide function value and gradient
* starting point X0 must be feasible or
  not too far away from the feasible set
* grad(f) must be Lipschitz continuous on a level set:
  L = { x : f(x)<=f(x0) }
* function must be defined everywhere on the feasible set F

USAGE:

Constrained optimization if far more complex than the unconstrained one.
Here we give very brief outline of the BLEIC optimizer. We strongly recommend
you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide
on optimization, which is available at http://www.alglib.net/optimization/

1. User initializes algorithm state with MinBLEICCreate() call

2. USer adds boundary and/or linear constraints by calling
   MinBLEICSetBC() and MinBLEICSetLC() functions.

3. User sets stopping conditions with MinBLEICSetCond().

4. User calls MinBLEICOptimize() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.

5. User calls MinBLEICResults() to get solution

6. Optionally user may call MinBLEICRestartFrom() to solve another problem
   with same N but another starting point.
   MinBLEICRestartFrom() allows to reuse already initialized structure.

NOTE: if you have box-only constraints (no  general  linear  constraints),
      then MinBC optimizer can be better option. It uses  special,  faster
      constraint activation method, which performs better on problems with
      multiple constraints active at the solution.

      On small-scale problems performance of MinBC is similar to  that  of
      MinBLEIC, but on large-scale ones (hundreds and thousands of  active
      constraints) it can be several times faster than MinBLEIC.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleiccreate(const ae_int_t n, const real_1d_array &x, minbleicstate &state, const xparams _xparams = alglib::xdefault);
void minbleiccreate(const real_1d_array &x, minbleicstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbleiccreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinBLEICCreate().  It  uses
finite differences in order to differentiate target function.

Description below contains information which is specific to  this function
only. We recommend to read comments on MinBLEICCreate() in  order  to  get
more information about creation of BLEIC optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinBLEICSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less  robust and precise. CG needs exact gradient values. Imprecise
   gradient may slow  down  convergence, especially  on  highly  nonlinear
   problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void minbleiccreatef(const ae_int_t n, const real_1d_array &x, const double diffstep, minbleicstate &state, const xparams _xparams = alglib::xdefault);
void minbleiccreatef(const real_1d_array &x, const double diffstep, minbleicstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minbleiciteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minbleiciteration(minbleicstate &state, const xparams _xparams = alglib::xdefault);

`minbleicoptguardgradient` function

/*************************************************************************
This  function  activates/deactivates verification  of  the  user-supplied
analytic gradient.

Upon  activation  of  this  option  OptGuard  integrity  checker  performs
numerical differentiation of your target function  at  the  initial  point
(note: future versions may also perform check  at  the  final  point)  and
compares numerical gradient with analytic one provided by you.

If difference is too large, an error flag is set and optimization  session
continues. After optimization session is over, you can retrieve the report
which  stores  both  gradients  and  specific  components  highlighted  as
suspicious by the OptGuard.

The primary OptGuard report can be retrieved with minbleicoptguardresults().

IMPORTANT: gradient check is a high-overhead option which  will  cost  you
           about 3*N additional function evaluations. In many cases it may
           cost as much as the rest of the optimization session.

           YOU SHOULD NOT USE IT IN THE PRODUCTION CODE UNLESS YOU WANT TO
           CHECK DERIVATIVES PROVIDED BY SOME THIRD PARTY.

NOTE: unlike previous incarnation of the gradient checking code,  OptGuard
      does NOT interrupt optimization even if it discovers bad gradient.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step used for numerical differentiation:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification
                    You should carefully choose TestStep. Value  which  is
                    too large (so large that  function  behavior  is  non-
                    cubic at this scale) will lead  to  false  alarms. Too
                    short step will result in rounding  errors  dominating
                    numerical derivative.

                    You may use different step for different parameters by
                    means of setting scale with minbleicsetscale().

=== EXPLANATION ==========================================================

In order to verify gradient algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void minbleicoptguardgradient(minbleicstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

`minbleicoptguardnonc1test0results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #0

Nonsmoothness (non-C1) test #0 studies  function  values  (not  gradient!)
obtained during line searches and monitors  behavior  of  the  directional
derivative estimate.

This test is less powerful than test #1, but it does  not  depend  on  the
gradient values and thus it is more robust against artifacts introduced by
numerical differentiation.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #0 "strong" report
    lngrep  -   C1 test #0 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbleicoptguardnonc1test0results(const minbleicstate &state, optguardnonc1test0report &strrep, optguardnonc1test0report &lngrep, const xparams _xparams = alglib::xdefault);

`minbleicoptguardnonc1test1results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #1

Nonsmoothness (non-C1)  test  #1  studies  individual  components  of  the
gradient computed during line search.

When precise analytic gradient is provided this test is more powerful than
test #0  which  works  with  function  values  and  ignores  user-provided
gradient.  However,  test  #0  becomes  more   powerful   when   numerical
differentiation is employed (in such cases test #1 detects  higher  levels
of numerical noise and becomes too conservative).

This test also tells specific components of the gradient which violate  C1
continuity, which makes it more informative than #0, which just tells that
continuity is violated.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* vidx - is an index of the variable in [0,N) with nonsmooth derivative
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], g[] - arrays of length CNT which store step lengths and  gradient
  values at these points; g[i] is evaluated in  x0+stp[i]*d  and  contains
  vidx-th component of the gradient.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #1 "strong" report
    lngrep  -   C1 test #1 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbleicoptguardnonc1test1results(minbleicstate &state, optguardnonc1test1report &strrep, optguardnonc1test1report &lngrep, const xparams _xparams = alglib::xdefault);

`minbleicoptguardresults` function

/*************************************************************************
Results of OptGuard integrity check, should be called  after  optimization
session is over.

=== PRIMARY REPORT =======================================================

OptGuard performs several checks which are intended to catch common errors
in the implementation of nonlinear function/gradient:
* incorrect analytic gradient
* discontinuous (non-C0) target functions (constraints)
* nonsmooth     (non-C1) target functions (constraints)

Each of these checks is activated with appropriate function:
* minbleicoptguardgradient() for gradient verification
* minbleicoptguardsmoothness() for C0/C1 checks

Following flags are set when these errors are suspected:
* rep.badgradsuspected, and additionally:
  * rep.badgradvidx for specific variable (gradient element) suspected
  * rep.badgradxbase, a point where gradient is tested
  * rep.badgraduser, user-provided gradient  (stored  as  2D  matrix  with
    single row in order to make  report  structure  compatible  with  more
    complex optimizers like MinNLC or MinLM)
  * rep.badgradnum,   reference    gradient    obtained    via   numerical
    differentiation (stored as  2D matrix with single row in order to make
    report structure compatible with more complex optimizers  like  MinNLC
    or MinLM)
* rep.nonc0suspected
* rep.nonc1suspected

=== ADDITIONAL REPORTS/LOGS ==============================================

Several different tests are performed to catch C0/C1 errors, you can  find
out specific test signaled error by looking to:
* rep.nonc0test0positive, for non-C0 test #0
* rep.nonc1test0positive, for non-C1 test #0
* rep.nonc1test1positive, for non-C1 test #1

Additional information (including line search logs)  can  be  obtained  by
means of:
* minbleicoptguardnonc1test0results()
* minbleicoptguardnonc1test1results()
which return detailed error reports, specific points where discontinuities
were found, and so on.

==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    rep     -   generic OptGuard report;  more  detailed  reports  can  be
                retrieved with other functions.

NOTE: false negatives (nonsmooth problems are not identified as  nonsmooth
      ones) are possible although unlikely.

      The reason  is  that  you  need  to  make several evaluations around
      nonsmoothness  in  order  to  accumulate  enough  information  about
      function curvature. Say, if you start right from the nonsmooth point,
      optimizer simply won't get enough data to understand what  is  going
      wrong before it terminates due to abrupt changes in the  derivative.
      It is also  possible  that  "unlucky"  step  will  move  us  to  the
      termination too quickly.

      Our current approach is to have less than 0.1%  false  negatives  in
      our test examples  (measured  with  multiple  restarts  from  random
      points), and to have exactly 0% false positives.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbleicoptguardresults(minbleicstate &state, optguardreport &rep, const xparams _xparams = alglib::xdefault);

`minbleicoptguardsmoothness` function

/*************************************************************************
This  function  activates/deactivates nonsmoothness monitoring  option  of
the  OptGuard  integrity  checker. Smoothness  monitor  silently  observes
solution process and tries to detect ill-posed problems, i.e. ones with:
a) discontinuous target function (non-C0)
b) nonsmooth     target function (non-C1)

Smoothness monitoring does NOT interrupt optimization  even if it suspects
that your problem is nonsmooth. It just sets corresponding  flags  in  the
OptGuard report which can be retrieved after optimization is over.

Smoothness monitoring is a moderate overhead option which often adds  less
than 1% to the optimizer running time. Thus, you can use it even for large
scale problems.

NOTE: OptGuard does  NOT  guarantee  that  it  will  always  detect  C0/C1
      continuity violations.

      First, minor errors are hard to  catch - say, a 0.0001 difference in
      the model values at two sides of the gap may be due to discontinuity
      of the model - or simply because the model has changed.

      Second, C1-violations  are  especially  difficult  to  detect  in  a
      noninvasive way. The optimizer usually  performs  very  short  steps
      near the nonsmoothness, and differentiation  usually   introduces  a
      lot of numerical noise.  It  is  hard  to  tell  whether  some  tiny
      discontinuity in the slope is due to real nonsmoothness or just  due
      to numerical noise alone.

      Our top priority was to avoid false positives, so in some rare cases
      minor errors may went unnoticed (however, in most cases they can  be
      spotted with restart from different initial point).

INPUT PARAMETERS:
    state   -   algorithm state
    level   -   monitoring level:
                * 0 - monitoring is disabled
                * 1 - noninvasive low-overhead monitoring; function values
                      and/or gradients are recorded, but OptGuard does not
                      try to perform additional evaluations  in  order  to
                      get more information about suspicious locations.

=== EXPLANATION ==========================================================

One major source of headache during optimization  is  the  possibility  of
the coding errors in the target function/constraints (or their gradients).
Such  errors   most   often   manifest   themselves  as  discontinuity  or
nonsmoothness of the target/constraints.

Another frequent situation is when you try to optimize something involving
lots of min() and max() operations, i.e. nonsmooth target. Although not  a
coding error, it is nonsmoothness anyway - and smooth  optimizers  usually
stop right after encountering nonsmoothness, well before reaching solution.

OptGuard integrity checker helps you to catch such situations: it monitors
function values/gradients being passed  to  the  optimizer  and  tries  to
errors. Upon discovering suspicious pair of points it  raises  appropriate
flag (and allows you to continue optimization). When optimization is done,
you can study OptGuard result.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minbleicoptguardsmoothness(minbleicstate &state, const ae_int_t level, const xparams _xparams = alglib::xdefault);
void minbleicoptguardsmoothness(minbleicstate &state, const xparams _xparams = alglib::xdefault);

`minbleicoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied gradient,  and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either  MinBLEICCreate() for analytical gradient or  MinBLEICCreateF()
   for numerical differentiation) you should choose appropriate variant of
   MinBLEICOptimize() - one  which  accepts  function  AND gradient or one
   which accepts function ONLY.

   Be careful to choose variant of MinBLEICOptimize() which corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinBLEICOptimize()  and specific
   function used to create optimizer.


                     |         USER PASSED TO MinBLEICOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinBLEICCreateF() |     work                FAIL
   MinBLEICCreate()  |     FAIL                work

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function  and  MinBLEICOptimize()  version.   Attemps   to   use   such
   combination (for  example,  to  create optimizer with MinBLEICCreateF()
   and  to  pass  gradient information to MinBLEICOptimize()) will lead to
   exception being thrown. Either  you  did  not pass gradient when it WAS
   needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicoptimize(minbleicstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minbleicoptimize(minbleicstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minbleicrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void minbleicrequesttermination(minbleicstate &state, const xparams _xparams = alglib::xdefault);

`minbleicrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinBLEICCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicrestartfrom(minbleicstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`minbleicresults` function

/*************************************************************************
BLEIC results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8    internal integrity control  detected  infinite or
                        NAN   values   in   function/gradient.   Abnormal
                        termination signalled.
                * -3   inconsistent constraints. Feasible point is
                       either nonexistent or too hard to find. Try to
                       restart optimizer with better initial approximation
                *  1   relative function improvement is no more than EpsF.
                *  2   scaled step is no more than EpsX.
                *  4   scaled gradient norm is no more than EpsG.
                *  5   MaxIts steps was taken
                *  8   terminated by user who called minbleicrequesttermination().
                       X contains point which was "current accepted"  when
                       termination request was submitted.
                More information about fields of this  structure  can  be
                found in the comments on MinBLEICReport datatype.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicresults(const minbleicstate &state, real_1d_array &x, minbleicreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minbleicresultsbuf` function

/*************************************************************************
BLEIC results

Buffered implementation of MinBLEICResults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicresultsbuf(const minbleicstate &state, real_1d_array &x, minbleicreport &rep, const xparams _xparams = alglib::xdefault);

`minbleicsetbc` function

/*************************************************************************
This function sets boundary constraints for BLEIC optimizer.

Boundary constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with MinBLEICRestartFrom().

NOTE: if you have box-only constraints (no  general  linear  constraints),
      then MinBC optimizer can be better option. It uses  special,  faster
      constraint activation method, which performs better on problems with
      multiple constraints active at the solution.

      On small-scale problems performance of MinBC is similar to  that  of
      MinBLEIC, but on large-scale ones (hundreds and thousands of  active
      constraints) it can be several times faster than MinBLEIC.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: this solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by  bound  constraints,
  even  when  numerical  differentiation is used (algorithm adjusts  nodes
  according to boundary constraints)

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetbc(minbleicstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minbleicsetcond` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinBLEICSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinBLEICSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection.

NOTE: when SetCond() called with non-zero MaxIts, BLEIC solver may perform
      slightly more than MaxIts iterations. I.e., MaxIts  sets  non-strict
      limit on iterations count.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetcond(minbleicstate &state, const double epsg, const double epsf, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minbleicsetlc` function

/*************************************************************************
This function sets linear constraints for BLEIC optimizer.

Linear constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with MinBLEICRestartFrom().

INPUT PARAMETERS:
    State   -   structure previously allocated with MinBLEICCreate call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE 1: linear (non-bound) constraints are satisfied only approximately:
* there always exists some minor violation (about Epsilon in magnitude)
  due to rounding errors
* numerical differentiation, if used, may  lead  to  function  evaluations
  outside  of the feasible  area,   because   algorithm  does  NOT  change
  numerical differentiation formula according to linear constraints.
If you want constraints to be  satisfied  exactly, try to reformulate your
problem  in  such  manner  that  all constraints will become boundary ones
(this kind of constraints is always satisfied exactly, both in  the  final
solution and in all intermediate points).

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetlc(minbleicstate &state, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minbleicsetlc(minbleicstate &state, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minbleicsetprecdefault` function

/*************************************************************************
Modification of the preconditioner: preconditioning is turned off.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetprecdefault(minbleicstate &state, const xparams _xparams = alglib::xdefault);

`minbleicsetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE 1: D[i] should be positive. Exception will be thrown otherwise.

NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetprecdiag(minbleicstate &state, const real_1d_array &d, const xparams _xparams = alglib::xdefault);

`minbleicsetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables  with  MinBLEICSetScale()
call  (before  or after MinBLEICSetPrecScale() call). Without knowledge of
the scale of your variables scale-based preconditioner will be  just  unit
matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetprecscale(minbleicstate &state, const xparams _xparams = alglib::xdefault);

`minbleicsetscale` function

/*************************************************************************
This function sets scaling coefficients for BLEIC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

In  most  optimizers  (and  in  the  BLEIC  too)  scaling is NOT a form of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner  by  separate  call  to  one  of  the  MinBLEICSetPrec...()
functions.

There is a special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minbleicsetscale(minbleicstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`minbleicsetstpmax` function

/*************************************************************************
This function sets maximum step length

IMPORTANT: this feature is hard to combine with preconditioning. You can't
set upper limit on step length, when you solve optimization  problem  with
linear (non-boundary) constraints AND preconditioner turned on.

When  non-boundary  constraints  are  present,  you  have to either a) use
preconditioner, or b) use upper limit on step length.  YOU CAN'T USE BOTH!
In this case algorithm will terminate with appropriate error code.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  lead   to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetstpmax(minbleicstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`minbleicsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinBLEICOptimize().

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetxrep(minbleicstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minbleic_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // subject to box constraints
        //
        //     -1<=x<=+1, -1<=y<=+1
        //
        // using BLEIC optimizer with:
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see minbleicsetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        // * OptGuard integrity check being used to check problem statement
        //   for some common errors like nonsmoothness or bad analytic gradient
        //
        // First, we create optimizer object and tune its properties:
        // * set box constraints
        // * set variable scales
        // * set stopping criteria
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minbleicstate state;
        minbleiccreate(x, state);
        minbleicsetbc(state, bndl, bndu);
        minbleicsetscale(state, s);
        minbleicsetcond(state, epsg, epsf, epsx, maxits);

        //
        // Then we activate OptGuard integrity checking.
        //
        // OptGuard monitor helps to catch common coding and problem statement
        // issues, like:
        // * discontinuity of the target function (C0 continuity violation)
        // * nonsmoothness of the target function (C1 continuity violation)
        // * erroneous analytic gradient, i.e. one inconsistent with actual
        //   change in the target/constraints
        //
        // OptGuard is essential for early prototyping stages because such
        // problems often result in premature termination of the optimizer
        // which is really hard to distinguish from the correct termination.
        //
        // IMPORTANT: GRADIENT VERIFICATION IS PERFORMED BY MEANS OF NUMERICAL
        //            DIFFERENTIATION. DO NOT USE IT IN PRODUCTION CODE!!!!!!!
        //
        //            Other OptGuard checks add moderate overhead, but anyway
        //            it is better to turn them off when they are not needed.
        //
        minbleicoptguardsmoothness(state);
        minbleicoptguardgradient(state, 0.001);

        //
        // Optimize and evaluate results
        //
        minbleicreport rep;
        alglib::minbleicoptimize(state, function1_grad);
        minbleicresults(state, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]

        //
        // Check that OptGuard did not report errors
        //
        // NOTE: want to test OptGuard? Try breaking the gradient - say, add
        //       1.0 to some of its components.
        //
        optguardreport ogrep;
        minbleicoptguardresults(state, ogrep);
        printf("%s\n", ogrep.badgradsuspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minbleic_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // subject to inequality constraints
        //
        // * x>=2 (posed as general linear constraint),
        // * x+y>=6
        //
        // using BLEIC optimizer with
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see minbleicsetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        // * OptGuard integrity check being used to check problem statement
        //   for some common errors like nonsmoothness or bad analytic gradient
        //
        // First, we create optimizer object and tune its properties:
        // * set linear constraints
        // * set variable scales
        // * set stopping criteria
        //
        real_1d_array x = "[5,5]";
        real_1d_array s = "[1,1]";
        real_2d_array c = "[[1,0,2],[1,1,6]]";
        integer_1d_array ct = "[1,1]";
        minbleicstate state;
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.000001;
        ae_int_t maxits = 0;

        minbleiccreate(x, state);
        minbleicsetlc(state, c, ct);
        minbleicsetscale(state, s);
        minbleicsetcond(state, epsg, epsf, epsx, maxits);

        //
        // Then we activate OptGuard integrity checking.
        //
        // OptGuard monitor helps to catch common coding and problem statement
        // issues, like:
        // * discontinuity of the target function (C0 continuity violation)
        // * nonsmoothness of the target function (C1 continuity violation)
        // * erroneous analytic gradient, i.e. one inconsistent with actual
        //   change in the target/constraints
        //
        // OptGuard is essential for early prototyping stages because such
        // problems often result in premature termination of the optimizer
        // which is really hard to distinguish from the correct termination.
        //
        // IMPORTANT: GRADIENT VERIFICATION IS PERFORMED BY MEANS OF NUMERICAL
        //            DIFFERENTIATION. DO NOT USE IT IN PRODUCTION CODE!!!!!!!
        //
        //            Other OptGuard checks add moderate overhead, but anyway
        //            it is better to turn them off when they are not needed.
        //
        minbleicoptguardsmoothness(state);
        minbleicoptguardgradient(state, 0.001);

        //
        // Optimize and evaluate results
        //
        minbleicreport rep;
        alglib::minbleicoptimize(state, function1_grad);
        minbleicresults(state, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,4]

        //
        // Check that OptGuard did not report errors
        //
        // NOTE: want to test OptGuard? Try breaking the gradient - say, add
        //       1.0 to some of its components.
        //
        optguardreport ogrep;
        minbleicoptguardresults(state, ogrep);
        printf("%s\n", ogrep.badgradsuspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minbleic_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // subject to box constraints
        //
        //     -1<=x<=+1, -1<=y<=+1
        //
        // using BLEIC optimizer with:
        // * numerical differentiation being used
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see minbleicsetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        // * OptGuard integrity check being used to check problem statement
        //   for some common errors like nonsmoothness or bad analytic gradient
        //
        // First, we create optimizer object and tune its properties:
        // * set box constraints
        // * set variable scales
        // * set stopping criteria
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        minbleicstate state;
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        double diffstep = 1.0e-6;

        minbleiccreatef(x, diffstep, state);
        minbleicsetbc(state, bndl, bndu);
        minbleicsetscale(state, s);
        minbleicsetcond(state, epsg, epsf, epsx, maxits);

        //
        // Then we activate OptGuard integrity checking.
        //
        // Numerical differentiation always produces "correct" gradient
        // (with some truncation error, but unbiased). Thus, we just have
        // to check smoothness properties of the target: C0 and C1 continuity.
        //
        // Sometimes user accidentally tries to solve nonsmooth problems
        // with smooth optimizer. OptGuard helps to detect such situations
        // early, at the prototyping stage.
        //
        minbleicoptguardsmoothness(state);

        //
        // Optimize and evaluate results
        //
        minbleicreport rep;
        alglib::minbleicoptimize(state, function1_func);
        minbleicresults(state, x, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]

        //
        // Check that OptGuard did not report errors
        //
        // Want to challenge OptGuard? Try to make your problem
        // nonsmooth by replacing 100*(x+3)^4 by 100*|x+3| and
        // re-run optimizer.
        //
        optguardreport ogrep;
        minbleicoptguardresults(state, ogrep);
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`mincg` subpackage

Classes

mincgreport
mincgstate

Functions

mincgcreate
mincgcreatef
mincgiteration
mincgoptguardgradient
mincgoptguardnonc1test0results
mincgoptguardnonc1test1results
mincgoptguardresults
mincgoptguardsmoothness
mincgoptimize
mincgrequesttermination
mincgrestartfrom
mincgresults
mincgresultsbuf
mincgsetcgtype
mincgsetcond
mincgsetprecdefault
mincgsetprecdiag
mincgsetprecscale
mincgsetscale
mincgsetstpmax
mincgsetxrep
mincgsuggeststep

Examples

mincg_d_1		Nonlinear optimization by CG
mincg_d_2		Nonlinear optimization with additional settings and restarts
mincg_numdiff		Nonlinear optimization by CG with numerical differentiation

`mincgreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated by user who called mincgrequesttermination(). X contains
        point which was "current accepted" when  termination  request  was
        submitted.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class mincgreport
{
public:
    mincgreport();
    mincgreport(const mincgreport &rhs);
    mincgreport& operator=(const mincgreport &rhs);
    virtual ~mincgreport();
    ae_int_t iterationscount;
    ae_int_t nfev;
    ae_int_t terminationtype;
};

`mincgstate` class

/*************************************************************************
This object stores state of the nonlinear CG optimizer.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class mincgstate
{
public:
    mincgstate();
    mincgstate(const mincgstate &rhs);
    mincgstate& operator=(const mincgstate &rhs);
    virtual ~mincgstate();
};

`mincgcreate` function

/*************************************************************************
        NONLINEAR CONJUGATE GRADIENT METHOD

DESCRIPTION:
The subroutine minimizes function F(x) of N arguments by using one of  the
nonlinear conjugate gradient methods.

These CG methods are globally convergent (even on non-convex functions) as
long as grad(f) is Lipschitz continuous in  a  some  neighborhood  of  the
L = { x : f(x)<=f(x0) }.


REQUIREMENTS:
Algorithm will request following information during its operation:
* function value F and its gradient G (simultaneously) at given point X


USAGE:
1. User initializes algorithm state with MinCGCreate() call
2. User tunes solver parameters with MinCGSetCond(), MinCGSetStpMax() and
   other functions
3. User calls MinCGOptimize() function which takes algorithm  state   and
   pointer (delegate, etc.) to callback function which calculates F/G.
4. User calls MinCGResults() to get solution
5. Optionally, user may call MinCGRestartFrom() to solve another  problem
   with same N but another starting point and/or another function.
   MinCGRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 25.03.2010 by Bochkanov Sergey
*************************************************************************/
void mincgcreate(const ae_int_t n, const real_1d_array &x, mincgstate &state, const xparams _xparams = alglib::xdefault);
void mincgcreate(const real_1d_array &x, mincgstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mincgcreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinCGCreate(). It uses
finite differences in order to differentiate target function.

Description below contains information which is specific to this function
only. We recommend to read comments on MinCGCreate() in order to get more
information about creation of CG optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinCGSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less  robust  and  precise.  L-BFGS  needs  exact  gradient values.
   Imprecise  gradient may slow down  convergence,  especially  on  highly
   nonlinear problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void mincgcreatef(const ae_int_t n, const real_1d_array &x, const double diffstep, mincgstate &state, const xparams _xparams = alglib::xdefault);
void mincgcreatef(const real_1d_array &x, const double diffstep, mincgstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mincgiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool mincgiteration(mincgstate &state, const xparams _xparams = alglib::xdefault);

`mincgoptguardgradient` function

/*************************************************************************
This  function  activates/deactivates verification  of  the  user-supplied
analytic gradient.

Upon  activation  of  this  option  OptGuard  integrity  checker  performs
numerical differentiation of your target function  at  the  initial  point
(note: future versions may also perform check  at  the  final  point)  and
compares numerical gradient with analytic one provided by you.

If difference is too large, an error flag is set and optimization  session
continues. After optimization session is over, you can retrieve the report
which  stores  both  gradients  and  specific  components  highlighted  as
suspicious by the OptGuard.

The primary OptGuard report can be retrieved with mincgoptguardresults().

IMPORTANT: gradient check is a high-overhead option which  will  cost  you
           about 3*N additional function evaluations. In many cases it may
           cost as much as the rest of the optimization session.

           YOU SHOULD NOT USE IT IN THE PRODUCTION CODE UNLESS YOU WANT TO
           CHECK DERIVATIVES PROVIDED BY SOME THIRD PARTY.

NOTE: unlike previous incarnation of the gradient checking code,  OptGuard
      does NOT interrupt optimization even if it discovers bad gradient.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step used for numerical differentiation:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification
                    You should carefully choose TestStep. Value  which  is
                    too large (so large that  function  behavior  is  non-
                    cubic at this scale) will lead  to  false  alarms. Too
                    short step will result in rounding  errors  dominating
                    numerical derivative.

                    You may use different step for different parameters by
                    means of setting scale with mincgsetscale().

=== EXPLANATION ==========================================================

In order to verify gradient algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void mincgoptguardgradient(mincgstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mincgoptguardnonc1test0results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #0

Nonsmoothness (non-C1) test #0 studies  function  values  (not  gradient!)
obtained during line searches and monitors  behavior  of  the  directional
derivative estimate.

This test is less powerful than test #1, but it does  not  depend  on  the
gradient values and thus it is more robust against artifacts introduced by
numerical differentiation.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #0 "strong" report
    lngrep  -   C1 test #0 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void mincgoptguardnonc1test0results(const mincgstate &state, optguardnonc1test0report &strrep, optguardnonc1test0report &lngrep, const xparams _xparams = alglib::xdefault);

`mincgoptguardnonc1test1results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #1

Nonsmoothness (non-C1)  test  #1  studies  individual  components  of  the
gradient computed during line search.

When precise analytic gradient is provided this test is more powerful than
test #0  which  works  with  function  values  and  ignores  user-provided
gradient.  However,  test  #0  becomes  more   powerful   when   numerical
differentiation is employed (in such cases test #1 detects  higher  levels
of numerical noise and becomes too conservative).

This test also tells specific components of the gradient which violate  C1
continuity, which makes it more informative than #0, which just tells that
continuity is violated.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* vidx - is an index of the variable in [0,N) with nonsmooth derivative
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], g[] - arrays of length CNT which store step lengths and  gradient
  values at these points; g[i] is evaluated in  x0+stp[i]*d  and  contains
  vidx-th component of the gradient.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #1 "strong" report
    lngrep  -   C1 test #1 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void mincgoptguardnonc1test1results(mincgstate &state, optguardnonc1test1report &strrep, optguardnonc1test1report &lngrep, const xparams _xparams = alglib::xdefault);

`mincgoptguardresults` function

/*************************************************************************
Results of OptGuard integrity check, should be called  after  optimization
session is over.

=== PRIMARY REPORT =======================================================

OptGuard performs several checks which are intended to catch common errors
in the implementation of nonlinear function/gradient:
* incorrect analytic gradient
* discontinuous (non-C0) target functions (constraints)
* nonsmooth     (non-C1) target functions (constraints)

Each of these checks is activated with appropriate function:
* mincgoptguardgradient() for gradient verification
* mincgoptguardsmoothness() for C0/C1 checks

Following flags are set when these errors are suspected:
* rep.badgradsuspected, and additionally:
  * rep.badgradvidx for specific variable (gradient element) suspected
  * rep.badgradxbase, a point where gradient is tested
  * rep.badgraduser, user-provided gradient  (stored  as  2D  matrix  with
    single row in order to make  report  structure  compatible  with  more
    complex optimizers like MinNLC or MinLM)
  * rep.badgradnum,   reference    gradient    obtained    via   numerical
    differentiation (stored as  2D matrix with single row in order to make
    report structure compatible with more complex optimizers  like  MinNLC
    or MinLM)
* rep.nonc0suspected
* rep.nonc1suspected

=== ADDITIONAL REPORTS/LOGS ==============================================

Several different tests are performed to catch C0/C1 errors, you can  find
out specific test signaled error by looking to:
* rep.nonc0test0positive, for non-C0 test #0
* rep.nonc1test0positive, for non-C1 test #0
* rep.nonc1test1positive, for non-C1 test #1

Additional information (including line search logs)  can  be  obtained  by
means of:
* mincgoptguardnonc1test0results()
* mincgoptguardnonc1test1results()
which return detailed error reports, specific points where discontinuities
were found, and so on.

==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    rep     -   generic OptGuard report;  more  detailed  reports  can  be
                retrieved with other functions.

NOTE: false negatives (nonsmooth problems are not identified as  nonsmooth
      ones) are possible although unlikely.

      The reason  is  that  you  need  to  make several evaluations around
      nonsmoothness  in  order  to  accumulate  enough  information  about
      function curvature. Say, if you start right from the nonsmooth point,
      optimizer simply won't get enough data to understand what  is  going
      wrong before it terminates due to abrupt changes in the  derivative.
      It is also  possible  that  "unlucky"  step  will  move  us  to  the
      termination too quickly.

      Our current approach is to have less than 0.1%  false  negatives  in
      our test examples  (measured  with  multiple  restarts  from  random
      points), and to have exactly 0% false positives.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void mincgoptguardresults(mincgstate &state, optguardreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mincgoptguardsmoothness` function

/*************************************************************************
This  function  activates/deactivates nonsmoothness monitoring  option  of
the  OptGuard  integrity  checker. Smoothness  monitor  silently  observes
solution process and tries to detect ill-posed problems, i.e. ones with:
a) discontinuous target function (non-C0)
b) nonsmooth     target function (non-C1)

Smoothness monitoring does NOT interrupt optimization  even if it suspects
that your problem is nonsmooth. It just sets corresponding  flags  in  the
OptGuard report which can be retrieved after optimization is over.

Smoothness monitoring is a moderate overhead option which often adds  less
than 1% to the optimizer running time. Thus, you can use it even for large
scale problems.

NOTE: OptGuard does  NOT  guarantee  that  it  will  always  detect  C0/C1
      continuity violations.

      First, minor errors are hard to  catch - say, a 0.0001 difference in
      the model values at two sides of the gap may be due to discontinuity
      of the model - or simply because the model has changed.

      Second, C1-violations  are  especially  difficult  to  detect  in  a
      noninvasive way. The optimizer usually  performs  very  short  steps
      near the nonsmoothness, and differentiation  usually   introduces  a
      lot of numerical noise.  It  is  hard  to  tell  whether  some  tiny
      discontinuity in the slope is due to real nonsmoothness or just  due
      to numerical noise alone.

      Our top priority was to avoid false positives, so in some rare cases
      minor errors may went unnoticed (however, in most cases they can  be
      spotted with restart from different initial point).

INPUT PARAMETERS:
    state   -   algorithm state
    level   -   monitoring level:
                * 0 - monitoring is disabled
                * 1 - noninvasive low-overhead monitoring; function values
                      and/or gradients are recorded, but OptGuard does not
                      try to perform additional evaluations  in  order  to
                      get more information about suspicious locations.

=== EXPLANATION ==========================================================

One major source of headache during optimization  is  the  possibility  of
the coding errors in the target function/constraints (or their gradients).
Such  errors   most   often   manifest   themselves  as  discontinuity  or
nonsmoothness of the target/constraints.

Another frequent situation is when you try to optimize something involving
lots of min() and max() operations, i.e. nonsmooth target. Although not  a
coding error, it is nonsmoothness anyway - and smooth  optimizers  usually
stop right after encountering nonsmoothness, well before reaching solution.

OptGuard integrity checker helps you to catch such situations: it monitors
function values/gradients being passed  to  the  optimizer  and  tries  to
errors. Upon discovering suspicious pair of points it  raises  appropriate
flag (and allows you to continue optimization). When optimization is done,
you can study OptGuard result.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void mincgoptguardsmoothness(mincgstate &state, const ae_int_t level, const xparams _xparams = alglib::xdefault);
void mincgoptguardsmoothness(mincgstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mincgoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied  gradient, and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either MinCGCreate()  for analytical gradient  or  MinCGCreateF()  for
   numerical differentiation) you should  choose  appropriate  variant  of
   MinCGOptimize() - one which accepts function AND gradient or one  which
   accepts function ONLY.

   Be careful to choose variant of MinCGOptimize()  which  corresponds  to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed  to  MinCGOptimize()  and  specific
   function used to create optimizer.


                  |         USER PASSED TO MinCGOptimize()
   CREATED WITH   |  function only   |  function and gradient
   ------------------------------------------------------------
   MinCGCreateF() |     work                FAIL
   MinCGCreate()  |     FAIL                work

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function and MinCGOptimize() version. Attemps to use  such  combination
   (for  example,  to create optimizer with  MinCGCreateF()  and  to  pass
   gradient information to MinCGOptimize()) will lead to  exception  being
   thrown. Either  you  did  not  pass  gradient when it WAS needed or you
   passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 20.04.2009 by Bochkanov Sergey
*************************************************************************/
void mincgoptimize(mincgstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void mincgoptimize(mincgstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`mincgrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void mincgrequesttermination(mincgstate &state, const xparams _xparams = alglib::xdefault);

`mincgrestartfrom` function

/*************************************************************************
This  subroutine  restarts  CG  algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used to store algorithm state.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void mincgrestartfrom(mincgstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mincgresults` function

/*************************************************************************
Conjugate gradient results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -8    internal integrity control  detected  infinite
                            or NAN values in  function/gradient.  Abnormal
                            termination signalled.
                    * -7    gradient verification failed.
                            See MinCGSetGradientCheck() for more information.
                    *  1    relative function improvement is no more than
                            EpsF.
                    *  2    relative step is no more than EpsX.
                    *  4    gradient norm is no more than EpsG
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible,
                            we return best X found so far
                    *  8    terminated by user
                * Rep.IterationsCount contains iterations count
                * NFEV countains number of function calculations

  -- ALGLIB --
     Copyright 20.04.2009 by Bochkanov Sergey
*************************************************************************/
void mincgresults(const mincgstate &state, real_1d_array &x, mincgreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`mincgresultsbuf` function

/*************************************************************************
Conjugate gradient results

Buffered implementation of MinCGResults(), which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 20.04.2009 by Bochkanov Sergey
*************************************************************************/
void mincgresultsbuf(const mincgstate &state, real_1d_array &x, mincgreport &rep, const xparams _xparams = alglib::xdefault);

`mincgsetcgtype` function

/*************************************************************************
This function sets CG algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    CGType  -   algorithm type:
                * -1    automatic selection of the best algorithm
                * 0     DY (Dai and Yuan) algorithm
                * 1     Hybrid DY-HS algorithm

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetcgtype(mincgstate &state, const ae_int_t cgtype, const xparams _xparams = alglib::xdefault);

`mincgsetcond` function

/*************************************************************************
This function sets stopping conditions for CG optimization algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinCGSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinCGSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
automatic stopping criterion selection (small EpsX).

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetcond(mincgstate &state, const double epsg, const double epsf, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`mincgsetprecdefault` function

/*************************************************************************
Modification of the preconditioner: preconditioning is turned off.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetprecdefault(mincgstate &state, const xparams _xparams = alglib::xdefault);

`mincgsetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

NOTE 2: D[i] should be positive. Exception will be thrown otherwise.

NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetprecdiag(mincgstate &state, const real_1d_array &d, const xparams _xparams = alglib::xdefault);

`mincgsetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables with MinCGSetScale() call
(before or after MinCGSetPrecScale() call). Without knowledge of the scale
of your variables scale-based preconditioner will be just unit matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetprecscale(mincgstate &state, const xparams _xparams = alglib::xdefault);

`mincgsetscale` function

/*************************************************************************
This function sets scaling coefficients for CG optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of CG optimizer  -  step
along I-th axis is equal to DiffStep*S[I].

In   most   optimizers  (and  in  the  CG  too)  scaling is NOT a form  of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner by separate call to one of the MinCGSetPrec...() functions.

There  is  special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void mincgsetscale(mincgstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`mincgsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetstpmax(mincgstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`mincgsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinCGOptimize().

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsetxrep(mincgstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`mincgsuggeststep` function

/*************************************************************************
This function allows to suggest initial step length to the CG algorithm.

Suggested  step  length  is used as starting point for the line search. It
can be useful when you have  badly  scaled  problem,  i.e.  when  ||grad||
(which is used as initial estimate for the first step) is many  orders  of
magnitude different from the desired step.

Line search  may  fail  on  such problems without good estimate of initial
step length. Imagine, for example, problem with ||grad||=10^50 and desired
step equal to 0.1 Line  search function will use 10^50  as  initial  step,
then  it  will  decrease step length by 2 (up to 20 attempts) and will get
10^44, which is still too large.

This function allows us to tell than line search should  be  started  from
some moderate step length, like 1.0, so algorithm will be able  to  detect
desired step length in a several searches.

Default behavior (when no step is suggested) is to use preconditioner,  if
it is available, to generate initial estimate of step length.

This function influences only first iteration of algorithm. It  should  be
called between MinCGCreate/MinCGRestartFrom() call and MinCGOptimize call.
Suggested step is ignored if you have preconditioner.

INPUT PARAMETERS:
    State   -   structure used to store algorithm state.
    Stp     -   initial estimate of the step length.
                Can be zero (no estimate).

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void mincgsuggeststep(mincgstate &state, const double stp, const xparams _xparams = alglib::xdefault);

mincg_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // using nonlinear conjugate gradient method with:
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see mincgsetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        // * OptGuard integrity check being used to check problem statement
        //   for some common errors like nonsmoothness or bad analytic gradient
        //
        // First, we create optimizer object and tune its properties
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.0000000001;
        ae_int_t maxits = 0;
        mincgstate state;
        mincgcreate(x, state);
        mincgsetcond(state, epsg, epsf, epsx, maxits);
        mincgsetscale(state, s);

        //
        // Activate OptGuard integrity checking.
        //
        // OptGuard monitor helps to catch common coding and problem statement
        // issues, like:
        // * discontinuity of the target function (C0 continuity violation)
        // * nonsmoothness of the target function (C1 continuity violation)
        // * erroneous analytic gradient, i.e. one inconsistent with actual
        //   change in the target/constraints
        //
        // OptGuard is essential for early prototyping stages because such
        // problems often result in premature termination of the optimizer
        // which is really hard to distinguish from the correct termination.
        //
        // IMPORTANT: GRADIENT VERIFICATION IS PERFORMED BY MEANS OF NUMERICAL
        //            DIFFERENTIATION. DO NOT USE IT IN PRODUCTION CODE!!!!!!!
        //
        //            Other OptGuard checks add moderate overhead, but anyway
        //            it is better to turn them off when they are not needed.
        //
        mincgoptguardsmoothness(state);
        mincgoptguardgradient(state, 0.001);

        //
        // Optimize and evaluate results
        //
        mincgreport rep;
        alglib::mincgoptimize(state, function1_grad);
        mincgresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

        //
        // Check that OptGuard did not report errors
        //
        // NOTE: want to test OptGuard? Try breaking the gradient - say, add
        //       1.0 to some of its components.
        //
        optguardreport ogrep;
        mincgoptguardresults(state, ogrep);
        printf("%s\n", ogrep.badgradsuspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

mincg_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // with nonlinear conjugate gradient method.
        //
        // Several advanced techniques are demonstrated:
        // * upper limit on step size
        // * restart from new point
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.0000000001;
        double stpmax = 0.1;
        ae_int_t maxits = 0;
        mincgstate state;
        mincgreport rep;

        // create and tune optimizer
        mincgcreate(x, state);
        mincgsetscale(state, s);
        mincgsetcond(state, epsg, epsf, epsx, maxits);
        mincgsetstpmax(state, stpmax);

        // Set up OptGuard integrity checker which catches errors
        // like nonsmooth targets or errors in the analytic gradient.
        //
        // OptGuard is essential at the early prototyping stages.
        //
        // NOTE: gradient verification needs 3*N additional function
        //       evaluations; DO NOT USE IT IN THE PRODUCTION CODE
        //       because it leads to unnecessary slowdown of your app.
        mincgoptguardsmoothness(state);
        mincgoptguardgradient(state, 0.001);

        // first run
        alglib::mincgoptimize(state, function1_grad);
        mincgresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

        // second run - algorithm is restarted with mincgrestartfrom()
        x = "[10,10]";
        mincgrestartfrom(state, x);
        alglib::mincgoptimize(state, function1_grad);
        mincgresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

        // check OptGuard integrity report. Why do we need it at all?
        // Well, try breaking the gradient by adding 1.0 to some
        // of its components - OptGuard should report it as error.
        // And it may also catch unintended errors too :)
        optguardreport ogrep;
        mincgoptguardresults(state, ogrep);
        printf("%s\n", ogrep.badgradsuspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

mincg_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // using numerical differentiation to calculate gradient.
        //
        // We also show how to use OptGuard integrity checker to catch common
        // problem statement errors like accidentally specifying nonsmooth target
        // function.
        //
        // First, we set up optimizer...
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.0000000001;
        double diffstep = 1.0e-6;
        ae_int_t maxits = 0;
        mincgstate state;
        mincgcreatef(x, diffstep, state);
        mincgsetcond(state, epsg, epsf, epsx, maxits);
        mincgsetscale(state, s);

        //
        // Then, we activate OptGuard integrity checking.
        //
        // Numerical differentiation always produces "correct" gradient
        // (with some truncation error, but unbiased). Thus, we just have
        // to check smoothness properties of the target: C0 and C1 continuity.
        //
        // Sometimes user accidentally tried to solve nonsmooth problems
        // with smooth optimizer. OptGuard helps to detect such situations
        // early, at the prototyping stage.
        //
        mincgoptguardsmoothness(state);

        //
        // Now we are ready to run the optimization
        //
        mincgreport rep;
        alglib::mincgoptimize(state, function1_func);
        mincgresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

        //
        // ...and to check OptGuard integrity report.
        //
        // Want to challenge OptGuard? Try to make your problem
        // nonsmooth by replacing 100*(x+3)^4 by 100*|x+3| and
        // re-run optimizer.
        //
        optguardreport ogrep;
        mincgoptguardresults(state, ogrep);
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`mincomp` subpackage

Classes

minasareport
minasastate

Functions

minasacreate
minasaiteration
minasaoptimize
minasarestartfrom
minasaresults
minasaresultsbuf
minasasetalgorithm
minasasetcond
minasasetstpmax
minasasetxrep
minbleicsetbarrierdecay
minbleicsetbarrierwidth
minlbfgssetcholeskypreconditioner
minlbfgssetdefaultpreconditioner

Examples

`minasareport` class

/*************************************************************************

*************************************************************************/
class minasareport
{
public:
    minasareport();
    minasareport(const minasareport &rhs);
    minasareport& operator=(const minasareport &rhs);
    virtual ~minasareport();
    ae_int_t iterationscount;
    ae_int_t nfev;
    ae_int_t terminationtype;
    ae_int_t activeconstraints;
};

`minasastate` class

/*************************************************************************

*************************************************************************/
class minasastate
{
public:
    minasastate();
    minasastate(const minasastate &rhs);
    minasastate& operator=(const minasastate &rhs);
    virtual ~minasastate();
};

`minasacreate` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 25.03.2010 by Bochkanov Sergey
*************************************************************************/
void minasacreate(const ae_int_t n, const real_1d_array &x, const real_1d_array &bndl, const real_1d_array &bndu, minasastate &state, const xparams _xparams = alglib::xdefault);
void minasacreate(const real_1d_array &x, const real_1d_array &bndl, const real_1d_array &bndu, minasastate &state, const xparams _xparams = alglib::xdefault);

`minasaiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minasaiteration(minasastate &state, const xparams _xparams = alglib::xdefault);

`minasaoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void minasaoptimize(minasastate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

`minasarestartfrom` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void minasarestartfrom(minasastate &state, const real_1d_array &x, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`minasaresults` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void minasaresults(const minasastate &state, real_1d_array &x, minasareport &rep, const xparams _xparams = alglib::xdefault);

`minasaresultsbuf` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void minasaresultsbuf(const minasastate &state, real_1d_array &x, minasareport &rep, const xparams _xparams = alglib::xdefault);

`minasasetalgorithm` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minasasetalgorithm(minasastate &state, const ae_int_t algotype, const xparams _xparams = alglib::xdefault);

`minasasetcond` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minasasetcond(minasastate &state, const double epsg, const double epsf, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`minasasetstpmax` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minasasetstpmax(minasastate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`minasasetxrep` function

/*************************************************************************
Obsolete optimization algorithm.
Was replaced by MinBLEIC subpackage.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minasasetxrep(minasastate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`minbleicsetbarrierdecay` function

/*************************************************************************
This is obsolete function which was used by previous version of the  BLEIC
optimizer. It does nothing in the current version of BLEIC.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetbarrierdecay(minbleicstate &state, const double mudecay, const xparams _xparams = alglib::xdefault);

`minbleicsetbarrierwidth` function

/*************************************************************************
This is obsolete function which was used by previous version of the  BLEIC
optimizer. It does nothing in the current version of BLEIC.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minbleicsetbarrierwidth(minbleicstate &state, const double mu, const xparams _xparams = alglib::xdefault);

`minlbfgssetcholeskypreconditioner` function

/*************************************************************************
Obsolete function, use MinLBFGSSetCholeskyPreconditioner() instead.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetcholeskypreconditioner(minlbfgsstate &state, const real_2d_array &p, const bool isupper, const xparams _xparams = alglib::xdefault);

`minlbfgssetdefaultpreconditioner` function

/*************************************************************************
Obsolete function, use MinLBFGSSetPrecDefault() instead.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetdefaultpreconditioner(minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

`mindf` subpackage

Classes

mindfreport
mindfstate

Functions

mindfcreate
mindfiteration
mindfoptimize
mindfrequesttermination
mindfresults
mindfresultsbuf
mindfsetalgogdemo
mindfsetalgogdemofixed
mindfsetbc
mindfsetcondf
mindfsetcondfx
mindfsetgdemopenalty
mindfsetgdemoprofilequick
mindfsetgdemoprofilerobust
mindfsetlc2dense
mindfsetnlc2
mindfsetscale
mindfsetseed
mindfsetxrep
mindfusetimers

Examples

mindf_gdemo_auto

Nonlinearly constrained differential evolution

`mindfreport` class

/*************************************************************************
This structure stores optimization report:
* f                         objective value at the solution
* iterationscount           total number of inner iterations
* nfev                      number of gradient evaluations
* terminationtype           termination type (see below)
* bcerr                     maximum violation of box constraints
* lcerr                     maximum violation of linear constraints
* nlcerr                    maximum violation of nonlinear constraints

If timers were activated, the structure also stores running times:
* timesolver                time (in seconds, stored as a floating-point
                            value) spent in the solver itself. Time spent
                            in the user callback is not included.
                            See 'TIMERS' below for more information.
* timecallback              time (in seconds, stored as a floating-point
                            value) spent in the user callback.
                            See 'TIMERS' below for more information.
* timetotal                 total time spent during the optimization,
                            including both the solver and callbacks.
                            See 'TIMERS' below for more information.
In  order  to  activate timers, the caller has to  call  mindfusetimers()
function.

Other fields of this structure are not documented and should not be used!

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -3    box constraints are inconsistent
  -1    inconsistent parameters were passed:
        * penalty parameter is zero, but we have nonlinear constraints
          set by mindfsetnlc2()
   1    function value has converged within epsf
   2    sampling radius decreased below epsx
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    User requested termination via mindfrequesttermination()

TIMERS

Starting from ALGLIB 4.04, many optimizers report time  spent  in  in  the
solver itself and in user  callbacks. The  time  is  reported  in seconds,
using floating-point (i.e. fractional-length intervals can  be  reported).

In  order  to  activate timers, the caller has to  call  mindfusetimers()
function.

The accuracy of   the  reported  value depends on the specific programming
language and OS being used:
* C++, no AE_OS is #defined - the accuracy is that of time() function, i.e.
  one second.
* C++, AE_OS=AE_WINDOWS is #defined - the accuracy is that of GetTickCount(),
  i.e. about 10-20ms
* C++, AE_OS=AE_POSIX is #defined - the accuracy is that of gettimeofday()
* C#, managed core, any OS - the accuracy is that of Environment.TickCount
* C#, HPC core, any OS - the accuracy is that of a corresponding C++ version
* any other language - the accuracy is that of a corresponding C++ version

Whilst modern operating systems provide more accurate timers, these timers
often have significant overhead or backward  compatibility  issues.  Thus,
ALGLIB stick to the most basic and efficient functions, even at  the  cost
of some accuracy being lost.
*************************************************************************/
class mindfreport
{
public:
    mindfreport();
    mindfreport(const mindfreport &rhs);
    mindfreport& operator=(const mindfreport &rhs);
    virtual ~mindfreport();
    double f;
    ae_int_t iterationscount;
    ae_int_t nfev;
    double bcerr;
    double lcerr;
    double nlcerr;
    ae_int_t terminationtype;
    double timetotal;
    double timesolver;
    double timecallback;
};

`mindfstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinDF subpackage to work  with  this
object
*************************************************************************/
class mindfstate
{
public:
    mindfstate();
    mindfstate(const mindfstate &rhs);
    mindfstate& operator=(const mindfstate &rhs);
    virtual ~mindfstate();
};

`mindfcreate` function

/*************************************************************************
                          GLOBAL OPTIMIZATION
               SUBJECT TO BOX/LINEAR/NONLINEAR CONSTRAINTS

The  subroutine  minimizes  function   F(x)  of N arguments subject to any
combination of:
* bound constraints
* linear inequality constraints
* linear equality constraints
* nonlinear generalized inequality constraints Li<=Ci(x)<=Ui, with one  of
  Li/Ui possibly being infinite

REQUIREMENTS:
* F() and C()  do  NOT  have  to be differentiable, locally Lipschitz   or
  continuous. Most solvers in this subpackage can deal with  nonsmoothness
  or minor discontinuities, although obviously smoother problems  are  the
  most easy ones.
* generally, F() and C() must be computable at any point which is feasible
  subject to box constraints

USAGE:

1. User  initializes  algorithm  state   with   mindfcreate()   call   and
   chooses specific solver to be used. There is some solver which is  used
   by default, with default settings, but  you  should  NOT  rely  on  the
   default choice. It may change in the future releases of ALGLIB  without
   notice, and no one can guarantee that the new solver will be  able   to
   solve your problem with default settings.

2. User adds boundary and/or linear and/or nonlinear constraints by  means
   of calling one of the following functions:
   a) mindfsetbc() for boundary constraints
   b) mindfsetlc2dense() for linear constraints
   c) mindfsetnlc2() for nonlinear constraints
   You may combine (a), (b) and (c) in one optimization problem.

3. User sets scale of the variables with mindfsetscale() function.  It  is
   VERY important to set  variable  scales  because  many  derivative-free
   algorithms refuse to work when variables are badly scaled.
   Scaling  helps  to seed initial  population,  control  convergence  and
   enforce penalties for constraint violation.

4. Finally, user calls mindfoptimize()  function which takes algorithm
   state and pointer (delegate, etc) to callback function which calculates
   F and G.

5. User calls mindfresults() to get a solution


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from  size  of  X
    X       -   starting point, array[N]. Some solvers can utilize  a good
                initial point to seed computations.

                As of ALGLIB 4.04, the initial point is:
                * used by GDEMO

                If the chosen solver does not need initial point, one  can
                supply zeros.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state


IMPORTANT: the  MINDF  optimizer   supports   parallel   model  evaluation
           ('callback  parallelism'). This feature, which  is  present  in
           commercial ALGLIB editions, greatly accelerates algorithms like
           differential evolution which usually issue  batch  requests  to
           user callbacks which can be efficiently parallelized.

           Callback parallelism  is  usually  beneficial  when  the  batch
           evalution requires more than several milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial  version'
           section,  and  comments  on  mindfoptimize()  function for more
           information.

  -- ALGLIB --
     Copyright 24.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfcreate(const ae_int_t n, const real_1d_array &x, mindfstate &state, const xparams _xparams = alglib::xdefault);
void mindfcreate(const real_1d_array &x, mindfstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool mindfiteration(mindfstate &state, const xparams _xparams = alglib::xdefault);

`mindfoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


CALLBACK PARALLELISM

The  MINDF  optimizer  supports   parallel   model  evaluation  ('callback
parallelism').  This  feature,  which  is  present  in  commercial  ALGLIB
editions, greatly accelerates  optimization  when  using  a  solver  which
issues batch requests, i.e. multiple requests  for  target  values,  which
can be computed independently by different threads.

Callback parallelism is  usually   beneficial  when   processing  a  batch
request requires more than several  milliseconds.  It  also  requires  the
solver which issues requests in convenient batches, e.g. the  differential
evolution solver.

See ALGLIB Reference Manual, 'Working with commercial version' section for
more information.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfoptimize(mindfstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of a running optimizer. It
should be called from a user-supplied callback when user decides  that  it
is time to "smoothly" terminate optimization process.  As  a  result,  the
optimizer  stops  at  the  point  which  was  "current  accepted" when the
termination request was submitted and returns  error  code  8  (successful
termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on an optimizer which is NOT running will have
      no effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfrequesttermination(mindfstate &state, const xparams _xparams = alglib::xdefault);

`mindfresults` function

/*************************************************************************
MinDF results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:

                rep.f stores objective value at the solution

                You  should     check  rep.terminationtype  in   order  to
                distinguish successful termination from unsuccessful one:
                * -8   internal integrity control  detected  infinite  or
                       NAN   values   in   function/gradient.    Abnormal
                       termination signalled.
                * -3   box constraints are inconsistent
                * -1   inconsistent parameters were passed:
                       * penalty parameter is zero, but we have nonlinear
                         constraints set by MinDFsetnlc2()
                *  1   successful termination according to a solver-specific
                       set of conditions
                *  8   User requested termination via minnsrequesttermination()

                If you activated timers  with  mindfusetimers(),  you  can
                also find out how much time was spent in various code parts:
                * rep.timetotal    - for a total time in seconds
                * rep.timesolver   - for a time spent in the solver itself
                * rep.timecallback - for a time spent in user callbacks
                See comments on mindfreport structure for more information
                about timers and their accuracy.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfresults(const mindfstate &state, real_1d_array &x, mindfreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfresultsbuf` function

/*************************************************************************

Buffered implementation of MinDFresults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfresultsbuf(const mindfstate &state, real_1d_array &x, mindfreport &rep, const xparams _xparams = alglib::xdefault);

`mindfsetalgogdemo` function

/*************************************************************************
This subroutine sets optimization algorithm to the differential  evolution
solver GDEMO (Generalized Differential Evolution Multiobjective)  with  an
automatic parameters selection.

NOTE: a version with manually tuned parameters can be activated by calling
      the mindfsetalgogdemofixed() function.

The primary stopping condition for the solver is to stop after the specified
number of iterations. You can also specify  additional  criteria  to  stop
early:
* stop when subpopulation target values (2N+1 best individuals) are within
  EPS from the best one so far (function values seem to converge)
* stop when both subpopulation target values  AND  subpopulation  variable
  values are within EPS from the best one so far

The first condition is specified with mindfsetcondf(), the second  one  is
activated with mindfsetcondfx().

Both conditions are heuristics which  may fail. Being 'within EPS from the
best value so far'  in  practice  means  that  we  are  somewhere   within
[0.1EPS,10EPS] from the true  solution;  however,  on  difficult  problems
this condition may fire too early.

Imposing an additional requirement that variable values have clustered  too
may prevent us from premature  stopping. However, on multi-extremal and/or
noisy problems too many individuals may be trapped away from the  optimum,
preventing this condition from activation.

ALGORITHM PROPERTIES:

* the solver uses a variant of  the  adaptive  parameter  tuning  strategy
  called 'Success-History  Based  Parameter  Adaptation  for  Differential
  Evolution Ensemble' (SHADE) by Ryoji Tanabe and Alex  Fukunaga.  You  do
  not have to specify crossover probability and differential weight,   the
  solver will automatically choose the most appropriate strategy.

* the solver can handle box, linear,  nonlinear  constraints.  Linear  and
  nonlinear constraints are  handled  by  means  of an L1/L2  penalty. The
  solver does not violate box constraints at any point,  but  may  violate
  linear and nonlinear ones. Penalty coefficient can be changed with   the
  mindfsetgdemopenalty() function.

* the solver heavily depends on variable scales being available (specified
  by means of mindfsetscale() call) and on box constraints with both lower
  and upper bounds being available which are used to determine the  search
  region. It will work without box constraints  and  without  scales,  but
  results are likely to be suboptimal.

* the solver is SIMD-optimized  and  parallelized  (in  commercial  ALGLIB
  editions), with nearly linear scalability of parallel processing.

* this solver is intended for finding solutions  with up to several digits
  of precision at best. Its primary purpose  is  to  find  at  least  some
  solution to an otherwise intractable problem.

IMPORTANT: derivative-free optimization is inherently less robust than the
           smooth nonlinear programming, especially when nonsmoothness and
           discontinuities are present.

           Derivative-free  algorithms  have  less  convergence guarantees
           than their smooth  counterparts.  It  is  considered  a  normal
           (although, obviously, undesirable) situation when a derivative-
           -free algorithm fails to converge with desired precision. Having
           2 digits of accurasy is already a  good  result,  on  difficult
           problems (high numerical noise, discontinuities) you  may  have
           even less than that.

INPUT PARAMETERS:
    State       -   solver
    EpochsCnt   -   iterations count, >0.  Usually   the  algorithm  needs
                    hundreds of iterations to converge.
    PopSize     -   population  size,  >=0.  Zero  value  means  that  the
                    default value (which is 10*N in the  current  version)
                    will be chosen. Good values are  in  5*N...20*N,  with
                    the smaller values being recommended for easy problems
                    and the larger  values  for  difficult  multi-extremal
                    and/or noisy tasks.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetalgogdemo(mindfstate &state, const ae_int_t epochscnt, const ae_int_t popsize, const xparams _xparams = alglib::xdefault);
void mindfsetalgogdemo(mindfstate &state, const ae_int_t epochscnt, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfsetalgogdemofixed` function

/*************************************************************************
This subroutine sets optimization algorithm to the differential  evolution
solver GDEMO (Generalized Differential Evolution Multiobjective)  with the
manual parameters selection.

Unlike DE with an automatic parameters selection  this  function  requires
user to manually specify  algorithm  parameters. In the general  case  the
full-auto GDEMO is better. However, it has to spend some time finding  out
properties of a problem being solved; furthermore, it is  not  allowed  to
try potentially dangerous values of  parameters  that  lead  to  premature
stopping. Manually tuning the solver to the specific problem at  hand  can
get 2x-3x better running time.

Aside from that, the algorithm is fully equivalent to automatic GDEMO, and
we   recommend  you  reading  comments  on  mindfsetalgogdemo()  for  more
information about algorithm properties and stopping criteria.

INPUT PARAMETERS:
    State       -   solver
    EpochsCnt   -   iterations count, >0.  Usually   the  algorithm  needs
                    hundreds of iterations to converge.
    Strategy    -   specific DE strategy to use:
                    * 0 for DE/rand/1
                    * 1 for DE/best/2
                    * 2 for DE/current-to-best/1
    CrossoverProb-  crossover probability, 0<CrossoverProb<1
    DifferentialWeight- weight, 0<DifferentialWeight<2
    PopSize     -   population  size,  >=0.  Zero  value  means  that  the
                    default value (which is 10*N in the  current  version)
                    will be chosen. Good values are  in  5*N...20*N,  with
                    the smaller values being recommended for easy problems
                    and the larger  values  for  difficult  multi-extremal
                    and/or noisy tasks.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetalgogdemofixed(mindfstate &state, const ae_int_t epochscnt, const ae_int_t strategy, const double crossoverprob, const double differentialweight, const ae_int_t popsize, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfsetbc` function

/*************************************************************************
This function sets box constraints.

Box constraints are inactive by default (after initial creation).

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

  -- ALGLIB --
     Copyright 24.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetbc(mindfstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`mindfsetcondf` function

/*************************************************************************
This function sets stopping condition for the optimizer: function   values
has converged to a neighborhood whose size is proportional to epsF.

Most derivarive-free solvers are heuristics, so the code used to implement
this stopping condition is an heuristic too. Usually 'proportional to EPS'
means that we are somewhere between Eps/10...Eps*10 away from the solution.
However, there are no warranties that the solver  has  actually  converged
to something, although in practice it works well.

The  specific  meaning  of  'converging'  is  algorithm-dependent.  It  is
possible that some future ALGLIB optimizers will  ignore  this  condition,
see comments on specific solvers for more info. This  condition  does  not
work for multi-objective problems.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsF    -   >=0:
                * zero value means no condition for F
                * EpsF>0 means stopping when the solver converged with
                  an error estimate less than EpsF*max(|F|,1)

  -- ALGLIB --
     Copyright 23.04.2024 by Bochkanov Sergey
*************************************************************************/
void mindfsetcondf(mindfstate &state, const double epsf, const xparams _xparams = alglib::xdefault);

`mindfsetcondfx` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

This function sets a  combined   stopping  condition:  stopping  when  two
criteria are met simultaneously:
* function   values   has  converged  to  a  neighborhood  whose  size  is
  proportional to epsF
* variable   values   has  converged  to  a  neighborhood  whose  size  is
  proportional to epsX
It is possible to use only one condition by setting another EPS to zero.

Most derivarive-free solvers are heuristics, so the code used to implement
this stopping condition is an heuristic too. Usually 'proportional to EPS'
means that we are somewhere between Eps/10...Eps*10 away from the solution.
However, there are no warranties that the solver  has  actually  converged
to something, although in practice it works well.

The  specific  meaning  of  'converging'  is  algorithm-dependent.  It  is
possible that some future ALGLIB optimizers will  ignore  this  condition,
see comments on specific solvers for more info. This  condition  does  not
work for multi-objective problems.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsF    -   >=0:
                * zero value means no condition for F
                * EpsF>0 means stopping when the solver converged with
                  an error estimate less than EpsF*max(|F|,1)
    EpsX    -   >=0:
                * zero value means no condition for X
                * EpsX>0 means stopping when the solver converged with
                  error in I-th variable less than EpsX*S[i], where S[i]
                  is a variable scale

  -- ALGLIB --
     Copyright 23.04.2024 by Bochkanov Sergey
*************************************************************************/
void mindfsetcondfx(mindfstate &state, const double epsf, const double epsx, const xparams _xparams = alglib::xdefault);

`mindfsetgdemopenalty` function

/*************************************************************************
This subroutine tells GDEMO differential  evolution  optimizer  to  handle
linear/nonlinear constraints by an L1/L2 penalty function.

IMPORTANT: this function does NOT change the  optimization  algorithm.  If
           want to activate differential evolution solver, you still  have
           to call a proper mindfsetalgo???() function.

INPUT PARAMETERS:
    State       -   solver
    Rho1, Rho2  -   penalty parameters for constraint violations:
                    * Rho1 is a multiplier for L1 penalty
                    * Rho2 is a multiplier for L2 penalty
                    * Rho1,Rho2>=0
                    * having both of them at zero means that some  default
                      value will be chosen.
                    Ignored for problems with box-only constraints.

                    L1 penalty is usually better at enforcing constraints,
                    but leads to slower convergence than L2 penalty. It is
                    possible to combine both kinds of penalties together.

                    There is a compromise between constraint  satisfaction
                    and  optimality:  high  values  of   Rho   mean   that
                    constraints are satisfied with high accuracy  but that
                    the target may  be  underoptimized  due  to  numerical
                    difficulties.  Small  values  of  Rho  mean  that  the
                    solution may  grossly  violate  constraints.  Choosing
                    good Rho is usually a matter of trial and error.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetgdemopenalty(mindfstate &state, const double rho1, const double rho2, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfsetgdemoprofilequick` function

/*************************************************************************
This subroutine tells GDEMO differential  evolution  optimizer  to  use  a
QUICK profile.

The QUICK profile is intended  to  facilitate accelerated  convergence  on
medium-complexity problems at the cost  of  (sometimes)  having  premature
convergence  on  difficult  and/or  multi-extremal  problems.  The  ROBUST
profile can be selected if you favor convergence  warranties  over  speed.
In most cases, the ROBUST profile is ~2x-3x slower than the QUICK one.

This function has effect only on adaptive GDEMO with automatic  parameters
selection.  It  has  no  effect  on  fixed-parameters  GDEMO  or any other
solvers.

IMPORTANT: this function does NOT change the  optimization  algorithm.  If
           you want to activate differential evolution solver,  you  still
           have to call a proper mindfsetalgo???() function.

INPUT PARAMETERS:
    State       -   solver

  -- ALGLIB --
     Copyright 25.04.2024 by Bochkanov Sergey
*************************************************************************/
void mindfsetgdemoprofilequick(mindfstate &state, const xparams _xparams = alglib::xdefault);

`mindfsetgdemoprofilerobust` function

/*************************************************************************
This subroutine tells GDEMO differential  evolution  optimizer  to  use  a
ROBUST profile (the default option).

The ROBUST profile is intended  to  facilitate  explorative  behavior  and
robust convergence even on difficult multi-extremal problems. It comes  at
the expense of increased running time  even  on  easy  problems. The QUICK
profile can be chosen if your problem is relatively easy to handle and you
prefer speed over robustness. In most cases, the QUICK profile  is  ~2x-3x
faster than the robust one.

This function has effect only on adaptive GDEMO with automatic  parameters
selection.  It  has  no  effect  on  fixed-parameters  GDEMO  or any other
solvers.

IMPORTANT: this function does NOT change the  optimization  algorithm.  If
           want to activate differential evolution solver, you still  have
           to call a proper mindfsetalgo???() function.

INPUT PARAMETERS:
    State       -   solver

  -- ALGLIB --
     Copyright 25.04.2024 by Bochkanov Sergey
*************************************************************************/
void mindfsetgdemoprofilerobust(mindfstate &state, const xparams _xparams = alglib::xdefault);

`mindfsetlc2dense` function

/*************************************************************************
This function sets two-sided linear constraints AL <= A*x <= AU with dense
constraint matrix A.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinDFcreate() call.
    A       -   linear constraints, array[K,N]. Each row of  A  represents
                one  constraint. One-sided  inequality   constraints, two-
                sided inequality  constraints,  equality  constraints  are
                supported (see below)
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number of equality/inequality constraints,  K>=0;  if  not
                given, inferred from sizes of A, AL, AU.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetlc2dense(mindfstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void mindfsetlc2dense(mindfstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfsetnlc2` function

/*************************************************************************
This function sets two-sided nonlinear constraints.

In fact, this function sets only  NUMBER  of  the  nonlinear  constraints.
Constraints  themselves  (constraint  functions)   are   passed   to   the
MinDFOptimize() method.

This method accepts user-defined vector function F[] where:
* first component of F[] corresponds to the objective
* subsequent NNLC components of F[] correspond to the two-sided  nonlinear
  constraints NL<=C(x)<=NU, where
  * NL[i]=NU[i] => I-th row is an equality constraint Ci(x)=NL
  * NL[i]<NU[i] => I-th tow is a  two-sided constraint NL[i]<=Ci(x)<=NU[i]
  * NL[i]=-INF  => I-th row is an one-sided constraint Ci(x)<=NU[i]
  * NU[i]=+INF  => I-th row is an one-sided constraint NL[i]<=Ci(x)
  * NL[i]=-INF, NU[i]=+INF => constraint is ignored

NOTE: you may combine nonlinear constraints with linear/boundary ones.  If
      your problem has mixed constraints, you  may explicitly specify some
      of them as linear or box ones.
      It helps optimizer to handle them more efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinDFCreate call.
    NL      -   array[NNLC], lower bounds, can contain -INF
    NU      -   array[NNLC], lower bounds, can contain +INF
    NNLC    -   constraints count, NNLC>=0

NOTE 1: nonlinear constraints are satisfied only  approximately!   It   is
        possible   that  algorithm  will  evaluate  function  outside   of
        feasible area!

NOTE 2: the algorithm scales variables according to the scale specified by
        MinDFSetScale() function, so it can handle problems with badly
        scaled variables (as long as we KNOW their scales).

        However,  there  is  no  way  to  automatically  scale   nonlinear
        constraints. Inappropriate scaling  of nonlinear  constraints  may
        ruin convergence. Solving problem with  constraint  "1000*G0(x)=0"
        is NOT the same as solving it with constraint "0.001*G0(x)=0".

        It means that YOU are  the  one who is responsible for the correct
        scaling of the nonlinear constraints. We recommend  you  to  scale
        nonlinear constraints in such  a  way  that  the  derivatives  (if
        constraints are differentiable) have approximately unit  magnitude
        (for problems  with  unit  variable  scales)  or  have  magnitudes
        approximately equal to 1/S[i] (where S is a variable scale set  by
        MinDFSetScale() function).

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetnlc2(mindfstate &state, const real_1d_array &nl, const real_1d_array &nu, const ae_int_t nnlc, const xparams _xparams = alglib::xdefault);
void mindfsetnlc2(mindfstate &state, const real_1d_array &nl, const real_1d_array &nu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfsetscale` function

/*************************************************************************
This function sets variable scales.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison  with  tolerances)  and  to
guide algorithm steps.

The scale of a variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 25.07.2023 by Bochkanov Sergey
*************************************************************************/
void mindfsetscale(mindfstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`mindfsetseed` function

/*************************************************************************
This function sets seed used by internal RNG.

By default, a random seed is used, i.e. every time you run the solver,  we
seed its generator with a new value obtained  from  the  system-wide  RNG.
Thus, the solver returns non-deterministic results. You  can  change  such
a behavior by specifying a fixed positive seed value.

INPUT PARAMETERS:
    S           -   optimizer structure
    SeedVal     -   seed value:
                    * positive values are used for  seeding  RNG  with  a
                      fixed  seed,  i.e.  subsequent  runs  on  the  same
                      objective will return the same results
                    * non-positive seed means that a random  seed is used
                      for every run, i.e. subsequent  runs  on  the  same
                      objective will return slightly different results

  -- ALGLIB --
     Copyright 26.04.2024 by Bochkanov Sergey
*************************************************************************/
void mindfsetseed(mindfstate &s, const ae_int_t seedval, const xparams _xparams = alglib::xdefault);

`mindfsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinDFOptimize().

NOTE: algorithm passes two parameters to rep() callback  - the best  point
      so far and a function value at the point. For unconstrained problems
      the function value is  non-increasing (the most recent best point is
      always at least not worse than the previous best one).  However,  it
      can increase between iterations when  solving  constrained  problems
      (a  better  point  may  have  higher  objective  value  but  smaller
      constraint violation).

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void mindfsetxrep(mindfstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`mindfusetimers` function

/*************************************************************************
This function activates/deactivates internal timers  used  to  track  time
spent in various parts of the solver (mostly, callbacks vs solver itself).

When activated with this function, the following timings are stored in the
mindfreport structure fields:
* total time spend in the optimization
* time spent in the callback
* time spent in the solver itself

See comments on mindfreport structure for more  information  about  timers
and their accuracy.

Timers are an essential part of reports that helps to find out  where  the
most time is spent and how to  optimize  the  code.  E.g.,  noticing  that
significant amount of time is spent  in  numerical  differentiation  makes
obvious that ALGLIB-provided parallel numerical differentiation is needed.

However, time measurements add noticeable  overhead,  about  50-100ns  per
function call. In some applications it results in a significant  slowdown,
that's why this option is inactive  by  default  and  should  be  manually
activated.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    UseTimers-  true or false

NOTE: when tracing is turned on with alglib::trace_file(), some derivative
      free  solvers   may  also  perform  internal,  more  detailed   time
      measurements, which are printed to the log file.

  -- ALGLIB --
     Copyright 23.04.2024 by Bochkanov Sergey
*************************************************************************/
void mindfusetimers(mindfstate &state, const bool usetimers, const xparams _xparams = alglib::xdefault);

mindf_gdemo_auto example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc2_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1,x2) = x0+x1
    //     f1(x0,x1,x2) = x2-exp(x0)
    //     f2(x0,x1,x2) = x0^2+x1^2-1
    //
    fi[0] = x[0]+x[1];
    fi[1] = x[2]-exp(x[0]);
    fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = x0+x1
        //
        // subject to nonlinear constraints
        //
        //    x0^2 + x1^2 - 1 <= 0
        //    x2-exp(x0) = 0
        //
        real_1d_array x0 = "[0,0,0]";
        real_1d_array s = "[1,1,1]";
        mindfstate state;
        mindfreport rep;
        real_1d_array x1;

        //
        // Create optimizer object
        //
        mindfcreate(x0, state);
        mindfsetscale(state, s);

        //
        // Choose  one of the nonlinear  programming  solvers  supported  by  MINDF
        // optimizer.
        //
        // This example shows how to use GDEMO (Generalized Differential Evolution,
        // MultiObjective) solver working in a single-objective mode. This solver
        // uses an adaptive choice of DE parameters (crossover, weight and strategy),
        // automatically choosing the most appropriate settings during the optimization.
        //
        // Thus, the only tunable parameters are:
        // * iterations count
        // * population size
        // * algorithm profile (ROBUST or QUICK)
        //
        // The latter two parameters can be omitted, the solver will use a default
        // population size (10*N in the current version) and a default profile (ROBUST
        // one).
        //
        // Metaheuristics typically need hundreds or thousands of iterations to converge,
        // in this example we set 200 iterations. Good values for population size
        // are between 5*N and 20*N, with 5*N being recommended for easy problems
        // and 20*N (or more) being recommended for difficult problems with multiple
        // extrema, many cross-interacting variables and/or noise.
        // 
        // In addition to these parameters it is also possible to specify a so-called
        // 'profile' - a set of recommendations regarding decisions algorithm is allowed
        // to make during parameters autotuning:
        // * the ROBUST profile is a default option. The solver tries only conservative
        //   strategies that have low probability of a failure (stagnation far away
        //   from the solution)
        // * the QUICK profile is intended  to  facilitate accelerated  convergence on
        //   medium-complexity problems at the cost of (sometimes) having premature
        //   convergence on difficult multi-extremal problems. It most often results
        //   in a 2x-3x higher convergence speed than the ROBUST profile.
        //
        // In this simple example we do not provide variable bounds and scales; however,
        // on real life problems it is very important to box as many variables as possible
        // (it helps to generate initial population and to avoid bad parameter values)
        // and to provide variable scales. Differential Evolution itself is scale-invariant;
        // however, penalties for constraint violation are scale-dependent.
        //
        ae_int_t maxits = 200;
        mindfsetalgogdemo(state, maxits);
        mindfsetgdemoprofilerobust(state);

        //
        // Unlike smooth solvers, metaheuristics do not have well-defined stopping criteria.
        // It is recommended to run the solver until iterations budget is exhausted.
        //
        // However, it is possible to let the solver stop early, when either:
        // * subpopulation target values (2N+1 best individuals) are within
        //   EPS from the best one so far (function values seem to converge)
        // * or 2N+1-subpopulation target values AND variable values are within EPS from
        //   the best solution so far
        //
        // Both conditions are heuristics that may fail. The fact that many candidate
        // objective values have clustered within EPS of the best objective value so far
        // usually means that we are somewhere within [0.1EPS,10EPS] away from the true
        // solution; however, on difficult problems this condition may fire too early.
        //
        // Imposing an additional requirement that variable values have clustered  too
        // may prevent us from premature  stopping. However, on multi-extremal and/or
        // noisy problems too many individuals may be trapped away from the  optimum,
        // preventing this condition from activation.
        //
        // The summary is that stopping criteria are heavily problem-dependent.
        //
        mindfsetcondf(state, 0.00001);

        //
        // Set nonlinear constraints.
        //
        // ALGLIB  supports  any  combination  of  box,  linear  and  nonlinear
        // constraints. This specific example uses only nonlinear ones.
        //
        // Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        // nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        // elements being possibly infinite (means that this specific bound  is
        // ignored). It includes equality constraints,  upper/lower  inequality
        // constraints, range constraints. In particular, a pair of constraints
        //
        //        x2-exp(x0)       = 0
        //        x0^2 + x1^2 - 1 <= 0
        //
        // can be specified by passing NL=[0,-INF], NU=[0,0] to mindfsetnlc2().
        // Constraining functions themselves are passed as a part  of a problem
        // target vector (see below).
        //
        //
        // Unlike smooth optimizers like SQP which naturally include linear and
        // nonlinear constraints into the  algorithm,  derivative-free  methods
        // often need special strategies to deal with them, with each  strategy
        // having its own limitations:
        //
        // * an L2 penalty,  which  has  good  global  constraint   enforcement
        //   properties, but usually allows some moderate constraint violation
        //
        // * an L1 penalty, which has potential to enforce constraints exactly,
        //   but has somewhat weaker ability to move iterations from  far  away
        //   points closer to the feasible area. It also  has  somewhat  harder
        //   numerical properties, needing more iterations to converge.
        //
        // * a combined L1/L2 penalty, which is a good compromise
        //
        // The code below sets constraints bounds and tells the solver  to  use
        // a mixed L1/L2 penalized strategy.
        //
        // NOTE: box constraints require no special handling.
        //
        real_1d_array nl = "[0,-inf]";
        real_1d_array nu = "[0,0]";
        double rho1 = 5;
        double rho2 = 5;
        mindfsetnlc2(state, nl, nu);
        mindfsetgdemopenalty(state, rho1, rho2);

        //
        // Optimize and test results.
        //
        // The optimizer object accepts vector function  with  its  first  component
        // being a target and subsequent components being nonlinear constraints.
        //
        // So, our vector function has the following form
        //
        //     {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
        //
        // with f0 being target function, f1 being equality constraint "f1=0",
        // f2 being inequality constraint "f2<=0".
        //
        //
        //
        // The commercial ALGLIB has two important improvements over the free edition:
        // * callback parallelism
        // * SIMD kernels for candidate points generation
        //
        // Differential evolution evaluates objectives/constraints in large batches.
        // When the batch takes more than several milliseconds to process, it makes
        // sense to compute function values at different points in parallel. It is
        // called 'callback parallelism'.
        //
        // Another useful performance improvement is an ability to utilize SIMD for
        // massive random numbers generation and mutation/crossover. The performance
        // impact of these operations can be noticeable when solving problems with
        // cheap objectives.
        //
        alglib::mindfoptimize(state, nlcfunc2_fvec);
        mindfresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [-0.70710,-0.70710,0.49306]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minlbfgs` subpackage

Classes

minlbfgsreport
minlbfgsstate

Functions

minlbfgscreate
minlbfgscreatef
minlbfgsiteration
minlbfgsoptguardgradient
minlbfgsoptguardnonc1test0results
minlbfgsoptguardnonc1test1results
minlbfgsoptguardresults
minlbfgsoptguardsmoothness
minlbfgsoptimize
minlbfgsrequesttermination
minlbfgsrestartfrom
minlbfgsresults
minlbfgsresultsbuf
minlbfgssetcond
minlbfgssetpreccholesky
minlbfgssetprecdefault
minlbfgssetprecdiag
minlbfgssetprecscale
minlbfgssetscale
minlbfgssetstpmax
minlbfgssetxrep

Examples

minlbfgs_d_1		Nonlinear optimization by L-BFGS
minlbfgs_d_2		Nonlinear optimization with additional settings and restarts
minlbfgs_numdiff		Nonlinear optimization by L-BFGS with numerical differentiation

`minlbfgsreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
   1    relative function improvement is no more than EpsF.
   2    relative step is no more than EpsX.
   4    gradient norm is no more than EpsG
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    terminated    by  user  who  called  minlbfgsrequesttermination().
        X contains point which was   "current accepted"  when  termination
        request was submitted.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minlbfgsreport
{
public:
    minlbfgsreport();
    minlbfgsreport(const minlbfgsreport &rhs);
    minlbfgsreport& operator=(const minlbfgsreport &rhs);
    virtual ~minlbfgsreport();
    ae_int_t iterationscount;
    ae_int_t nfev;
    ae_int_t terminationtype;
};

`minlbfgsstate` class

/*************************************************************************

*************************************************************************/
class minlbfgsstate
{
public:
    minlbfgsstate();
    minlbfgsstate(const minlbfgsstate &rhs);
    minlbfgsstate& operator=(const minlbfgsstate &rhs);
    virtual ~minlbfgsstate();
};

`minlbfgscreate` function

/*************************************************************************
        LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION

DESCRIPTION:
The subroutine minimizes function F(x) of N arguments by  using  a  quasi-
Newton method (LBFGS scheme) which is optimized to use  a  minimum  amount
of memory.
The subroutine generates the approximation of an inverse Hessian matrix by
using information about the last M steps of the algorithm  (instead of N).
It lessens a required amount of memory from a value  of  order  N^2  to  a
value of order 2*N*M.


REQUIREMENTS:
Algorithm will request following information during its operation:
* function value F and its gradient G (simultaneously) at given point X


USAGE:
1. User initializes algorithm state with MinLBFGSCreate() call
2. User tunes solver parameters with MinLBFGSSetCond() MinLBFGSSetStpMax()
   and other functions
3. User calls MinLBFGSOptimize() function which takes algorithm  state and
   pointer (delegate, etc.) to callback function which calculates F/G.
4. User calls MinLBFGSResults() to get solution
5. Optionally user may call MinLBFGSRestartFrom() to solve another problem
   with same N/M but another starting point and/or another function.
   MinLBFGSRestartFrom() allows to reuse already initialized structure.

INPUT PARAMETERS:
    N       -   problem dimension. N>0
    M       -   number of corrections in the BFGS scheme of Hessian
                approximation update. Recommended value:  3<=M<=7. The smaller
                value causes worse convergence, the bigger will  not  cause  a
                considerably better convergence, but will cause a fall in  the
                performance. M<=N.
    X       -   initial solution approximation, array[0..N-1].


OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

IMPORTANT: the   LBFGS  optimizer  supports  parallel  parallel  numerical
           differentiation  ('callback    parallelism').   This   feature,
           which  is  present  in  commercial   ALGLIB   editions  greatly
           accelerates  optimization  with  numerical  differentiation  of
           an expensive target functions.

           Callback parallelism is usually  beneficial  when  computing  a
           numerical gradient requires  more  than  several  milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial version'
           section,  and  comments  on  minlbfgsoptimize() function for
           more information.

NOTES:
1. you may tune stopping conditions with MinLBFGSSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLBFGSSetStpMax() function to bound algorithm's  steps.  However,
   L-BFGS rarely needs such a tuning.


  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgscreate(const ae_int_t n, const ae_int_t m, const real_1d_array &x, minlbfgsstate &state, const xparams _xparams = alglib::xdefault);
void minlbfgscreate(const ae_int_t m, const real_1d_array &x, minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minlbfgscreatef` function

/*************************************************************************
The subroutine is finite difference variant of MinLBFGSCreate().  It  uses
finite differences in order to differentiate target function.

Description below contains information which is specific to  this function
only. We recommend to read comments on MinLBFGSCreate() in  order  to  get
more information about creation of LBFGS optimizer.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of corrections in the BFGS scheme of Hessian
                approximation update. Recommended value:  3<=M<=7. The smaller
                value causes worse convergence, the bigger will  not  cause  a
                considerably better convergence, but will cause a fall in  the
                performance. M<=N.
    X       -   starting point, array[0..N-1].
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


IMPORTANT: the   LBFGS  optimizer  supports  parallel  parallel  numerical
           differentiation  ('callback    parallelism').   This   feature,
           which  is  present  in  commercial   ALGLIB   editions  greatly
           accelerates  optimization  with  numerical  differentiation  of
           an expensive target functions.

           Callback parallelism is usually  beneficial  when  computing  a
           numerical gradient requires  more  than  several  milliseconds.

           See ALGLIB Reference Manual, 'Working with commercial version'
           section,  and  comments  on  minlbfgsoptimize() function for
           more information.

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinLBFGSSetScale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large truncation  errors, while too small
   step will result in too large numerical  errors.  1.0E-6  can  be  good
   value to start with.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is   less  robust  and  precise.  LBFGS  needs  exact  gradient values.
   Imprecise gradient may slow  down  convergence,  especially  on  highly
   nonlinear problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 16.05.2011 by Bochkanov Sergey
*************************************************************************/
void minlbfgscreatef(const ae_int_t n, const ae_int_t m, const real_1d_array &x, const double diffstep, minlbfgsstate &state, const xparams _xparams = alglib::xdefault);
void minlbfgscreatef(const ae_int_t m, const real_1d_array &x, const double diffstep, minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlbfgsiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minlbfgsiteration(minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

`minlbfgsoptguardgradient` function

/*************************************************************************
This  function  activates/deactivates verification  of  the  user-supplied
analytic gradient.

Upon  activation  of  this  option  OptGuard  integrity  checker  performs
numerical differentiation of your target function  at  the  initial  point
(note: future versions may also perform check  at  the  final  point)  and
compares numerical gradient with analytic one provided by you.

If difference is too large, an error flag is set and optimization  session
continues. After optimization session is over, you can retrieve the report
which  stores  both  gradients  and  specific  components  highlighted  as
suspicious by the OptGuard.

The primary OptGuard report can be retrieved with minlbfgsoptguardresults().

IMPORTANT: gradient check is a high-overhead option which  will  cost  you
           about 3*N additional function evaluations. In many cases it may
           cost as much as the rest of the optimization session.

           YOU SHOULD NOT USE IT IN THE PRODUCTION CODE UNLESS YOU WANT TO
           CHECK DERIVATIVES PROVIDED BY SOME THIRD PARTY.

NOTE: unlike previous incarnation of the gradient checking code,  OptGuard
      does NOT interrupt optimization even if it discovers bad gradient.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step used for numerical differentiation:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification
                    You should carefully choose TestStep. Value  which  is
                    too large (so large that  function  behavior  is  non-
                    cubic at this scale) will lead  to  false  alarms. Too
                    short step will result in rounding  errors  dominating
                    numerical derivative.

                    You may use different step for different parameters by
                    means of setting scale with minlbfgssetscale().

=== EXPLANATION ==========================================================

In order to verify gradient algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptguardgradient(minlbfgsstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlbfgsoptguardnonc1test0results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #0

Nonsmoothness (non-C1) test #0 studies  function  values  (not  gradient!)
obtained during line searches and monitors  behavior  of  the  directional
derivative estimate.

This test is less powerful than test #1, but it does  not  depend  on  the
gradient values and thus it is more robust against artifacts introduced by
numerical differentiation.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #0 "strong" report
    lngrep  -   C1 test #0 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptguardnonc1test0results(const minlbfgsstate &state, optguardnonc1test0report &strrep, optguardnonc1test0report &lngrep, const xparams _xparams = alglib::xdefault);

`minlbfgsoptguardnonc1test1results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #1

Nonsmoothness (non-C1)  test  #1  studies  individual  components  of  the
gradient computed during line search.

When precise analytic gradient is provided this test is more powerful than
test #0  which  works  with  function  values  and  ignores  user-provided
gradient.  However,  test  #0  becomes  more   powerful   when   numerical
differentiation is employed (in such cases test #1 detects  higher  levels
of numerical noise and becomes too conservative).

This test also tells specific components of the gradient which violate  C1
continuity, which makes it more informative than #0, which just tells that
continuity is violated.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* vidx - is an index of the variable in [0,N) with nonsmooth derivative
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], g[] - arrays of length CNT which store step lengths and  gradient
  values at these points; g[i] is evaluated in  x0+stp[i]*d  and  contains
  vidx-th component of the gradient.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #1 "strong" report
    lngrep  -   C1 test #1 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptguardnonc1test1results(minlbfgsstate &state, optguardnonc1test1report &strrep, optguardnonc1test1report &lngrep, const xparams _xparams = alglib::xdefault);

`minlbfgsoptguardresults` function

/*************************************************************************
Results of OptGuard integrity check, should be called  after  optimization
session is over.

=== PRIMARY REPORT =======================================================

OptGuard performs several checks which are intended to catch common errors
in the implementation of nonlinear function/gradient:
* incorrect analytic gradient
* discontinuous (non-C0) target functions (constraints)
* nonsmooth     (non-C1) target functions (constraints)

Each of these checks is activated with appropriate function:
* minlbfgsoptguardgradient() for gradient verification
* minlbfgsoptguardsmoothness() for C0/C1 checks

Following flags are set when these errors are suspected:
* rep.badgradsuspected, and additionally:
  * rep.badgradvidx for specific variable (gradient element) suspected
  * rep.badgradxbase, a point where gradient is tested
  * rep.badgraduser, user-provided gradient  (stored  as  2D  matrix  with
    single row in order to make  report  structure  compatible  with  more
    complex optimizers like MinNLC or MinLM)
  * rep.badgradnum,   reference    gradient    obtained    via   numerical
    differentiation (stored as  2D matrix with single row in order to make
    report structure compatible with more complex optimizers  like  MinNLC
    or MinLM)
* rep.nonc0suspected
* rep.nonc1suspected

=== ADDITIONAL REPORTS/LOGS ==============================================

Several different tests are performed to catch C0/C1 errors, you can  find
out specific test signaled error by looking to:
* rep.nonc0test0positive, for non-C0 test #0
* rep.nonc1test0positive, for non-C1 test #0
* rep.nonc1test1positive, for non-C1 test #1

Additional information (including line search logs)  can  be  obtained  by
means of:
* minlbfgsoptguardnonc1test0results()
* minlbfgsoptguardnonc1test1results()
which return detailed error reports, specific points where discontinuities
were found, and so on.

==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    rep     -   generic OptGuard report;  more  detailed  reports  can  be
                retrieved with other functions.

NOTE: false negatives (nonsmooth problems are not identified as  nonsmooth
      ones) are possible although unlikely.

      The reason  is  that  you  need  to  make several evaluations around
      nonsmoothness  in  order  to  accumulate  enough  information  about
      function curvature. Say, if you start right from the nonsmooth point,
      optimizer simply won't get enough data to understand what  is  going
      wrong before it terminates due to abrupt changes in the  derivative.
      It is also  possible  that  "unlucky"  step  will  move  us  to  the
      termination too quickly.

      Our current approach is to have less than 0.1%  false  negatives  in
      our test examples  (measured  with  multiple  restarts  from  random
      points), and to have exactly 0% false positives.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptguardresults(minlbfgsstate &state, optguardreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlbfgsoptguardsmoothness` function

/*************************************************************************
This  function  activates/deactivates nonsmoothness monitoring  option  of
the  OptGuard  integrity  checker. Smoothness  monitor  silently  observes
solution process and tries to detect ill-posed problems, i.e. ones with:
a) discontinuous target function (non-C0)
b) nonsmooth     target function (non-C1)

Smoothness monitoring does NOT interrupt optimization  even if it suspects
that your problem is nonsmooth. It just sets corresponding  flags  in  the
OptGuard report which can be retrieved after optimization is over.

Smoothness monitoring is a moderate overhead option which often adds  less
than 1% to the optimizer running time. Thus, you can use it even for large
scale problems.

NOTE: OptGuard does  NOT  guarantee  that  it  will  always  detect  C0/C1
      continuity violations.

      First, minor errors are hard to  catch - say, a 0.0001 difference in
      the model values at two sides of the gap may be due to discontinuity
      of the model - or simply because the model has changed.

      Second, C1-violations  are  especially  difficult  to  detect  in  a
      noninvasive way. The optimizer usually  performs  very  short  steps
      near the nonsmoothness, and differentiation  usually   introduces  a
      lot of numerical noise.  It  is  hard  to  tell  whether  some  tiny
      discontinuity in the slope is due to real nonsmoothness or just  due
      to numerical noise alone.

      Our top priority was to avoid false positives, so in some rare cases
      minor errors may went unnoticed (however, in most cases they can  be
      spotted with restart from different initial point).

INPUT PARAMETERS:
    state   -   algorithm state
    level   -   monitoring level:
                * 0 - monitoring is disabled
                * 1 - noninvasive low-overhead monitoring; function values
                      and/or gradients are recorded, but OptGuard does not
                      try to perform additional evaluations  in  order  to
                      get more information about suspicious locations.

=== EXPLANATION ==========================================================

One major source of headache during optimization  is  the  possibility  of
the coding errors in the target function/constraints (or their gradients).
Such  errors   most   often   manifest   themselves  as  discontinuity  or
nonsmoothness of the target/constraints.

Another frequent situation is when you try to optimize something involving
lots of min() and max() operations, i.e. nonsmooth target. Although not  a
coding error, it is nonsmoothness anyway - and smooth  optimizers  usually
stop right after encountering nonsmoothness, well before reaching solution.

OptGuard integrity checker helps you to catch such situations: it monitors
function values/gradients being passed  to  the  optimizer  and  tries  to
errors. Upon discovering suspicious pair of points it  raises  appropriate
flag (and allows you to continue optimization). When optimization is done,
you can study OptGuard result.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptguardsmoothness(minlbfgsstate &state, const ae_int_t level, const xparams _xparams = alglib::xdefault);
void minlbfgsoptguardsmoothness(minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlbfgsoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    grad    -   callback which calculates function (or merit function)
                value func and gradient grad at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


CALLBACK PARALLELISM:

The LBFGS optimizer supports parallel numerical differentiation ('callback
parallelism').  This  feature,  which  is  present  in  commercial  ALGLIB
editions,  greatly  accelerates  numerical  differentiation  of  expensive
targets.

Callback parallelism is usually beneficial computing a numerical  gradient
requires more than several milliseconds. In this case the job of computing
individual gradient components can be split between multiple threads. Even
inexpensive  targets  can  benefit  from  parallelism, if  you  have  many
variables.

ALGLIB Reference Manual, 'Working with commercial  version' section, tells
how to activate callback parallelism for your programming language.

CALLBACKS ACCEPTED

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied gradient,  and one which uses function value
   only  and  numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   (either MinLBFGSCreate() for analytical gradient  or  MinLBFGSCreateF()
   for numerical differentiation) you should choose appropriate variant of
   MinLBFGSOptimize() - one  which  accepts  function  AND gradient or one
   which accepts function ONLY.

   Be careful to choose variant of MinLBFGSOptimize() which corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinLBFGSOptimize()  and specific
   function used to create optimizer.


                     |         USER PASSED TO MinLBFGSOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinLBFGSCreateF() |     work                FAIL
   MinLBFGSCreate()  |     FAIL                work

   Here "FAIL" denotes inappropriate combinations  of  optimizer  creation
   function  and  MinLBFGSOptimize()  version.   Attemps   to   use   such
   combination (for example, to create optimizer with MinLBFGSCreateF() and
   to pass gradient information to MinCGOptimize()) will lead to exception
   being thrown. Either  you  did  not pass gradient when it WAS needed or
   you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlbfgsoptimize(minlbfgsstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlbfgsoptimize(minlbfgsstate &state,
    void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minlbfgsrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void minlbfgsrequesttermination(minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

`minlbfgsrestartfrom` function

/*************************************************************************
This  subroutine restarts LBFGS algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used to store algorithm state
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgsrestartfrom(minlbfgsstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlbfgsresults` function

/*************************************************************************
L-BFGS algorithm results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -8    internal integrity control  detected  infinite
                            or NAN values in  function/gradient.  Abnormal
                            termination signalled.
                    * -2    rounding errors prevent further improvement.
                            X contains best point found.
                    * -1    incorrect parameters were specified
                    *  1    relative function improvement is no more than
                            EpsF.
                    *  2    relative step is no more than EpsX.
                    *  4    gradient norm is no more than EpsG
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible
                    *  8    terminated by user who called minlbfgsrequesttermination().
                            X contains point which was "current accepted" when
                            termination request was submitted.
                * Rep.IterationsCount contains iterations count
                * NFEV countains number of function calculations

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgsresults(const minlbfgsstate &state, real_1d_array &x, minlbfgsreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minlbfgsresultsbuf` function

/*************************************************************************
L-BFGS algorithm results

Buffered implementation of MinLBFGSResults which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgsresultsbuf(const minlbfgsstate &state, real_1d_array &x, minlbfgsreport &rep, const xparams _xparams = alglib::xdefault);

`minlbfgssetcond` function

/*************************************************************************
This function sets stopping conditions for L-BFGS optimization algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinLBFGSSetScale()
    EpsF    -   >=0
                The  subroutine  finishes  its work if on k+1-th iteration
                the  condition  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
                is satisfied.
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinLBFGSSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to
automatic stopping criterion selection (small EpsX).

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetcond(minlbfgsstate &state, const double epsg, const double epsf, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minlbfgssetpreccholesky` function

/*************************************************************************
Modification of the preconditioner: Cholesky factorization of  approximate
Hessian is used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    P       -   triangular preconditioner, Cholesky factorization of
                the approximate Hessian. array[0..N-1,0..N-1],
                (if larger, only leading N elements are used).
    IsUpper -   whether upper or lower triangle of P is given
                (other triangle is not referenced)

After call to this function preconditioner is changed to P  (P  is  copied
into the internal buffer).

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

NOTE 2:  P  should  be nonsingular. Exception will be thrown otherwise.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetpreccholesky(minlbfgsstate &state, const real_2d_array &p, const bool isupper, const xparams _xparams = alglib::xdefault);

`minlbfgssetprecdefault` function

/*************************************************************************
Modification  of  the  preconditioner:  default  preconditioner    (simple
scaling, same for all elements of X) is used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetprecdefault(minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

`minlbfgssetprecdiag` function

/*************************************************************************
Modification  of  the  preconditioner:  diagonal of approximate Hessian is
used.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    D       -   diagonal of the approximate Hessian, array[0..N-1],
                (if larger, only leading N elements are used).

NOTE:  you  can  change  preconditioner  "on  the  fly",  during algorithm
iterations.

NOTE 2: D[i] should be positive. Exception will be thrown otherwise.

NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE.

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetprecdiag(minlbfgsstate &state, const real_1d_array &d, const xparams _xparams = alglib::xdefault);

`minlbfgssetprecscale` function

/*************************************************************************
Modification of the preconditioner: scale-based diagonal preconditioning.

This preconditioning mode can be useful when you  don't  have  approximate
diagonal of Hessian, but you know that your  variables  are  badly  scaled
(for  example,  one  variable is in [1,10], and another in [1000,100000]),
and most part of the ill-conditioning comes from different scales of vars.

In this case simple  scale-based  preconditioner,  with H[i] = 1/(s[i]^2),
can greatly improve convergence.

IMPRTANT: you should set scale of your variables  with  MinLBFGSSetScale()
call  (before  or after MinLBFGSSetPrecScale() call). Without knowledge of
the scale of your variables scale-based preconditioner will be  just  unit
matrix.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 13.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetprecscale(minlbfgsstate &state, const xparams _xparams = alglib::xdefault);

`minlbfgssetscale` function

/*************************************************************************
This function sets scaling coefficients for LBFGS optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

In  most  optimizers  (and  in  the  LBFGS  too)  scaling is NOT a form of
preconditioning. It just  affects  stopping  conditions.  You  should  set
preconditioner  by  separate  call  to  one  of  the  MinLBFGSSetPrec...()
functions.

There  is  special  preconditioning  mode, however,  which  uses   scaling
coefficients to form diagonal preconditioning matrix. You  can  turn  this
mode on, if you want.   But  you should understand that scaling is not the
same thing as preconditioning - these are two different, although  related
forms of tuning solver.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetscale(minlbfgsstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`minlbfgssetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0 (default),  if
                you don't want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetstpmax(minlbfgsstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`minlbfgssetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinLBFGSOptimize().


  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlbfgssetxrep(minlbfgsstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minlbfgs_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // using LBFGS method, with:
        // * initial point x=[0,0]
        // * unit scale being set for all variables (see minlbfgssetscale for more info)
        // * stopping criteria set to "terminate after short enough step"
        //
        // First, we create optimizer object and tune its properties.
        //
        // IMPORTANT: the  LBFGS  optimizer  supports  parallel parallel numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is  present  in  commercial  ALGLIB  editions greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  minlbfgsoptimize() function for
        //            more information.
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.0000000001;
        ae_int_t maxits = 0;
        minlbfgsstate state;
        minlbfgscreate(1, x, state);
        minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        minlbfgssetscale(state, s);

        //
        // Optimize and examine results.
        //
        minlbfgsreport rep;
        alglib::minlbfgsoptimize(state, function1_grad);
        minlbfgsresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minlbfgs_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr) 
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    // and its derivatives df/d0 and df/dx1
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
    grad[0] = 400*pow(x[0]+3,3);
    grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
        // using LBFGS method.
        //
        // Several advanced techniques are demonstrated:
        // * upper limit on step size
        // * restart from new point
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsg = 0;
        double epsf = 0;
        double epsx = 0.0000000001;
        double stpmax = 0.1;
        ae_int_t maxits = 0;
        minlbfgsstate state;
        minlbfgsreport rep;

        // create and tune optimizer
        minlbfgscreate(1, x, state);
        minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        minlbfgssetstpmax(state, stpmax);
        minlbfgssetscale(state, s);

        // Set up OptGuard integrity checker which catches errors
        // like nonsmooth targets or errors in the analytic gradient.
        //
        // OptGuard is essential at the early prototyping stages.
        //
        // NOTE: gradient verification needs 3*N additional function
        //       evaluations; DO NOT USE IT IN THE PRODUCTION CODE
        //       because it leads to unnecessary slowdown of your app.
        minlbfgsoptguardsmoothness(state);
        minlbfgsoptguardgradient(state, 0.001);

        // first run
        alglib::minlbfgsoptimize(state, function1_grad);
        minlbfgsresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

        // second run - algorithm is restarted
        x = "[10,10]";
        minlbfgsrestartfrom(state, x);
        alglib::minlbfgsoptimize(state, function1_grad);
        minlbfgsresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]

        // check OptGuard integrity report. Why do we need it at all?
        // Well, try breaking the gradient by adding 1.0 to some
        // of its components - OptGuard should report it as error.
        // And it may also catch unintended errors too :)
        optguardreport ogrep;
        minlbfgsoptguardresults(state, ogrep);
        printf("%s\n", ogrep.badgradsuspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc0suspected ? "true" : "false"); // EXPECTED: false
        printf("%s\n", ogrep.nonc1suspected ? "true" : "false"); // EXPECTED: false
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minlbfgs_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
    //
    // this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
    //
    func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x,y) = 100*(x+3)^4+(y-3)^4
        //
        // using numerical differentiation to calculate gradient.
        //
        // IMPORTANT: the  LBFGS  optimizer  supports  parallel parallel numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is  present  in  commercial  ALGLIB  editions greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  minlbfgsoptimize() function for
        //            more information.
        //
        real_1d_array x = "[0,0]";
        double epsg = 0.0000000001;
        double epsf = 0;
        double epsx = 0;
        double diffstep = 1.0e-6;
        ae_int_t maxits = 0;
        minlbfgsstate state;
        minlbfgsreport rep;

        minlbfgscreatef(1, x, diffstep, state);
        minlbfgssetcond(state, epsg, epsf, epsx, maxits);
        alglib::minlbfgsoptimize(state, function1_func);
        minlbfgsresults(state, x, rep);

        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minlm` subpackage

Classes

minlmreport
minlmstate

Functions

minlmcreatev
minlmcreatevj
minlmiteration
minlmoptguardgradient
minlmoptguardresults
minlmoptimize
minlmrequesttermination
minlmrestartfrom
minlmresults
minlmresultsbuf
minlmsetacctype
minlmsetbc
minlmsetcond
minlmsetlc
minlmsetnonmonotonicsteps
minlmsetnumdiff
minlmsetscale
minlmsetstpmax
minlmsetxrep

Examples

minlm_d_restarts		Efficient restarts of LM optimizer
minlm_d_v		Nonlinear least squares optimization using function vector only
minlm_d_vb		Bound constrained nonlinear least squares optimization
minlm_d_vj		Nonlinear least squares optimization using function vector and Jacobian

`minlmreport` class

/*************************************************************************
Optimization report, filled by MinLMResults() function

FIELDS:
* TerminationType, completetion code:
    * -8    optimizer detected NAN/INF values either in the function itself,
            or in its Jacobian
    * -5    inappropriate solver was used:
            * solver created with minlmcreatefgh() used  on  problem  with
              general linear constraints (set with minlmsetlc() call).
    * -3    constraints are inconsistent
    *  2    relative step is no more than EpsX.
    *  5    MaxIts steps was taken
    *  7    stopping conditions are too stringent,
            further improvement is impossible
    *  8    terminated   by  user  who  called  MinLMRequestTermination().
            X contains point which was "current accepted" when termination
            request was submitted.
* F, objective value, SUM(f[i]^2)
* IterationsCount, contains iterations count
* NFunc, number of function calculations
* NJac, number of Jacobi matrix calculations
* NGrad, number of gradient calculations
* NHess, number of Hessian calculations
* NCholesky, number of Cholesky decomposition calculations
*************************************************************************/
class minlmreport
{
public:
    minlmreport();
    minlmreport(const minlmreport &rhs);
    minlmreport& operator=(const minlmreport &rhs);
    virtual ~minlmreport();
    ae_int_t iterationscount;
    ae_int_t terminationtype;
    double f;
    ae_int_t nfunc;
    ae_int_t njac;
    ae_int_t ngrad;
    ae_int_t nhess;
    ae_int_t ncholesky;
};

`minlmstate` class

/*************************************************************************
Levenberg-Marquardt optimizer.

This structure should be created using one of the MinLMCreate???()
functions. You should not access its fields directly; use ALGLIB functions
to work with it.
*************************************************************************/
class minlmstate
{
public:
    minlmstate();
    minlmstate(const minlmstate &rhs);
    minlmstate& operator=(const minlmstate &rhs);
    virtual ~minlmstate();
};

`minlmcreatev` function

/*************************************************************************
                IMPROVED LEVENBERG-MARQUARDT METHOD FOR
                 NON-LINEAR LEAST SQUARES OPTIMIZATION

DESCRIPTION:
This function is used to find minimum of function which is represented  as
sum of squares:
    F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])
using value of function vector f[] only. Finite differences  are  used  to
calculate Jacobian.


REQUIREMENTS:
This algorithm will request following information during its operation:
* function vector f[] at given point X

There are several overloaded versions of  MinLMOptimize()  function  which
correspond  to  different LM-like optimization algorithms provided by this
unit. You should choose version which accepts fvec() callback.

You can try to initialize MinLMState structure with VJ  function and  then
use incorrect version  of  MinLMOptimize()  (for  example,  version  which
works with general form function and does not accept function vector), but
it will  lead  to  exception being thrown after first attempt to calculate
Jacobian.


USAGE:
1. User initializes algorithm state with MinLMCreateV() call
2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
   other functions
3. User calls MinLMOptimize() function which  takes algorithm  state   and
   callback functions.
4. User calls MinLMResults() to get solution
5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
   with same N/M but another starting point and/or another function.
   MinLMRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   dimension, N>1
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of functions f[i]
    X       -   initial solution, array[0..N-1]
    DiffStep-   differentiation step, >0. By  default,  symmetric  3-point
                formula which provides good accuracy is used.  It  can  be
                changed to a faster but  less  precise  2-point  one  with
                minlmsetnumdiff() function.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

See also MinLMIteration, MinLMResults.

NOTES:
1. you may tune stopping conditions with MinLMSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLMSetStpMax() function to bound algorithm's steps.

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlmcreatev(const ae_int_t n, const ae_int_t m, const real_1d_array &x, const double diffstep, minlmstate &state, const xparams _xparams = alglib::xdefault);
void minlmcreatev(const ae_int_t m, const real_1d_array &x, const double diffstep, minlmstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minlmcreatevj` function

/*************************************************************************
                IMPROVED LEVENBERG-MARQUARDT METHOD FOR
                 NON-LINEAR LEAST SQUARES OPTIMIZATION

DESCRIPTION:
This function is used to find minimum of function which is represented  as
sum of squares:
    F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1])
using value of function vector f[] and Jacobian of f[].


REQUIREMENTS:
This algorithm will request following information during its operation:

* function vector f[] at given point X
* function vector f[] and Jacobian of f[] (simultaneously) at given point

There are several overloaded versions of  MinLMOptimize()  function  which
correspond  to  different LM-like optimization algorithms provided by this
unit. You should choose version which accepts fvec()  and jac() callbacks.
First  one  is used to calculate f[] at given point, second one calculates
f[] and Jacobian df[i]/dx[j].

You can try to initialize MinLMState structure with VJ  function and  then
use incorrect version  of  MinLMOptimize()  (for  example,  version  which
works  with  general  form function and does not provide Jacobian), but it
will  lead  to  exception  being  thrown  after first attempt to calculate
Jacobian.


USAGE:
1. User initializes algorithm state with MinLMCreateVJ() call
2. User tunes solver parameters with MinLMSetCond(),  MinLMSetStpMax() and
   other functions
3. User calls MinLMOptimize() function which  takes algorithm  state   and
   callback functions.
4. User calls MinLMResults() to get solution
5. Optionally, user may call MinLMRestartFrom() to solve  another  problem
   with same N/M but another starting point and/or another function.
   MinLMRestartFrom() allows to reuse already initialized structure.


INPUT PARAMETERS:
    N       -   dimension, N>1
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of functions f[i]
    X       -   initial solution, array[0..N-1]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

NOTES:
1. you may tune stopping conditions with MinLMSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use MinLMSetStpMax() function to bound algorithm's steps.

  -- ALGLIB --
     Copyright 30.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlmcreatevj(const ae_int_t n, const ae_int_t m, const real_1d_array &x, minlmstate &state, const xparams _xparams = alglib::xdefault);
void minlmcreatevj(const ae_int_t m, const real_1d_array &x, minlmstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlmiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minlmiteration(minlmstate &state, const xparams _xparams = alglib::xdefault);

`minlmoptguardgradient` function

/*************************************************************************
This  function  activates/deactivates verification  of  the  user-supplied
analytic Jacobian.

Upon  activation  of  this  option  OptGuard  integrity  checker  performs
numerical differentiation of your target function vector  at  the  initial
point (note: future versions may also perform check  at  the final  point)
and compares numerical Jacobian with analytic one provided by you.

If difference is too large, an error flag is set and optimization  session
continues. After optimization session is over, you can retrieve the report
which stores  both  Jacobians,  and  specific  components  highlighted  as
suspicious by the OptGuard.

The OptGuard report can be retrieved with minlmoptguardresults().

IMPORTANT: gradient check is a high-overhead option which  will  cost  you
           about 3*N additional function evaluations. In many cases it may
           cost as much as the rest of the optimization session.

           YOU SHOULD NOT USE IT IN THE PRODUCTION CODE UNLESS YOU WANT TO
           CHECK DERIVATIVES PROVIDED BY SOME THIRD PARTY.

NOTE: unlike previous incarnation of the gradient checking code,  OptGuard
      does NOT interrupt optimization even if it discovers bad gradient.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step used for numerical differentiation:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification
                    You should carefully choose TestStep. Value  which  is
                    too large (so large that  function  behavior  is  non-
                    cubic at this scale) will lead  to  false  alarms. Too
                    short step will result in rounding  errors  dominating
                    numerical derivative.

                    You may use different step for different parameters by
                    means of setting scale with minlmsetscale().

=== EXPLANATION ==========================================================

In order to verify gradient algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void minlmoptguardgradient(minlmstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlmoptguardresults` function

/*************************************************************************
Results of OptGuard integrity check, should be called  after  optimization
session is over.

OptGuard checks analytic Jacobian  against  reference  value  obtained  by
numerical differentiation with user-specified step.

NOTE: other optimizers perform additional OptGuard checks for things  like
      C0/C1-continuity violations. However, LM optimizer  can  check  only
      for incorrect Jacobian.

      The reason is that unlike line search methods LM optimizer does  not
      perform extensive evaluations along the line. Thus, we simply do not
      have enough data to catch C0/C1-violations.

This check is activated with  minlmoptguardgradient() function.

Following flags are set when these errors are suspected:
* rep.badgradsuspected, and additionally:
  * rep.badgradfidx for specific function (Jacobian row) suspected
  * rep.badgradvidx for specific variable (Jacobian column) suspected
  * rep.badgradxbase, a point where gradient/Jacobian is tested
  * rep.badgraduser, user-provided gradient/Jacobian
  * rep.badgradnum, reference gradient/Jacobian obtained via numerical
    differentiation

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    rep     -   OptGuard report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minlmoptguardresults(minlmstate &state, optguardreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlmoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


CALLBACK PARALLELISM

The MINLM optimizer supports parallel parallel  numerical  differentiation
('callback parallelism'). This feature, which  is  present  in  commercial
ALGLIB  editions,  greatly   accelerates   optimization   with   numerical
differentiation of an expensive target functions.

Callback parallelism is usually  beneficial  when  computing  a  numerical
gradient requires more than several  milliseconds.  In this case  the  job
of computing individual gradient components can be split between  multiple
threads. Even inexpensive targets can benefit  from  parallelism,  if  you
have many variables.

If you solve a curve fitting problem, i.e. the function vector is actually
the same function computed at different points of a data points space,  it
may  be  better  to  use  an LSFIT curve fitting solver, which offers more
fine-grained parallelism due to knowledge of  the  problem  structure.  In
particular, it can accelerate both numerical differentiation  and problems
with user-supplied gradients.

ALGLIB Reference Manual, 'Working with commercial  version' section, tells
how to activate callback parallelism for your programming language.

  -- ALGLIB --
     Copyright 03.12.2023 by Bochkanov Sergey
*************************************************************************/
void minlmoptimize(minlmstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minlmoptimize(minlmstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minlmrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void minlmrequesttermination(minlmstate &state, const xparams _xparams = alglib::xdefault);

`minlmrestartfrom` function

/*************************************************************************
This  subroutine  restarts  LM  algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used for reverse communication previously
                allocated with MinLMCreateXXX call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void minlmrestartfrom(minlmstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlmresults` function

/*************************************************************************
Levenberg-Marquardt algorithm results

NOTE: if you activated OptGuard integrity checking functionality and  want
      to get OptGuard report,  it  can  be  retrieved  with  the  help  of
      minlmoptguardresults() function.

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization  report;  includes  termination   codes   and
                additional information. Termination codes are listed below,
                see comments for this structure for more info.

                Termination code is stored in rep.terminationtype field:
                * -8    optimizer detected NAN/INF values either in the
                        function itself, or in its Jacobian
                * -3    constraints are inconsistent
                *  2    relative step is no more than EpsX.
                *  5    MaxIts steps was taken
                *  7    stopping conditions are too stringent,
                        further improvement is impossible
                *  8    terminated by user who called minlmrequesttermination().
                        X contains point which was "current accepted" when
                        termination request was submitted.

                rep.f contains SUM(f[i]^2) at X

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlmresults(const minlmstate &state, real_1d_array &x, minlmreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minlmresultsbuf` function

/*************************************************************************
Levenberg-Marquardt algorithm results

Buffered implementation of MinLMResults(), which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void minlmresultsbuf(const minlmstate &state, real_1d_array &x, minlmreport &rep, const xparams _xparams = alglib::xdefault);

`minlmsetacctype` function

/*************************************************************************
This function is used to change acceleration settings

You can choose between three acceleration strategies:
* AccType=0, no acceleration.
* AccType=1, secant updates are used to update quadratic model after  each
  iteration. After fixed number of iterations (or after  model  breakdown)
  we  recalculate  quadratic  model  using  analytic  Jacobian  or  finite
  differences. Number of secant-based iterations depends  on  optimization
  settings: about 3 iterations - when we have analytic Jacobian, up to 2*N
  iterations - when we use finite differences to calculate Jacobian.

AccType=1 is recommended when Jacobian  calculation  cost is prohibitively
high (several Mx1 function vector calculations  followed  by  several  NxN
Cholesky factorizations are faster than calculation of one M*N  Jacobian).
It should also be used when we have no Jacobian, because finite difference
approximation takes too much time to compute.

Table below list  optimization  protocols  (XYZ  protocol  corresponds  to
MinLMCreateXYZ) and acceleration types they support (and use by  default).

ACCELERATION TYPES SUPPORTED BY OPTIMIZATION PROTOCOLS:

protocol    0   1   comment
V           +   +
VJ          +   +
FGH         +

DEFAULT VALUES:

protocol    0   1   comment
V               x   without acceleration it is so slooooooooow
VJ          x
FGH         x

NOTE: this  function should be called before optimization. Attempt to call
it during algorithm iterations may result in unexpected behavior.

NOTE: attempt to call this function with unsupported protocol/acceleration
combination will result in exception being thrown.

  -- ALGLIB --
     Copyright 14.10.2010 by Bochkanov Sergey
*************************************************************************/
void minlmsetacctype(minlmstate &state, const ae_int_t acctype, const xparams _xparams = alglib::xdefault);

`minlmsetbc` function

/*************************************************************************
This function sets boundary constraints for LM optimizer

Boundary constraints are inactive by default (after initial creation).
They are preserved until explicitly turned off with another SetBC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (latter is recommended because
                it will allow solver to use better algorithm).
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (latter is recommended because
                it will allow solver to use better algorithm).

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: this solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by bound constraints
  or at its boundary

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minlmsetbc(minlmstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`minlmsetcond` function

/*************************************************************************
This function sets stopping conditions for Levenberg-Marquardt optimization
algorithm.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - ste pvector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinLMSetScale()
                Recommended values: 1E-9 ... 1E-12.
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations   is    unlimited.   Only   Levenberg-Marquardt
                iterations  are  counted  (L-BFGS/CG  iterations  are  NOT
                counted because their cost is very low compared to that of
                LM).

Passing  EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to automatic
stopping criterion selection (small EpsX).

NOTE: it is not recommended to set large EpsX (say, 0.001). Because LM  is
      a second-order method, it performs very precise steps anyway.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlmsetcond(minlmstate &state, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minlmsetlc` function

/*************************************************************************
This function sets general linear constraints for LM optimizer

Linear constraints are inactive by default (after initial creation).  They
are preserved until explicitly turned off with another minlmsetlc() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

IMPORTANT: if you have linear constraints, it is strongly  recommended  to
           set scale of variables with minlmsetscale(). QP solver which is
           used to calculate linearly constrained steps heavily relies  on
           good scaling of input problems.

IMPORTANT: solvers created with minlmcreatefgh()  do  not  support  linear
           constraints.

NOTE: linear  (non-bound)  constraints are satisfied only approximately  -
      there  always  exists some violation due  to  numerical  errors  and
      algorithmic limitations.

NOTE: general linear constraints  add  significant  overhead  to  solution
      process. Although solver performs roughly same amount of  iterations
      (when compared  with  similar  box-only  constrained  problem), each
      iteration   now    involves  solution  of  linearly  constrained  QP
      subproblem, which requires ~3-5 times more Cholesky  decompositions.
      Thus, if you can reformulate your problem in such way  this  it  has
      only box constraints, it may be beneficial to do so.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minlmsetlc(minlmstate &state, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minlmsetlc(minlmstate &state, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

`minlmsetnonmonotonicsteps` function

/*************************************************************************
This function is used  to  activate/deactivate  nonmonotonic  steps.  Such
steps may improve  convergence  on  noisy  problems  or  ones  with  minor
smoothness defects.

In its standard mode, LM solver compares value at the trial   point   f[1]
with the value at the current point f[0]. Only steps that decrease f() are
accepted.

When the nonmonotonic mode is activated, f[1]  is  compared  with  maximum
over several previous  locations:  max(f[0],f[-1],...,f[-CNT]).  We  still
accept only steps that decrease  f(),  however  our  reference  value  has
changed. The net results is that f[1]>f[0] are now allowed.

Nonmonotonic steps can help to handle minor defects in the objective (e.g.
small  noise,  discontinuous  jumps  or  nonsmoothness).  However,  it  is
important  that  the  overall  shape  of  the  problem  is  still  smooth.
It  may  also  help  to  minimize  perfectly  smooth  targets with complex
geometries by allowing to jump through curved valleys.

However, sometimes nonmonotonic steps degrade convergence by  allowing  an
optimizer to wander too far away from the solution, so this feature should
be used only after careful testing.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    Cnt     -   nonmonotonic memory length, Cnt>=0:
                * 0 for traditional monotonic steps
                * 2..3 is recommended for the nonmonotonic optimization

  -- ALGLIB --
     Copyright 07.04.2024 by Bochkanov Sergey
*************************************************************************/
void minlmsetnonmonotonicsteps(minlmstate &state, const ae_int_t cnt, const xparams _xparams = alglib::xdefault);

`minlmsetnumdiff` function

/*************************************************************************
This function sets specific finite  difference  formula  to  be  used  for
numerical differentiation.

It works only for optimizers created  with  minlmcreatev()  function;   in
other cases it has no effect.

INPUT PARAMETERS:
    State       -   structure previously allocated with MinLMCreateV() call.
    FormulaType -   formula type:
                    * 3 for a 3-point formula, which is also  known  as  a
                      symmetric difference quotient (the formula  actually
                      uses only two function values per variable:  at  x+h
                      and x-h). A good choice for medium-accuracy  setups,
                      a default option.
                    * 2 for a forward (or backward, depending  on variable
                      bounds)  finite   difference  (f(x+h)-f(x))/h.  This
                      formula has the lowest accuracy. However, it  is  4x
                      faster than the 5-point formula and 2x  faster  than
                      the 3-point one because, in addition to the  central
                      value f(x), it needs only  one  additional  function
                      evaluation per variable.


  -- ALGLIB --
     Copyright 03.12.2024 by Bochkanov Sergey
*************************************************************************/
void minlmsetnumdiff(minlmstate &state, const ae_int_t formulatype, const xparams _xparams = alglib::xdefault);

`minlmsetscale` function

/*************************************************************************
This function sets scaling coefficients for LM optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Generally, scale is NOT considered to be a form of preconditioner.  But LM
optimizer is unique in that it uses scaling matrix both  in  the  stopping
condition tests and as Marquardt damping factor.

Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it  is
easier  to  converge  when  variables  are  properly  scaled, so premature
stopping is possible when very badly scalled variables are  combined  with
relaxed stopping conditions).

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minlmsetscale(minlmstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`minlmsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

NOTE: non-zero StpMax leads to moderate  performance  degradation  because
intermediate  step  of  preconditioned L-BFGS optimization is incompatible
with limits on step size.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlmsetstpmax(minlmstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`minlmsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinLMOptimize(). Both Levenberg-Marquardt and internal  L-BFGS
iterations are reported.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minlmsetxrep(minlmstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minlm_d_restarts example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
void  function2_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = x0^2+1
    // f1(x0,x1) = x1-1
    //
    fi[0] = x[0]*x[0]+1;
    fi[1] = x[1]-1;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // using several starting points and efficient restarts.
        //
        real_1d_array x;
        double epsx = 0.0000000001;
        ae_int_t maxits = 0;
        minlmstate state;
        minlmreport rep;

        //
        // create optimizer using minlmcreatev()
        //
        x = "[10,10]";
        minlmcreatev(2, x, 0.0001, state);
        minlmsetcond(state, epsx, maxits);
        alglib::minlmoptimize(state, function1_fvec);
        minlmresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]

        //
        // restart optimizer using minlmrestartfrom()
        //
        // we can use different starting point, different function,
        // different stopping conditions, but the problem size
        // must remain unchanged.
        //
        x = "[4,4]";
        minlmrestartfrom(state, x);
        alglib::minlmoptimize(state, function2_fvec);
        minlmresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [0,1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minlm_d_v example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // using "V" mode of the Levenberg-Marquardt optimizer (function values only,
        // no Jacobian information). The optimization algorithm uses function vector
        //
        //     f[] = {f1,f2}
        //
        // No other information (Jacobian, gradient, etc.) is needed.
        //
        // IMPORTANT: the  MINLM  optimizer  supports  parallel parallel numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            If  you  solve  a  curve fitting problem, i.e. the  function
        //            vector is actually the same function computed  at  different
        //            points of a data points space, then it may be better to  use
        //            an LSFIT curve fitting solver, which offers more fine-grained
        //            parallelism due to knowledge of the  problem  structure.  In
        //            particular, it can accelerate both numerical differentiation
        //            and problems with user-supplied gradients.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minlmoptimize()  function  for
        //            more information.
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsx = 0.0000000001;
        ae_int_t maxits = 0;
        minlmstate state;
        minlmreport rep;

        //
        // Create optimizer, tell it to:
        // * use numerical differentiation with step equal to 0.0001
        // * use unit scale for all variables (s is a unit vector)
        // * stop after short enough step (less than epsx)
        //
        minlmcreatev(2, x, 0.0001, state);
        minlmsetcond(state, epsx, maxits);
        minlmsetscale(state, s);

        //
        // A new feature: nonmonotonic steps!
        //
        // In theory, LM solver should be used only for smooth and continuous objectives. In practice,
        // real life objectives are often results of some long numerical simulation and have defects
        // like small discontinuous jumps, small noise or minor nonsmoothness. Such defects often
        // prevent optimization progress because an uphill step may be required to move past the
        // defect.
        //
        // Nonmonotonic steps allow to tolerate a minor and temporary increase in the objective,
        // allowing progress beyond an obstacle. This feature is also essential for ill-conditioned
        // targets - it allows the solver to jump through a curved valley instead of navigating along
        // its bottom.
        //
        // However, sometimes nonmonotonic steps degrade convergence by  allowing  an optimizer to
        // wander too far away from the solution, so this feature should be used only after careful
        // testing.
        //
        // The code below sets the nonmonotonic memory length to 0 (which means traditional monotonic
        // optimization). If you want to try a nonmonotonic optimization, use 2 or 3 as a recommended
        // memory length.
        //
        minlmsetnonmonotonicsteps(state, 0);

        //
        // Optimize
        //
        alglib::minlmoptimize(state, function1_fvec);

        //
        // Test optimization results
        //
        // NOTE: because we use numerical differentiation, we do not
        //       verify Jacobian correctness - it is always "correct".
        //       However, if you switch to analytic gradient, consider
        //       checking it with OptGuard (see other examples).
        //
        minlmresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minlm_d_vb example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // with box constraints
        //
        //     -1 <= x0 <= +1
        //     -1 <= x1 <= +1
        //
        // using "V" mode of the Levenberg-Marquardt optimizer.  The  optimization
        // algorithm uses function  vector  f[] = {f1,f2}.  No  other  information
        // (Jacobian, gradient, etc.) is needed.
        //
        // IMPORTANT: the  MINLM  optimizer  supports  parallel parallel numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            If  you  solve  a  curve fitting problem, i.e. the  function
        //            vector is actually the same function computed  at  different
        //            points of a data points space, then it may be better to  use
        //            an LSFIT curve fitting solver, which offers more fine-grained
        //            parallelism due to knowledge of the  problem  structure.  In
        //            particular, it can accelerate both numerical differentiation
        //            and problems with user-supplied gradients.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minlmoptimize()  function  for
        //            more information.
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        double epsx = 0.0000000001;
        ae_int_t maxits = 0;
        minlmstate state;

        //
        // Create optimizer, tell it to:
        // * use numerical differentiation with step equal to 1.0
        // * use unit scale for all variables (s is a unit vector)
        // * stop after short enough step (less than epsx)
        // * set box constraints
        //
        minlmcreatev(2, x, 0.0001, state);
        minlmsetbc(state, bndl, bndu);
        minlmsetcond(state, epsx, maxits);
        minlmsetscale(state, s);

        //
        // A new feature: nonmonotonic steps!
        //
        // In theory, LM solver should be used only for smooth and continuous objectives. In practice,
        // real life objectives are often results of some long numerical simulation and have defects
        // like small discontinuous jumps, small noise or minor nonsmoothness. Such defects often
        // prevent optimization progress because an uphill step may be required to move past the
        // defect.
        //
        // Nonmonotonic steps allow to tolerate a minor and temporary increase in the objective,
        // allowing progress beyond an obstacle. This feature is also essential for ill-conditioned
        // targets - it allows the solver to jump through a curved valley instead of navigating along
        // its bottom.
        //
        // However, sometimes nonmonotonic steps degrade convergence by  allowing  an optimizer to
        // wander too far away from the solution, so this feature should be used only after careful
        // testing.
        //
        // The code below sets the nonmonotonic memory length to 0 (which means traditional monotonic
        // optimization). If you want to try a nonmonotonic optimization, use 2 or 3 as a recommended
        // memory length.
        //
        minlmsetnonmonotonicsteps(state, 0);

        //
        // Optimize
        //
        alglib::minlmoptimize(state, function1_fvec);

        //
        // Test optimization results
        //
        // NOTE: because we use numerical differentiation, we do not
        //       verify Jacobian correctness - it is always "correct".
        //       However, if you switch to analytic gradient, consider
        //       checking it with OptGuard (see other examples).
        //
        minlmreport rep;
        minlmresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,+1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minlm_d_vj example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
void  function1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
    jac[0][0] = 20*(x[0]+3);
    jac[0][1] = 0;
    jac[1][0] = 0;
    jac[1][1] = 2*(x[1]-3);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // using "VJ" mode of the Levenberg-Marquardt optimizer.  The optimization
        // algorithm uses the  function  vector  f[] = {f1,f2}  and  the  Jacobian
        // matrix J = {dfi/dxj}, both of them provided by user.
        //
        // IMPORTANT: the   MINLM   optimizer  supports     parallel     numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            If  you  solve  a  curve fitting problem, i.e. the  function
        //            vector is actually the same function computed  at  different
        //            points of a data points space, then it may be better to  use
        //            an LSFIT curve fitting solver, which offers more fine-grained
        //            parallelism due to knowledge of the  problem  structure.  In
        //            particular, it can accelerate both numerical differentiation
        //            and problems with user-supplied gradients.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minlmoptimize()  function  for
        //            more information.
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsx = 0.0000000001;
        ae_int_t maxits = 0;
        minlmstate state;

        //
        // Create optimizer, tell it to:
        // * use analytic gradient provided by user
        // * use unit scale for all variables (s is a unit vector)
        // * stop after short enough step (less than epsx)
        //
        minlmcreatevj(2, x, state);
        minlmsetcond(state, epsx, maxits);
        minlmsetscale(state, s);

        //
        // A new feature: nonmonotonic steps!
        //
        // In theory, LM solver should be used only for smooth and continuous objectives. In practice,
        // real life objectives are often results of some long numerical simulation and have defects
        // like small discontinuous jumps, small noise or minor nonsmoothness. Such defects often
        // prevent optimization progress because an uphill step may be required to move past the
        // defect.
        //
        // Nonmonotonic steps allow to tolerate a minor and temporary increase in the objective,
        // allowing progress beyond an obstacle. This feature is also essential for ill-conditioned
        // targets - it allows the solver to jump through a curved valley instead of navigating along
        // its bottom.
        //
        // However, sometimes nonmonotonic steps degrade convergence by  allowing  an optimizer to
        // wander too far away from the solution, so this feature should be used only after careful
        // testing.
        //
        // The code below sets the nonmonotonic memory length to 0 (which means traditional monotonic
        // optimization). If you want to try a nonmonotonic optimization, use 2 or 3 as a recommended
        // memory length.
        //
        minlmsetnonmonotonicsteps(state, 0);

        //
        // Optimize
        //
        alglib::minlmoptimize(state, function1_fvec, function1_jac);

        //
        // Test optimization results
        //
        minlmreport rep;
        minlmresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minlp` subpackage

Classes

minlpreport
minlpstate

Functions

minlpaddlc2
minlpaddlc2dense
minlpcreate
minlpoptimize
minlpresults
minlpresultsbuf
minlpsetalgodss
minlpsetalgoipm
minlpsetbc
minlpsetbcall
minlpsetbci
minlpsetcost
minlpsetlc
minlpsetlc2
minlpsetlc2dense
minlpsetscale

Examples

minlp_basic

Basic linear programming example

`minlpreport` class

/*************************************************************************
This structure stores optimization report:
* f                         target function value
* lagbc                     Lagrange coefficients for box constraints
* laglc                     Lagrange coefficients for linear constraints
* y                         dual variables
* stats                     array[N+M], statuses of box (N) and linear (M)
                            constraints. This array is filled only by  DSS
                            algorithm because IPM always stops at INTERIOR
                            point:
                            * stats[i]>0  =>  constraint at upper bound
                                              (also used for free non-basic
                                              variables set to zero)
                            * stats[i]<0  =>  constraint at lower bound
                            * stats[i]=0  =>  constraint is inactive, basic
                                              variable
* primalerror               primal feasibility error
* dualerror                 dual feasibility error
* slackerror                complementary slackness error
* iterationscount           iteration count
* terminationtype           completion code (see below)

COMPLETION CODES

Completion codes:
* -4    LP problem is primal unbounded (dual infeasible)
* -3    LP problem is primal infeasible (dual unbounded)
*  1..4 successful completion
*  5    MaxIts steps was taken
*  7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.

LAGRANGE COEFFICIENTS

Positive Lagrange coefficient means that constraint is at its upper bound.
Negative coefficient means that constraint is at its lower  bound.  It  is
expected that at the solution the dual feasibility condition holds:

    C + SUM(Ei*LagBC[i],i=0..n-1) + SUM(Ai*LagLC[i],i=0..m-1) ~ 0

where
* C is a cost vector (linear term)
* Ei is a vector with 1.0 at position I and 0 in other positions
* Ai is an I-th row of linear constraint matrix
*************************************************************************/
class minlpreport
{
public:
    minlpreport();
    minlpreport(const minlpreport &rhs);
    minlpreport& operator=(const minlpreport &rhs);
    virtual ~minlpreport();
    double f;
    real_1d_array lagbc;
    real_1d_array laglc;
    real_1d_array y;
    integer_1d_array stats;
    double primalerror;
    double dualerror;
    double slackerror;
    ae_int_t iterationscount;
    ae_int_t terminationtype;
};

`minlpstate` class

/*************************************************************************
This object stores linear solver state.
You should use functions provided by MinLP subpackage to work with this
object
*************************************************************************/
class minlpstate
{
public:
    minlpstate();
    minlpstate(const minlpstate &rhs);
    minlpstate& operator=(const minlpstate &rhs);
    virtual ~minlpstate();
};

`minlpaddlc2` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present constraints.

Constraint is passed in compressed format: as list of non-zero entries  of
coefficient vector A. Such approach is more efficient than  dense  storage
for highly sparse constraint vectors.

INPUT PARAMETERS:
    State   -   structure previously allocated with minlpcreate() call.
    IdxA    -   array[NNZ], indexes of non-zero elements of A:
                * can be unsorted
                * can include duplicate indexes (corresponding entries  of
                  ValA[] will be summed)
    ValA    -   array[NNZ], values of non-zero elements of A
    NNZ     -   number of non-zero coefficients in A
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpaddlc2(minlpstate &state, const integer_1d_array &idxa, const real_1d_array &vala, const ae_int_t nnz, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minlpaddlc2dense` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present constraints.

This version accepts dense constraint vector as input, but  sparsifies  it
for internal storage and processing. Thus, time to add one  constraint  in
is O(N) - we have to scan entire array of length N. Sparse version of this
function is order of magnitude faster for  constraints  with  just  a  few
nonzeros per row.

INPUT PARAMETERS:
    State   -   structure previously allocated with minlpcreate() call.
    A       -   linear constraint coefficient, array[N], right side is NOT
                included.
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint Ai*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint Ai*x<=AU
                * AU=+INF  => one-sided constraint AL<=Ai*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpaddlc2dense(minlpstate &state, const real_1d_array &a, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minlpcreate` function

/*************************************************************************
                            LINEAR PROGRAMMING

The subroutine creates LP  solver.  After  initial  creation  it  contains
default optimization problem with zero cost vector and all variables being
fixed to zero values and no constraints.

In order to actually solve something you should:
* set cost vector with minlpsetcost()
* set variable bounds with minlpsetbc() or minlpsetbcall()
* specify constraint matrix with one of the following functions:
  [*] minlpsetlc()        for dense one-sided constraints
  [*] minlpsetlc2dense()  for dense two-sided constraints
  [*] minlpsetlc2()       for sparse two-sided constraints
  [*] minlpaddlc2dense()  to add one dense row to constraint matrix
  [*] minlpaddlc2()       to add one row to constraint matrix (compressed format)
* call minlpoptimize() to run the solver and  minlpresults()  to  get  the
  solution vector and additional information.

By  default,  LP  solver uses best algorithm available. As of ALGLIB 3.17,
sparse interior point (barrier) solver is used. Future releases of  ALGLIB
may introduce other solvers.

User may choose specific LP algorithm by calling:
* minlpsetalgodss() for revised dual simplex method with DSE  pricing  and
  bounds flipping ratio test (aka long dual step).  Large-scale  sparse LU
  solverwith  Forest-Tomlin update is used internally  as  linear  algebra
  driver.
* minlpsetalgoipm() for sparse interior point method

INPUT PARAMETERS:
    N       -   problem size

OUTPUT PARAMETERS:
    State   -   optimizer in the default state

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpcreate(const ae_int_t n, minlpstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpoptimize` function

/*************************************************************************
This function solves LP problem.

INPUT PARAMETERS:
    State   -   algorithm state

You should use minlpresults() function to access results  after  calls  to
this function.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey.
*************************************************************************/
void minlpoptimize(minlpstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpresults` function

/*************************************************************************
LP solver results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution (on failure: last trial point)
    Rep     -   optimization report. You should check Rep.TerminationType,
                which contains completion code, and you may check  another
                fields which contain another information  about  algorithm
                functioning.

                Failure codes returned by algorithm are:
                * -4    LP problem is primal unbounded (dual infeasible)
                * -3    LP problem is primal infeasible (dual unbounded)
                * -2    IPM solver detected that problem is either
                        infeasible or unbounded

                Success codes:
                *  1..4 successful completion
                *  5    MaxIts steps was taken

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minlpresults(const minlpstate &state, real_1d_array &x, minlpreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpresultsbuf` function

/*************************************************************************
LP results

Buffered implementation of MinLPResults() which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minlpresultsbuf(const minlpstate &state, real_1d_array &x, minlpreport &rep, const xparams _xparams = alglib::xdefault);

`minlpsetalgodss` function

/*************************************************************************
This function sets LP algorithm to revised dual simplex method.

ALGLIB implementation of dual simplex method supports advanced performance
and stability improvements like DSE pricing , bounds flipping  ratio  test
(aka long dual step), Forest-Tomlin update, shifting.

INPUT PARAMETERS:
    State   -   optimizer
    Eps     -   stopping condition, Eps>=0:
                * should be small number about 1E-6 or 1E-7.
                * zero value means that solver automatically selects good
                  value (can be different in different ALGLIB versions)
                * default value is zero
                Algorithm stops when relative error is less than Eps.

===== TRACING DSS SOLVER =================================================

DSS solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'DSS'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'DSS.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'DSS'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("DSS,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 08.11.2020 by Bochkanov Sergey
*************************************************************************/
void minlpsetalgodss(minlpstate &state, const double eps, const xparams _xparams = alglib::xdefault);

`minlpsetalgoipm` function

/*************************************************************************
This function sets LP algorithm to sparse interior point method.

ALGORITHM INFORMATION:

* this  algorithm  is  our implementation  of  interior  point  method  as
  formulated by  R.J.Vanderbei, with minor modifications to the  algorithm
  (damped Newton directions are extensively used)
* like all interior point methods, this algorithm  tends  to  converge  in
  roughly same number of iterations (between 15 and 50) independently from
  the problem dimensionality

INPUT PARAMETERS:
    State   -   optimizer
    Eps     -   stopping condition, Eps>=0:
                * should be small number about 1E-6 or 1E-8.
                * zero value means that solver automatically selects good
                  value (can be different in different ALGLIB versions)
                * default value is zero
                Algorithm  stops  when  primal  error  AND  dual error AND
                duality gap are less than Eps.

===== TRACING IPM SOLVER =================================================

IPM solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'IPM'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'IPM.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'IPM'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("IPM,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 08.11.2020 by Bochkanov Sergey
*************************************************************************/
void minlpsetalgoipm(minlpstate &state, const double eps, const xparams _xparams = alglib::xdefault);
void minlpsetalgoipm(minlpstate &state, const xparams _xparams = alglib::xdefault);

`minlpsetbc` function

/*************************************************************************
This function sets box constraints for LP solver (all variables  at  once,
different constraints for different variables).

The default state of constraints is to have all variables fixed  at  zero.
You have to overwrite it by your own constraint vector. Constraint  status
is preserved until constraints are  explicitly  overwritten  with  another
minlpsetbc()  call,   overwritten   with  minlpsetbcall(),  or   partially
overwritten with minlmsetbci() call.

Following types of constraints are supported:

    DESCRIPTION         CONSTRAINT              HOW TO SPECIFY
    fixed variable      x[i]=Bnd[i]             BndL[i]=BndU[i]
    lower bound         BndL[i]<=x[i]           BndU[i]=+INF
    upper bound         x[i]<=BndU[i]           BndL[i]=-INF
    range               BndL[i]<=x[i]<=BndU[i]  ...
    free variable       -                       BndL[I]=-INF, BndU[I]+INF

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
    BndU    -   upper bounds, array[N].

NOTE: infinite values can be specified by means of Double.PositiveInfinity
      and  Double.NegativeInfinity  (in  C#)  and  alglib::fp_posinf   and
      alglib::fp_neginf (in C++).

NOTE: you may replace infinities by very small/very large values,  but  it
      is not recommended because large numbers may introduce large numerical
      errors in the algorithm.

NOTE: if constraints for all variables are same you may use minlpsetbcall()
      which allows to specify constraints without using arrays.

NOTE: BndL>BndU will result in LP problem being recognized as infeasible.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetbc(minlpstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpsetbcall` function

/*************************************************************************
This function sets box constraints for LP solver (all variables  at  once,
same constraints for all variables)

The default state of constraints is to have all variables fixed  at  zero.
You have to overwrite it by your own constraint vector. Constraint  status
is preserved until constraints are  explicitly  overwritten  with  another
minlpsetbc() call or partially overwritten with minlpsetbcall().

Following types of constraints are supported:

    DESCRIPTION         CONSTRAINT              HOW TO SPECIFY
    fixed variable      x[i]=Bnd[i]             BndL[i]=BndU[i]
    lower bound         BndL[i]<=x[i]           BndU[i]=+INF
    upper bound         x[i]<=BndU[i]           BndL[i]=-INF
    range               BndL[i]<=x[i]<=BndU[i]  ...
    free variable       -                       BndL[I]=-INF, BndU[I]+INF

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bound, same for all variables
    BndU    -   upper bound, same for all variables

NOTE: infinite values can be specified by means of Double.PositiveInfinity
      and  Double.NegativeInfinity  (in  C#)  and  alglib::fp_posinf   and
      alglib::fp_neginf (in C++).

NOTE: you may replace infinities by very small/very large values,  but  it
      is not recommended because large numbers may introduce large numerical
      errors in the algorithm.

NOTE: minlpsetbc() can  be  used  to  specify  different  constraints  for
      different variables.

NOTE: BndL>BndU will result in LP problem being recognized as infeasible.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetbcall(minlpstate &state, const double bndl, const double bndu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpsetbci` function

/*************************************************************************
This function sets box constraints for I-th variable (other variables are
not modified).

The default state of constraints is to have all variables fixed  at  zero.
You have to overwrite it by your own constraint vector.

Following types of constraints are supported:

    DESCRIPTION         CONSTRAINT              HOW TO SPECIFY
    fixed variable      x[i]=Bnd[i]             BndL[i]=BndU[i]
    lower bound         BndL[i]<=x[i]           BndU[i]=+INF
    upper bound         x[i]<=BndU[i]           BndL[i]=-INF
    range               BndL[i]<=x[i]<=BndU[i]  ...
    free variable       -                       BndL[I]=-INF, BndU[I]+INF

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    I       -   variable index, in [0,N)
    BndL    -   lower bound for I-th variable
    BndU    -   upper bound for I-th variable

NOTE: infinite values can be specified by means of Double.PositiveInfinity
      and  Double.NegativeInfinity  (in  C#)  and  alglib::fp_posinf   and
      alglib::fp_neginf (in C++).

NOTE: you may replace infinities by very small/very large values,  but  it
      is not recommended because large numbers may introduce large numerical
      errors in the algorithm.

NOTE: minlpsetbc() can  be  used  to  specify  different  constraints  for
      different variables.

NOTE: BndL>BndU will result in LP problem being recognized as infeasible.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetbci(minlpstate &state, const ae_int_t i, const double bndl, const double bndu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpsetcost` function

/*************************************************************************
This function sets cost term for LP solver.

By default, cost term is zero.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    C       -   cost term, array[N].

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetcost(minlpstate &state, const real_1d_array &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpsetlc` function

/*************************************************************************
This function sets one-sided linear constraints A*x ~ AU, where "~" can be
a mix of "<=", "=" and ">=".

IMPORTANT: this function is provided here for compatibility with the  rest
           of ALGLIB optimizers which accept constraints  in  format  like
           this one. Many real-life problems feature two-sided constraints
           like a0 <= a*x <= a1. It is really inefficient to add them as a
           pair of one-sided constraints.

           Use minlpsetlc2dense(), minlpsetlc2(), minlpaddlc2()  (or   its
           sparse version) wherever possible.

INPUT PARAMETERS:
    State   -   structure previously allocated with minlpcreate() call.
    A       -   linear constraints, array[K,N+1]. Each row of A represents
                one constraint, with first N elements being linear coefficients,
                and last element being right side.
    CT      -   constraint types, array[K]:
                * if CT[i]>0, then I-th constraint is A[i,*]*x >= A[i,n]
                * if CT[i]=0, then I-th constraint is A[i,*]*x  = A[i,n]
                * if CT[i]<0, then I-th constraint is A[i,*]*x <= A[i,n]
    K       -   number of equality/inequality constraints,  K>=0;  if  not
                given, inferred from sizes of A and CT.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetlc(minlpstate &state, const real_2d_array &a, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minlpsetlc(minlpstate &state, const real_2d_array &a, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpsetlc2` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
sparse constraining matrix A. Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

INPUT PARAMETERS:
    State   -   structure previously allocated with minlpcreate() call.
    A       -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetlc2(minlpstate &state, const sparsematrix &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`minlpsetlc2dense` function

/*************************************************************************
This function sets two-sided linear constraints AL <= A*x <= AU.

This version accepts dense matrix as  input;  internally  LP  solver  uses
sparse storage  anyway  (most  LP  problems  are  sparse),  but  for  your
convenience it may accept dense inputs. This  function  overwrites  linear
constraints set by previous calls (if such calls were made).

We recommend you to use sparse version of this function unless  you  solve
small-scale LP problem (less than few hundreds of variables).

NOTE: there also exist several versions of this function:
      * one-sided dense version which  accepts  constraints  in  the  same
        format as one used by QP and  NLP solvers
      * two-sided sparse version which accepts sparse matrix
      * two-sided dense  version which allows you to add constraints row by row
      * two-sided sparse version which allows you to add constraints row by row

INPUT PARAMETERS:
    State   -   structure previously allocated with minlpcreate() call.
    A       -   linear constraints, array[K,N]. Each row of  A  represents
                one  constraint. One-sided  inequality   constraints, two-
                sided inequality  constraints,  equality  constraints  are
                supported (see below)
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number of equality/inequality constraints,  K>=0;  if  not
                given, inferred from sizes of A, AL, AU.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetlc2dense(minlpstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minlpsetlc2dense(minlpstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minlpsetscale` function

/*************************************************************************
This function sets scaling coefficients.

ALGLIB optimizers use scaling matrices to test stopping  conditions and as
preconditioner.

Scale of the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the
   function

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minlpsetscale(minlpstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

minlp_basic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates how to minimize
        //
        //     F(x0,x1) = -0.1*x0 - x1
        //
        // subject to box constraints
        //
        //     -1 <= x0,x1 <= +1 
        //
        // and general linear constraints
        //
        //     x0 - x1 >= -1
        //     x0 + x1 <=  1
        //
        // We use dual simplex solver provided by ALGLIB for this task. Box
        // constraints are specified by means of constraint vectors bndl and
        // bndu (we have bndl<=x<=bndu). General linear constraints are
        // specified as AL<=A*x<=AU, with AL/AU being 2x1 vectors and A being
        // 2x2 matrix.
        //
        // NOTE: some/all components of AL/AU can be +-INF, same applies to
        //       bndl/bndu. You can also have AL[I]=AU[i] (as well as
        //       BndL[i]=BndU[i]).
        //
        real_2d_array a = "[[1,-1],[1,+1]]";
        real_1d_array al = "[-1,-inf]";
        real_1d_array au = "[+inf,+1]";
        real_1d_array c = "[-0.1,-1]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        real_1d_array x;
        minlpstate state;
        minlpreport rep;

        minlpcreate(2, state);

        //
        // Set cost vector, box constraints, general linear constraints.
        //
        // Box constraints can be set in one call to minlpsetbc() or minlpsetbcall()
        // (latter sets same constraints for all variables and accepts two scalars
        // instead of two vectors).
        //
        // General linear constraints can be specified in several ways:
        // * minlpsetlc2dense() - accepts dense 2D array as input; sometimes this
        //   approach is more convenient, although less memory-efficient.
        // * minlpsetlc2() - accepts sparse matrix as input
        // * minlpaddlc2dense() - appends one row to the current set of constraints;
        //   row being appended is specified as dense vector
        // * minlpaddlc2() - appends one row to the current set of constraints;
        //   row being appended is specified as sparse set of elements
        // Independently from specific function being used, LP solver uses sparse
        // storage format for internal representation of constraints.
        //
        minlpsetcost(state, c);
        minlpsetbc(state, bndl, bndu);
        minlpsetlc2dense(state, a, al, au, 2);

        //
        // Set scale of the parameters.
        //
        // It is strongly recommended that you set scale of your variables.
        // Knowing their scales is essential for evaluation of stopping criteria
        // and for preconditioning of the algorithm steps.
        // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
        //
        minlpsetscale(state, s);

        //
        // Solve with the sparse IPM.
        //
        // Commercial ALGLIB can parallelize sparse Cholesky factorization which is the
        // most time-consuming part of the algorithm. See the ALGLIB Reference Manual for
        // more information on how to activate parallelism support.
        //
        minlpsetalgoipm(state);
        minlpoptimize(state);
        minlpresults(state, x, rep);
        printf("%s\n", x.tostring(3).c_str()); // EXPECTED: [0,1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minmo` subpackage

Classes

minmoreport
minmostate

Functions

minmoaddlc2
minmoaddlc2dense
minmoaddlc2sparsefromdense
minmocreate
minmocreatef
minmoiteration
minmooptimize
minmorequesttermination
minmorestartfrom
minmoresults
minmosetalgonbi
minmosetbc
minmosetcond
minmosetlc2
minmosetlc2dense
minmosetlc2mixed
minmosetnlc2
minmosetscale
minmosetxrep

Examples

minmo_biobjective		Unconstrained biobjective optimization
minmo_biobjective_constr		Nonlinearly constrained biobjective optimization

`minmoreport` class

/*************************************************************************
These fields store optimization report:
* inneriterationscount      total number of inner iterations
* outeriterationscount      number of internal optimization sessions performed
* nfev                      number of gradient evaluations
* terminationtype           termination type (see below)

Scaled constraint violations (maximum over all Pareto points) are reported:
* bcerr                     maximum violation of the box constraints
* bcidx                     index of the most violated box  constraint (or
                            -1, if all box constraints  are  satisfied  or
                            there are no box constraint)
* lcerr                     maximum violation of the  linear  constraints,
                            computed as maximum  scaled  distance  between
                            final point and constraint boundary.
* lcidx                     index of the most violated  linear  constraint
                            (or -1, if all constraints  are  satisfied  or
                            there are no general linear constraints)
* nlcerr                    maximum violation of the nonlinear constraints
* nlcidx                    index of the most violated nonlinear constraint
                            (or -1, if all constraints  are  satisfied  or
                            there are no nonlinear constraints)

Violations  of  the  box  constraints  are  scaled  on per-component basis
according to  the  scale  vector s[]  as specified by the minmosetscale().
Violations of the general linear  constraints  are  also   computed  using
user-supplied variable scaling. Violations of  the  nonlinear  constraints
are computed "as is"

TERMINATION CODES

TerminationType field contains completion code, which can be either:

=== FAILURE CODE ===
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signaled.
  -3    box  constraints  are  infeasible.  Note: infeasibility of non-box
        constraints does NOT trigger emergency  completion;  you  have  to
        examine  bcerr/lcerr/nlcerr   to  detect   possibly   inconsistent
        constraints.

=== SUCCESS CODE ===
   2    relative step is no more than EpsX.
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.

NOTE: The solver internally performs many optimization sessions:  one  for
      each Pareto point, and some  amount  of  preparatory  optimizations.
      Different optimization  sessions  may  return  different  completion
      codes. If at least one of internal optimizations failed, its failure
      code is returned. If none of them failed, the most frequent code  is
      returned.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minmoreport
{
public:
    minmoreport();
    minmoreport(const minmoreport &rhs);
    minmoreport& operator=(const minmoreport &rhs);
    virtual ~minmoreport();
    ae_int_t inneriterationscount;
    ae_int_t outeriterationscount;
    ae_int_t nfev;
    ae_int_t terminationtype;
    double bcerr;
    ae_int_t bcidx;
    double lcerr;
    ae_int_t lcidx;
    double nlcerr;
    ae_int_t nlcidx;
};

`minmostate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinMO subpackage to work  with  this
object
*************************************************************************/
class minmostate
{
public:
    minmostate();
    minmostate(const minmostate &rhs);
    minmostate& operator=(const minmostate &rhs);
    virtual ~minmostate();
};

`minmoaddlc2` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of sparse constraints.

Constraint is passed in the compressed  format:  as  a  list  of  non-zero
entries of the coefficient vector A. Such approach is more efficient  than
the dense storage for highly sparse constraint vectors.

INPUT PARAMETERS:
    State   -   structure previously allocated with minmocreate() call.
    IdxA    -   array[NNZ], indexes of non-zero elements of A:
                * can be unsorted
                * can include duplicate indexes (corresponding entries  of
                  ValA[] will be summed)
    ValA    -   array[NNZ], values of non-zero elements of A
    NNZ     -   number of non-zero coefficients in A
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minmoaddlc2(minmostate &state, const integer_1d_array &idxa, const real_1d_array &vala, const ae_int_t nnz, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minmoaddlc2dense` function

/*************************************************************************
This function appends two-sided linear  constraint  AL<=A*x<=AU  to  dense
constraints list.

INPUT PARAMETERS:
    State   -   structure previously allocated with minmocreate() call.
    A       -   linear constraint coefficient, array[N], right side is NOT
                included.
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint Ai*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint Ai*x<=AU
                * AU=+INF  => one-sided constraint AL<=Ai*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minmoaddlc2dense(minmostate &state, const real_1d_array &a, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minmoaddlc2sparsefromdense` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present sparse constraints.

Constraint vector A is  passed  as  a  dense  array  which  is  internally
sparsified by this function.

INPUT PARAMETERS:
    State   -   structure previously allocated with minmocreate() call.
    DA      -   array[N], constraint vector
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minmoaddlc2sparsefromdense(minmostate &state, const real_1d_array &da, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minmocreate` function

/*************************************************************************
                    MULTI-OBJECTIVE  OPTIMIZATION

DESCRIPTION:

The  solver  minimizes an M-dimensional vector function F(x) of N arguments
subject to any combination of:
* box constraints
* two-sided linear equality/inequality constraints AL<=A*x<=AU, where some
  of AL/AU can be infinite (i.e. missing)
* two-sided nonlinear equality/inequality constraints NL<=C(x)<=NU,  where
  some of NL/NU can be infinite (i.e. missing)

REQUIREMENTS:
* F(), C() are continuously differentiable on the feasible set and on  its
  neighborhood

USAGE:

1. User initializes algorithm state using either:
   * minmocreate()  to perform optimization with user-supplied Jacobian
   * minmocreatef() to perform optimization with numerical differentiation

2. User chooses which multi-objective solver to use. At the present moment
   only NBI (Normal Boundary Intersection) solver is implemented, which is
   activated by calling minmosetalgonbi().

3. User adds boundary and/or linear and/or nonlinear constraints by  means
   of calling one of the following functions:
   a) minmosetbc() for boundary constraints
   b) minmosetlc2()      for two-sided sparse linear constraints;
      minmosetlc2dense() for two-sided dense  linear constraints;
      minmosetlc2mixed() for two-sided mixed sparse/dense constraints
   c) minmosetnlc2()     for two-sided nonlinear constraints
   You may combine (a), (b) and (c) in one optimization problem.

4. User sets scale of the variables with minmosetscale() function.  It  is
   VERY important to set  scale  of  the  variables,  because  nonlinearly
   constrained problems are hard to solve when variables are badly scaled.

5. User sets  stopping  conditions  with  minmosetcond().

6. Finally, user calls minmooptimize()   function  which  takes  algorithm
   state  and  pointers  (delegate, etc.) to the callback functions  which
   calculate F/C

7. User calls  minmoresults()  to  get the solution

8. Optionally user may call minmorestartfrom() to solve  another   problem
   with same M,N but another starting point. minmorestartfrom() allows  to
   reuse an already initialized optimizer structure.


INPUT PARAMETERS:
    N       -   variables count, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from the size of X
    M       -   objectives count, M>0.
                M=1 is possible, although makes little sense - it is better
                to use MinNLC directly.
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to reinforce feasibility during  initial  stages  of  the
                  optimization

OUTPUT PARAMETERS:
    State   -   structure that stores algorithm state

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmocreate(const ae_int_t n, const ae_int_t m, const real_1d_array &x, minmostate &state, const xparams _xparams = alglib::xdefault);
void minmocreate(const ae_int_t m, const real_1d_array &x, minmostate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minmocreatef` function

/*************************************************************************
This subroutine is a finite  difference variant of minmocreate().  It uses
finite differences in order to differentiate target function.

Description below contains information which is specific to this  function
only. We recommend to read comments on minmocreate() too.

INPUT PARAMETERS:
    N       -   variables count, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from the size of X
    M       -   objectives count, M>0.
                M=1 is possible, although makes little sense - it is better
                to use MinNLC directly.
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to reinforce feasibility during  initial  stages  of  the
                  optimization
    DiffStep-   differentiation step, >0

OUTPUT PARAMETERS:
    State   -   structure that stores algorithm state

NOTES:
1. algorithm uses 4-point central formula for differentiation.
2. differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is a scaling vector which can be set by minmosetscale() call.
3. we recommend you to use moderate values of  differentiation  step.  Too
   large step means too large TRUNCATION errors,  whilst  too  small  step
   means too large NUMERICAL errors.
   1.0E-4 can be good value to start from for a unit-scaled problem.
4. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs 4*N function evaluations. This function will work for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.
   We should also say that code which relies on numerical  differentiation
   is  less   robust   and  precise.  Imprecise  gradient  may  slow  down
   convergence, especially on highly nonlinear problems.
   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmocreatef(const ae_int_t n, const ae_int_t m, const real_1d_array &x, const double diffstep, minmostate &state, const xparams _xparams = alglib::xdefault);
void minmocreatef(const ae_int_t m, const real_1d_array &x, const double diffstep, minmostate &state, const xparams _xparams = alglib::xdefault);

`minmoiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minmoiteration(minmostate &state, const xparams _xparams = alglib::xdefault);

`minmooptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied Jacobian, and one which uses  only  function
   vector and numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   you should choose appropriate variant of MinMOOptimize() -  one   which
   needs function vector AND Jacobian or one which needs ONLY function.

   Be careful to choose variant of MinMOOptimize()  which  corresponds  to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinMOOptimize()   and   specific
   function used to create optimizer.


                     |         USER PASSED TO MinMOOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinMOCreateF()    |     works               FAILS
   MinMOCreate()     |     FAILS               works

   Here "FAILS" denotes inappropriate combinations  of  optimizer creation
   function  and  MinMOOptimize()  version.   Attemps   to    use     such
   combination will lead to exception. Either  you  did  not pass gradient
   when it WAS needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmooptimize(minmostate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minmooptimize(minmostate &state,
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minmorequesttermination` function

/*************************************************************************
This subroutine  submits  request  for  the  termination  of  the  running
optimizer.

It should be called from the user-supplied callback when user decides that
it is time to "smoothly" terminate optimization process, or from some other
thread. As a result, optimizer stops  at  the  state  which  was  "current
accepted" when termination request was submitted and returns error code  8
(successful termination).

Usually it results in an incomplete Pareto front being returned.

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmorequesttermination(minmostate &state, const xparams _xparams = alglib::xdefault);

`minmorestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from the new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinMOCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmorestartfrom(minmostate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`minmoresults` function

/*************************************************************************
MinMO results:  the  solution  found,  completion  codes  and   additional
information.

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    ParetoFront-array[FrontSize,N+M], approximate Pareto front.
                Its columns have the following structure:
                * first N columns are variable values
                * next  M columns are objectives at these points
                Its rows have the following structure:
                * first M rows contain solutions to single-objective tasks
                  with I-th row storing result for  I-th  objective  being
                  minimized ignoring other ones.
                  Thus, ParetoFront[I,N+I] for  0<=I<M  stores  so  called
                  'ideal objective vector'.
                * subsequent FrontSize-M rows  store  variables/objectives
                  at  various  randomly  and  nearly   uniformly   sampled
                  locations of the Pareto front.

    FrontSize-  front size, >=0.
                * no larger than the number passed to setalgo()
                * for  a  single-objective  task,  FrontSize=1  is  ALWAYS
                  returned, no matter what was specified during setalgo()
                  call.
                * if  the   solver   was   prematurely   terminated   with
                  minnorequesttermination(), an  incomplete  Pareto  front
                  will be returned (it may even have less than M rows)
                * if a  failure (negative completion code) was   signaled,
                  FrontSize=0 will be returned

    Rep     -   optimization report, contains information about completion
                code, constraint violation at the solution and so on.

                You   should   check   rep.terminationtype  in  order   to
                distinguish successful termination from unsuccessful one:

                === FAILURE CODES ===
                * -8    internal  integrity control  detected  infinite or
                        NAN   values    in   function/gradient.   Abnormal
                        termination signalled.
                * -3    constraint bounds are  infeasible,  i.e.  we  have
                        box/linear/nonlinear constraint  with  two  bounds
                        present, and a lower one being  greater  than  the
                        upper one.
                        Note: less obvious infeasibilities of  constraints
                              do NOT  trigger  emergency  completion;  you
                              have to examine rep.bcerr/rep.lcerr/rep.nlcerr
                              to detect possibly inconsistent constraints.

                === SUCCESS CODES ===
                *  2   scaled step is no more than EpsX.
                *  5   MaxIts steps were taken.
                *  8   user   requested    algorithm    termination    via
                       minmorequesttermination(), last accepted point   is
                       returned.

                More information about fields of this  structure  can  be
                found in the comments on minmoreport datatype.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmoresults(const minmostate &state, real_2d_array &paretofront, ae_int_t &frontsize, minmoreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minmosetalgonbi` function

/*************************************************************************
Use the NBI (Normal Boundary Intersection)  algorithm  for  multiobjective
optimization.

NBI is a simple yet powerful multiobjective  optimization  algorithm  that
has the following attractive properties:
* it generates nearly uniformly distributed Pareto points
* it is applicable to problems with more than 2 objectives
* it naturally supports a mix of box, linear and nonlinear constraints
* it is less sensitive to the bad scaling of the targets

The only drawback of the algorithm is that for more than 2  objectives  it
can miss some small parts of the Pareto front that are  located  near  its
boundaries.

INPUT PARAMETERS:
    State       -   structure which stores algorithm state
    FrontSize   -   desired Pareto front size, FrontSize>=M,
                    where M is an objectives count
    PolishSolutions-whether additional solution improving phase is needed
                    or not:
                    * if False, the original NBI as formulated  by Das and
                      Dennis is used. It quickly produces  good solutions,
                      but these solutions can be suboptimal (usually within
                      0.1% of the optimal values).
                      The reason is that the original NBI formulation does
                      not account for  degeneracies that allow significant
                      progress for one objective with no deterioration for
                      other objectives.
                    * if True,  the  original  NBI  is  followed  by   the
                      additional solution  polishing  phase.  This  solver
                      mode is several times slower than the original  NBI,
                      but produces better solutions.

  -- ALGLIB --
     Copyright 20.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetalgonbi(minmostate &state, const ae_int_t frontsize, const bool polishsolutions, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minmosetbc` function

/*************************************************************************
This function sets boundary constraints for the MO optimizer.

Boundary constraints are inactive by  default  (after  initial  creation).
They are preserved after algorithm restart with MinMORestartFrom().

You may combine boundary constraints with  general  linear ones - and with
nonlinear ones! Boundary constraints are  handled  more  efficiently  than
other types.  Thus,  if  your  problem  has  mixed  constraints,  you  may
explicitly specify some of them as boundary and save some time/space.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                a very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                a very large number or +INF.

NOTE 1:  it is possible to specify  BndL[i]=BndU[i].  In  this  case  I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetbc(minmostate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`minmosetcond` function

/*************************************************************************
This function sets stopping conditions for inner iterations of the optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinMOSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsX=0 and MaxIts=0 (simultaneously) will lead to an automatic
selection of the stopping condition.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetcond(minmostate &state, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`minmosetlc2` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
sparse constraining matrix A. Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

INPUT PARAMETERS:
    State   -   structure previously allocated with minmocreate() call.
    A       -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minmosetlc2(minmostate &state, const sparsematrix &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`minmosetlc2dense` function

/*************************************************************************
This function sets two-sided linear constraints AL <= A*x <= AU with dense
constraint matrix A.

NOTE: knowing  that  constraint  matrix  is dense may help some MO solvers
      to utilize efficient dense Level 3  BLAS  for  dense  parts  of  the
      problem. If your problem has both dense and sparse constraints,  you
      can use minmosetlc2mixed() function.

INPUT PARAMETERS:
    State   -   structure previously allocated with minmocreate() call.
    A       -   linear constraints, array[K,N]. Each row of  A  represents
                one  constraint. One-sided  inequality   constraints, two-
                sided inequality  constraints,  equality  constraints  are
                supported (see below)
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number of equality/inequality constraints,  K>=0;  if  not
                given, inferred from sizes of A, AL, AU.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetlc2dense(minmostate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minmosetlc2dense(minmostate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

`minmosetlc2mixed` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
mixed constraining matrix A including sparse part (first SparseK rows) and
dense part (last DenseK rows). Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

This function may be useful if constraint matrix includes large number  of
both types of rows - dense and sparse. If you have just a few sparse rows,
you  may  represent  them  in  dense  format  without losing  performance.
Similarly, if you have just a few dense rows,  you  can  store them in the
sparse format with almost same performance.

INPUT PARAMETERS:
    State   -   structure previously allocated with minmocreate() call.
    SparseA -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    SparseK -   number of sparse constraints, SparseK>=0
    DenseA  -   linear constraints, array[K,N], set of dense constraints.
                Each row of A represents one general linear constraint.
    DenseK  -   number of dense constraints, DenseK>=0
    AL, AU  -   lower and upper bounds, array[SparseK+DenseK], with former
                SparseK elements corresponding to sparse constraints,  and
                latter DenseK elements corresponding to dense constraints;
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minmosetlc2mixed(minmostate &state, const sparsematrix &sparsea, const ae_int_t ksparse, const real_2d_array &densea, const ae_int_t kdense, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

`minmosetnlc2` function

/*************************************************************************
This function sets two-sided nonlinear constraints for MinMO optimizer.

In fact, this function sets  only  constraints  COUNT  and  their  BOUNDS.
Constraints  themselves  (constraint  functions)   are   passed   to   the
MinMOOptimize() method as callbacks.

MinMOOptimize() method accepts a user-defined vector function F[]  and its
Jacobian J[], where:
* first M components of F[]  and  first  M rows  of  J[]   correspond   to
  multiple objectives
* subsequent NNLC components of F[] (and rows of J[]) correspond  to  two-
  sided nonlinear constraints NL<=C(x)<=NU, where
  * NL[i]=NU[i] => I-th row is an equality constraint Ci(x)=NL
  * NL[i]<NU[i] => I-th tow is a  two-sided constraint NL[i]<=Ci(x)<=NU[i]
  * NL[i]=-INF  => I-th row is an one-sided constraint Ci(x)<=NU[i]
  * NU[i]=+INF  => I-th row is an one-sided constraint NL[i]<=Ci(x)
  * NL[i]=-INF, NU[i]=+INF => constraint is ignored

NOTE: you may combine nonlinear constraints with linear/boundary ones.  If
      your problem has mixed constraints, you  may explicitly specify some
      of them as linear or box ones.
      It helps optimizer to handle them more efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinMOCreate call.
    NL      -   array[NNLC], lower bounds, can contain -INF
    NU      -   array[NNLC], lower bounds, can contain +INF
    NNLC    -   constraints count, NNLC>=0

NOTE 1: nonlinear constraints are satisfied only  approximately!   It   is
        possible that the algorithm will evaluate the function  outside of
        the feasible area!

NOTE 2: algorithm scales variables  according  to the scale  specified by
        MinMOSetScale()  function,  so  it can handle problems with badly
        scaled variables (as long as we KNOW their scales).

        However,  there  is  no  way  to  automatically  scale   nonlinear
        constraints. Inappropriate scaling  of nonlinear  constraints  may
        ruin convergence. Solving problem with  constraint  "1000*G0(x)=0"
        is NOT the same as solving it with constraint "0.001*G0(x)=0".

        It means that YOU are  the  one who is responsible for the correct
        scaling of the nonlinear constraints Gi(x) and Hi(x). We recommend
        you to scale nonlinear constraints in such a way that the Jacobian
        rows have approximately unit magnitude  (for  problems  with  unit
        scale) or have magnitude approximately equal to 1/S[i] (where S is
        a scale set by MinMOSetScale() function).

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetnlc2(minmostate &state, const real_1d_array &nl, const real_1d_array &nu, const ae_int_t nnlc, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minmosetscale` function

/*************************************************************************
This function sets scaling coefficients for the MO optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetscale(minmostate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`minmosetxrep` function

/*************************************************************************
This function turns on/off reporting of the Pareto front points.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback  function  (if  it
was provided to MinMOOptimize) every time we find a Pareto front point.

NOTE: according to the communication protocol used by ALGLIB,  the  solver
      passes two parameters to the rep() callback - a current point and  a
      target value at the current point.
      However, because  we solve a  multi-objective  problem,  the  target
      parameter is not used and set to zero.

  -- ALGLIB --
     Copyright 01.03.2023 by Bochkanov Sergey
*************************************************************************/
void minmosetxrep(minmostate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minmo_biobjective example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  multiobjective2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates the bi-objective target
    //
    //     f0(x0,x1) = x0^2 + (x1-1)^2
    //     f1(x0,x1) = (x0-1(^2 + x1^2
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = x[0]*x[0]+(x[1]-1)*(x[1]-1);
    fi[1] = (x[0]-1)*(x[0]-1)+x[1]*x[1];
    jac[0][0] = 2*x[0];
    jac[0][1] = 2*(x[1]-1);
    jac[1][0] = 2*(x[0]-1);
    jac[1][1] = 2*x[1];
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of two targets
        //
        //     f0(x0,x1) = x0^2 + (x1-1)^2
        //     f1(x0,x1) = (x0-1(^2 + x1^2
        //
        // These targets are Euclidean distances to (0,1) and (1,0) respectively, thus solutions
        // to this problem occupy the straight line segment connecting these points. (Points
        // outside of the line are Pareto non-optimal because one can always decrease both distances
        // by moving closer to the line).
        //
        ae_int_t nvars = 2;
        ae_int_t nobjectives = 2;
        real_1d_array x0 = "[0,0]";
        ae_int_t frontsize = 10;
        bool polishsolutions = true;
        minmostate state;
        minmocreate(nvars, nobjectives, x0, state);

        //
        // The solver is configured to compute 10 points approximating the Pareto front,
        // and to polish solutions (i.e. use an additional optimization phase that improves
        // accuracy on degenerate problems; not actually necessary for this simple example).
        //
        minmosetalgonbi(state, frontsize, polishsolutions);

        //
        // Optimize and test results.
        //
        // The optimization is performed using analytic (user-provided) Jacobian matrix.
        // Use minmocreatef(), if you do not know analytic form of the Jacobian and want
        // ALGLIB to perform numerical differentiation.
        //
        // We requested 10 Pareto-optimal points and we expect solver to compute all points
        // (it is possible to return less if the solver was terminated)
        //
        minmoreport rep;
        real_2d_array paretofront;
        alglib::minmooptimize(state, multiobjective2_jac);
        minmoresults(state, paretofront, frontsize, rep);
        printf("%d\n", int(frontsize)); // EXPECTED: 10
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minmo_biobjective_constr example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  multiobjective2constr_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates the bi-objective target
    //
    //     f0(x0,x1) = x0^2 + (x1-1)^2
    //     f1(x0,x1) = (x0-1(^2 + x1^2
    //
    // nonlinear constraint function
    //
    //     f2(x0,x1) = x0^2 + x1^2
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = x[0]*x[0]+(x[1]-1)*(x[1]-1);
    fi[1] = (x[0]-1)*(x[0]-1)+x[1]*x[1];
    fi[2] = x[0]*x[0]+x[1]*x[1];
    jac[0][0] = 2*x[0];
    jac[0][1] = 2*(x[1]-1);
    jac[1][0] = 2*(x[0]-1);
    jac[1][1] = 2*x[1];
    jac[2][0] = 2*x[0];
    jac[2][1] = 2*x[1];
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of two targets
        //
        //     f0(x0,x1) = x0^2 + (x1-1)^2
        //     f1(x0,x1) = (x0-1(^2 + x1^2
        //
        // subject to a nonlinear constraint
        //
        //     f2(x0,x1) = x0^2 + x1^2 >= 1
        //
        // These targets are Euclidean distances to (0,1) and (1,0) respectively, thus solutions to this
        // problem should occupy the straight line segment connecting these points. However, due to the
        // nonlinear constraint being present, Pareto front has another shape.
        //
        ae_int_t nvars = 2;
        ae_int_t nobjectives = 2;
        real_1d_array x0 = "[0,0]";
        ae_int_t frontsize = 10;
        bool polishsolutions = true;
        real_1d_array lowerbnd = "[1]";
        real_1d_array upperbnd = "[+inf]";
        minmostate state;
        minmocreate(nvars, nobjectives, x0, state);
        minmosetnlc2(state, lowerbnd, upperbnd, 1);

        //
        // The solver is configured to compute 10 points approximating the Pareto front,
        // and to polish solutions (i.e. use an additional optimization phase that improves
        // accuracy on degenerate problems; not actually necessary for this simple example).
        //
        minmosetalgonbi(state, frontsize, polishsolutions);

        //
        // Optimize and test results.
        //
        // The optimization is performed using analytic (user-provided) Jacobian matrix.
        // Use minmocreatef(), if you do not know analytic form of the Jacobian and want
        // ALGLIB to perform numerical differentiation.
        //
        // We requested 10 Pareto-optimal points and we expect solver to compute all points
        // (it is possible to return less if the solver was terminated)
        //
        minmoreport rep;
        real_2d_array paretofront;
        alglib::minmooptimize(state, multiobjective2constr_jac);
        minmoresults(state, paretofront, frontsize, rep);
        printf("%d\n", int(frontsize)); // EXPECTED: 10
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minnlc` subpackage

Classes

minnlcreport
minnlcstate

Functions

minnlcaddlc2
minnlcaddlc2dense
minnlcaddlc2sparsefromdense
minnlccreate
minnlccreatebuf
minnlccreatef
minnlccreatefbuf
minnlciteration
minnlcoptguardgradient
minnlcoptguardnonc1test0results
minnlcoptguardnonc1test1results
minnlcoptguardresults
minnlcoptguardsmoothness
minnlcoptimize
minnlcrequesttermination
minnlcrestartfrom
minnlcresults
minnlcresultsbuf
minnlcsetalgoaul2
minnlcsetalgoorbit
minnlcsetalgosl1qp
minnlcsetalgosl1qpbfgs
minnlcsetalgosqp
minnlcsetalgosqpbfgs
minnlcsetbc
minnlcsetcond
minnlcsetcond3
minnlcsetlc
minnlcsetlc2
minnlcsetlc2dense
minnlcsetlc2mixed
minnlcsetnlc
minnlcsetnlc2
minnlcsetnumdiff
minnlcsetscale
minnlcsetstpmax
minnlcsetxrep

Examples

minnlc_d_equality		Nonlinearly constrained optimization (equality constraints)
minnlc_d_inequality		Nonlinearly constrained optimization (inequality constraints)
minnlc_d_mixed		Nonlinearly constrained optimization with mixed equality/inequality constraints
minnlc_d_numdiff		Nonlinearly constrained optimization with numerical differentiation
minnlc_d_sparse		Nonlinearly constrained optimization with sparse Jacobian
minnlc_modelbased		Nonlinearly constrained optimization for expensive objectives/constraints using surrogate models

`minnlcreport` class

/*************************************************************************
These fields store optimization report:
* f                         objective value at the solution
* iterationscount           total number of inner iterations
* nfev                      number of gradient evaluations
* terminationtype           termination type (see below)

Scaled constraint violations are reported:
* bcerr                     maximum violation of the box constraints
* bcidx                     index of the most violated box  constraint (or
                            -1, if all box constraints  are  satisfied  or
                            there is no box constraint)
* lcerr                     maximum violation of the  linear  constraints,
                            computed as maximum  scaled  distance  between
                            final point and constraint boundary.
* lcidx                     index of the most violated  linear  constraint
                            (or -1, if all constraints  are  satisfied  or
                            there is no general linear constraints)
* nlcerr                    maximum violation of the nonlinear constraints
* nlcidx                    index of the most violated nonlinear constraint
                            (or -1, if all constraints  are  satisfied  or
                            there is no nonlinear constraints)

Violations of box constraints are scaled on per-component basis  according
to  the  scale  vector s[] as specified by minnlcsetscale(). Violations of
the general linear  constraints  are  also  computed  using  user-supplied
variable scaling. Violations of nonlinear constraints are computed "as is"

LAGRANGE COEFFICIENTS

SQP solver (one activated by setalgosqp()/setalgosqpbfgs(), but  not  with
legacy functions) sets the following fields (other solvers  fill  them  by
zeros):

* lagbc[]                   array[N],   Lagrange   multipliers   for   box
                            constraints. IMPORTANT: COEFFICIENTS FOR FIXED
                            VARIABLES ARE SET TO ZERO. See  below  for  an
                            explanation.
                            This  parameter   stores    the  same  results
                            independently of whether analytic gradient  is
                            provided or numerical differentiation is used.

* lagbcnz[]                 array[N],   Lagrange   multipliers   for   box
                            constraints, behaves differently depending  on
                            whether  analytic  gradient  is   provided  or
                            numerical differentiation is used:
                            * for analytic Jacobian,   lagbcnz[]  contains
                              correct   coefficients   for  all  kinds  of
                              variables - fixed or not.
                            * for numerical Jacobian, it is  the  same  as
                              lagbc[], i.e. components corresponding  to
                              fixed vars are zero.
                            See below for an explanation.

* laglc[]                   array[Mlin], coeffs for linear constraints

* lagnlc[]                  array[Mnlc], coeffs for nonlinear constraints


Positive Lagrange coefficient means that constraint is at its upper bound.
Negative coefficient means that constraint is at its lower  bound.  It  is
expected that at the solution the dual feasibility condition holds:

    grad + SUM(Ei*LagBC[i],i=0..n-1) +
        SUM(Ai*LagLC[i],i=0..mlin-1) +
        SUM(Ni*LagNLC[i],i=0..mnlc-1) ~ 0

    (except for fixed variables which are handled specially)

where
* grad is a gradient at the solution
* Ei is a vector with 1.0 at position I and 0 in other positions
* Ai is an I-th row of linear constraint matrix
* Ni is an gradient of I-th nonlinear constraint

Fixed variables have two sets of Lagrange multipliers  for  the  following
reasons:
* analytic gradient and numerical gradient behave  differently  for  fixed
  vars. Numerical differentiation does not violate box  constraints,  thus
  gradient components corresponding to fixed vars are zero because we have
  no way of  differentiating  for  these  vars   without   violating   box
  constraints.
  Contrary to that, analytic gradient usually returns correct values  even
  for fixed vars.
* ideally,  we  would  like  numerical  gradient  to  be an almost perfect
  replacement for an analytic one.  Thus,  we  need  Lagrange  multipliers
  which do not change when we change the gradient type.
* on the other hand, we do not want to  lose  the  possibility  of  having
  a full set of Lagrange multipliers for problems with analytic gradient.
  Thus, there is a special field lagbcnz[] whose contents depends  on  the
  information available to us.

TERMINATION CODES

TerminationType field contains completion code, which can be either FAILURE
code, SUCCESS code, or SUCCESS code + ADDITIONAL code.  The  latter option
is used for more detailed reporting.

=== FAILURE CODE ===
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient, recovery was impossible.  Abnormal  termination
        signaled.
  -3    box  constraints  are  infeasible.  Note: infeasibility of non-box
        constraints does NOT trigger emergency  completion;  you  have  to
        examine  bcerr/lcerr/nlcerr   to  detect   possibly   inconsistent
        constraints.

=== SUCCESS CODE ===
   2    relative step is no more than EpsX.
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    user requested algorithm termination via minnlcrequesttermination(),
        last accepted point is returned

=== ADDITIONAL CODES ===
* +800      if   during   algorithm   execution   the   solver encountered
            NAN/INF values in the target or  constraints  but  managed  to
            recover by reducing trust region radius,  the  solver  returns
            one of SUCCESS codes but adds +800 to the code.

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minnlcreport
{
public:
    minnlcreport();
    minnlcreport(const minnlcreport &rhs);
    minnlcreport& operator=(const minnlcreport &rhs);
    virtual ~minnlcreport();
    double f;
    ae_int_t iterationscount;
    ae_int_t nfev;
    ae_int_t terminationtype;
    double bcerr;
    ae_int_t bcidx;
    double lcerr;
    ae_int_t lcidx;
    double nlcerr;
    ae_int_t nlcidx;
    real_1d_array lagbc;
    real_1d_array lagbcnz;
    real_1d_array laglc;
    real_1d_array lagnlc;
    ae_int_t dbgphase0its;
};

`minnlcstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinNLC subpackage to work  with  this
object
*************************************************************************/
class minnlcstate
{
public:
    minnlcstate();
    minnlcstate(const minnlcstate &rhs);
    minnlcstate& operator=(const minnlcstate &rhs);
    virtual ~minnlcstate();
};

`minnlcaddlc2` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present sparse constraints.

Constraint is passed in compressed format: as list of non-zero entries  of
coefficient vector A. Such approach is more efficient than  dense  storage
for highly sparse constraint vectors.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnlccreate() call.
    IdxA    -   array[NNZ], indexes of non-zero elements of A:
                * can be unsorted
                * can include duplicate indexes (corresponding entries  of
                  ValA[] will be summed)
    ValA    -   array[NNZ], values of non-zero elements of A
    NNZ     -   number of non-zero coefficients in A
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minnlcaddlc2(minnlcstate &state, const integer_1d_array &idxa, const real_1d_array &vala, const ae_int_t nnz, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minnlcaddlc2dense` function

/*************************************************************************
This function appends a two-sided linear constraint AL <= A*x <= AU to the
matrix of dense constraints.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnlccreate() call.
    A       -   linear constraint coefficient, array[N], right side is NOT
                included.
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint Ai*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint Ai*x<=AU
                * AU=+INF  => one-sided constraint AL<=Ai*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 15.04.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcaddlc2dense(minnlcstate &state, const real_1d_array &a, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minnlcaddlc2sparsefromdense` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present sparse constraints.

Constraint vector A is  passed  as  a  dense  array  which  is  internally
sparsified by this function.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnlccreate() call.
    DA      -   array[N], constraint vector
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minnlcaddlc2sparsefromdense(minnlcstate &state, const real_1d_array &da, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minnlccreate` function

/*************************************************************************
                  NONLINEARLY  CONSTRAINED  OPTIMIZATION

DESCRIPTION:
The  subroutine  minimizes a function  F(x)  of N arguments subject to the
any combination of the:
* bound constraints
* linear inequality constraints
* linear equality constraints
* nonlinear equality constraints Gi(x)=0
* nonlinear inequality constraints Hi(x)<=0

REQUIREMENTS:
* the user must provide callback calculating F(), H(), G()  -  either both
  value and gradient, or merely a value (numerical differentiation will be
  used)
* F(), G(), H() are continuously differentiable on the  feasible  set  and
  its neighborhood
* starting point X0, which can be infeasible

USAGE:

Here we give the very brief outline  of  the MinNLC optimizer. We strongly
recommend you to study examples in the ALGLIB Reference Manual and to read
ALGLIB User Guide: https://www.alglib.net/nonlinear-programming/

1. The user initializes the solver with minnlccreate() or  minnlccreatef()
   (the latter is used for numerical  differentiation)  call  and  chooses
   which NLC solver to use.

   In the current release the following solvers can be used:

   * sparse large-scale filter-based SQP solver, recommended  for problems
     of any size (from several variables to thousands  of  variables).
     Activated with minnlcsetalgosqp() function.

   * dense SQP-BFGS solver, recommended  for small-scale problems  (up  to
     several hundreds of variables) with a very expensive target function.
     Requires less function  evaluations than SQP, but has more  expensive
     iteration. Activated with minnlcsetalgosqpbfgs() function.

   * ORBIT, a model-based derivative  free  solver  that  uses  local  RBF
     models to optimize expensive objectives.  This  solver  is  activated
     with minnlcsetalgoorbit() function.

   * several other solvers, including legacy ones

2. [optional] user activates OptGuard  integrity checker  which  tries  to
   detect possible errors in the user-supplied callbacks:
   * discontinuity/nonsmoothness of the target/nonlinear constraints
   * errors in the analytic gradient provided by user
   This feature is essential for early prototyping stages because it helps
   to catch common coding and problem statement errors.
   OptGuard can be activated with following functions (one per each  check
   performed):
   * minnlcoptguardsmoothness()
   * minnlcoptguardgradient()

3. User adds boundary and/or linear and/or nonlinear constraints by  means
   of calling one of the following functions:
   a) minnlcsetbc() for boundary constraints
   b) minnlcsetlc2() for sparse two-sided linear constraints,
      minnlcsetlc2dense() for dense two-sided linear constraints,
      minnlcsetlc2mixed() for mixed sparse/dense two-sided linear constraints
    * minqpaddlc2dense()  to add one dense row to the dense constraint submatrix
    * minqpaddlc2()       to add one sparse row to the sparse constraint submatrix
    * minqpaddlc2sparsefromdense() to add one sparse row (passed as a dense array) to the sparse constraint submatrix
   c) minnlcsetnlc2() for nonlinear constraints
   You may combine (a), (b) and (c) in one optimization problem.

4. User sets scale of the variables with minnlcsetscale() function. It  is
   VERY important to set  scale  of  the  variables,  because  nonlinearly
   constrained problems are hard to solve when variables are badly scaled.
   Knowing  variable   scales   helps   to  check  stopping  criteria  and
   precondition the solver.

5. User sets stopping conditions with minnlcsetcond3() or minnlcsetcond().
   If NLC solver uses inner/outer  iteration  layout,  this  function sets
   stopping conditions for INNER iterations.

6. Finally, user calls minnlcoptimize()  function  which  takes  algorithm
   state and pointer (delegate, etc.) to callback function which calculates
   F/G/H.

7. User calls  minnlcresults()  to  get  solution;  additionally  you  can
   retrieve OptGuard report with minnlcoptguardresults(), and get detailed
   report about purported errors in the target function with:
   * minnlcoptguardnonc1test0results()
   * minnlcoptguardnonc1test1results()

8. Optionally user may call minnlcrestartfrom() to solve  another  problem
   with same N but another starting point. minnlcrestartfrom()  allows  to
   reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlccreate(const ae_int_t n, const real_1d_array &x, minnlcstate &state, const xparams _xparams = alglib::xdefault);
void minnlccreate(const real_1d_array &x, minnlcstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlccreatebuf` function

/*************************************************************************
Buffered version of minnlccreate() which reuses already  allocated  memory
as much as possible.

  -- ALGLIB --
     Copyright 06.10.2024 by Bochkanov Sergey
*************************************************************************/
void minnlccreatebuf(const ae_int_t n, const real_1d_array &x, minnlcstate &state, const xparams _xparams = alglib::xdefault);
void minnlccreatebuf(const real_1d_array &x, minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlccreatef` function

/*************************************************************************
This subroutine is a finite  difference variant of MinNLCCreate(). It uses
finite differences in order to differentiate target function.

Description below contains information which is specific to this  function
only. We recommend to read comments on MinNLCCreate() in order to get more
information about creation of the NLC optimizer.

CALLBACK PARALLELISM

The MINNLC optimizer supports parallel parallel  numerical differentiation
('callback parallelism'). This feature, which  is  present  in  commercial
ALGLIB  editions,  greatly   accelerates   optimization   with   numerical
differentiation of an expensive target functions.

Callback parallelism is usually  beneficial  when  computing  a  numerical
gradient requires more than several  milliseconds.  In this case  the  job
of computing individual gradient components can be split between  multiple
threads. Even inexpensive targets can benefit  from  parallelism,  if  you
have many variables.

ALGLIB Reference Manual, 'Working with commercial  version' section, tells
how to activate callback parallelism for your programming language.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size ofX
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.
    DiffStep-   differentiation step, >0.
                By default, a 5-point formula is used  (actually,  only  4
                function values per variable are used because  the central
                one has zero coefficient due to symmetry; that's why  this
                formula is often called a 4-point one). It can be  changed
                with minnlcsetnumdiff() function.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

NOTES:

1. the differentiation step along I-th axis is equal to DiffStep*S[I] where
   S[] is scaling vector which can be set by MinNLCSetScale() call.

2. we recommend you to use moderate values of  differentiation  step.  Too
   large step will result in too large TRUNCATION  errors, while too small
   step will result in too large NUMERICAL  errors.  1.0E-4  can  be  good
   value to start from.

3. Numerical  differentiation  is   very   inefficient  -   one   gradient
   calculation needs ~N function evaluations. This function will work  for
   any N - either small (1...10), moderate (10...100) or  large  (100...).
   However, performance penalty will be too severe for any N's except  for
   small ones.

   We should also say that code which relies on numerical  differentiation
   is  less   robust   and  precise.  Imprecise  gradient  may  slow  down
   convergence, especially  on  highly  nonlinear  problems  or  near  the
   solution.

   Thus  we  recommend to use this function for fast prototyping on small-
   dimensional problems only, and to implement analytical gradient as soon
   as possible.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlccreatef(const ae_int_t n, const real_1d_array &x, const double diffstep, minnlcstate &state, const xparams _xparams = alglib::xdefault);
void minnlccreatef(const real_1d_array &x, const double diffstep, minnlcstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlccreatefbuf` function

/*************************************************************************
Buffered version of minnlccreatef() which reuses already allocated memory
as much as possible.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlccreatefbuf(const ae_int_t n, const real_1d_array &x, const double diffstep, minnlcstate &state, const xparams _xparams = alglib::xdefault);
void minnlccreatefbuf(const real_1d_array &x, const double diffstep, minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlciteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minnlciteration(minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlcoptguardgradient` function

/*************************************************************************
This  function  activates/deactivates verification  of  the  user-supplied
analytic gradient/Jacobian.

Upon  activation  of  this  option  OptGuard  integrity  checker  performs
numerical differentiation of your target  function  (constraints)  at  the
initial point (note: future versions may also perform check  at  the final
point) and compares numerical gradient/Jacobian with analytic one provided
by you.

If difference is too large, an error flag is set and optimization  session
continues. After optimization session is over, you can retrieve the report
which stores both gradients/Jacobians, and specific components highlighted
as suspicious by the OptGuard.

The primary OptGuard report can be retrieved with minnlcoptguardresults().

IMPORTANT: gradient check is a high-overhead option which  will  cost  you
           about 3*N additional function evaluations. In many cases it may
           cost as much as the rest of the optimization session.

           YOU SHOULD NOT USE IT IN THE PRODUCTION CODE UNLESS YOU WANT TO
           CHECK DERIVATIVES PROVIDED BY SOME THIRD PARTY.

NOTE: unlike previous incarnation of the gradient checking code,  OptGuard
      does NOT interrupt optimization even if it discovers bad gradient.

INPUT PARAMETERS:
    State       -   structure used to store algorithm state
    TestStep    -   verification step used for numerical differentiation:
                    * TestStep=0 turns verification off
                    * TestStep>0 activates verification
                    You should carefully choose TestStep. Value  which  is
                    too large (so large that  function  behavior  is  non-
                    cubic at this scale) will lead  to  false  alarms. Too
                    short step will result in rounding  errors  dominating
                    numerical derivative.

                    You may use different step for different parameters by
                    means of setting scale with minnlcsetscale().

=== EXPLANATION ==========================================================

In order to verify gradient algorithm performs following steps:
  * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i],
    where X[i] is i-th component of the initial point and S[i] is a  scale
    of i-th parameter
  * F(X) is evaluated at these trial points
  * we perform one more evaluation in the middle point of the interval
  * we  build  cubic  model using function values and derivatives at trial
    points and we compare its prediction with actual value in  the  middle
    point

  -- ALGLIB --
     Copyright 15.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcoptguardgradient(minnlcstate &state, const double teststep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minnlcoptguardnonc1test0results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #0

Nonsmoothness (non-C1) test #0 studies  function  values  (not  gradient!)
obtained during line searches and monitors  behavior  of  the  directional
derivative estimate.

This test is less powerful than test #1, but it does  not  depend  on  the
gradient values and thus it is more robust against artifacts introduced by
numerical differentiation.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* fidx - is an index of the function (0 for  target  function, 1 or higher
  for nonlinear constraints) which is suspected of being "non-C1"
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #0 "strong" report
    lngrep  -   C1 test #0 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minnlcoptguardnonc1test0results(const minnlcstate &state, optguardnonc1test0report &strrep, optguardnonc1test0report &lngrep, const xparams _xparams = alglib::xdefault);

`minnlcoptguardnonc1test1results` function

/*************************************************************************
Detailed results of the OptGuard integrity check for nonsmoothness test #1

Nonsmoothness (non-C1)  test  #1  studies  individual  components  of  the
gradient computed during line search.

When precise analytic gradient is provided this test is more powerful than
test #0  which  works  with  function  values  and  ignores  user-provided
gradient.  However,  test  #0  becomes  more   powerful   when   numerical
differentiation is employed (in such cases test #1 detects  higher  levels
of numerical noise and becomes too conservative).

This test also tells specific components of the gradient which violate  C1
continuity, which makes it more informative than #0, which just tells that
continuity is violated.

Two reports are returned:
* a "strongest" one, corresponding  to  line   search  which  had  highest
  value of the nonsmoothness indicator
* a "longest" one, corresponding to line search which  had  more  function
  evaluations, and thus is more detailed

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* fidx - is an index of the function (0 for  target  function, 1 or higher
  for nonlinear constraints) which is suspected of being "non-C1"
* vidx - is an index of the variable in [0,N) with nonsmooth derivative
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], g[] - arrays of length CNT which store step lengths and  gradient
  values at these points; g[i] is evaluated in  x0+stp[i]*d  and  contains
  vidx-th component of the gradient.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.

==========================================================================
= SHORTLY SPEAKING: build a 2D plot of (stp,f) and look at it -  you  will
=                   see where C1 continuity is violated.
==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    strrep  -   C1 test #1 "strong" report
    lngrep  -   C1 test #1 "long" report

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minnlcoptguardnonc1test1results(minnlcstate &state, optguardnonc1test1report &strrep, optguardnonc1test1report &lngrep, const xparams _xparams = alglib::xdefault);

`minnlcoptguardresults` function

/*************************************************************************
Results of OptGuard integrity check, should be called  after  optimization
session is over.

=== PRIMARY REPORT =======================================================

OptGuard performs several checks which are intended to catch common errors
in the implementation of nonlinear function/gradient:
* incorrect analytic gradient
* discontinuous (non-C0) target functions (constraints)
* nonsmooth     (non-C1) target functions (constraints)

Each of these checks is activated with appropriate function:
* minnlcoptguardgradient() for gradient verification
* minnlcoptguardsmoothness() for C0/C1 checks

Following flags are set when these errors are suspected:
* rep.badgradsuspected, and additionally:
  * rep.badgradfidx for specific function (Jacobian row) suspected
  * rep.badgradvidx for specific variable (Jacobian column) suspected
  * rep.badgradxbase, a point where gradient/Jacobian is tested
  * rep.badgraduser, user-provided gradient/Jacobian
  * rep.badgradnum, reference gradient/Jacobian obtained via numerical
    differentiation
* rep.nonc0suspected, and additionally:
  * rep.nonc0fidx - an index of specific function violating C0 continuity
* rep.nonc1suspected, and additionally
  * rep.nonc1fidx - an index of specific function violating C1 continuity
Here function index 0 means  target function, index 1  or  higher  denotes
nonlinear constraints.

=== ADDITIONAL REPORTS/LOGS ==============================================

Several different tests are performed to catch C0/C1 errors, you can  find
out specific test signaled error by looking to:
* rep.nonc0test0positive, for non-C0 test #0
* rep.nonc1test0positive, for non-C1 test #0
* rep.nonc1test1positive, for non-C1 test #1

Additional information (including line search logs)  can  be  obtained  by
means of:
* minnlcoptguardnonc1test0results()
* minnlcoptguardnonc1test1results()
which return detailed error reports, specific points where discontinuities
were found, and so on.

==========================================================================

INPUT PARAMETERS:
    state   -   algorithm state

OUTPUT PARAMETERS:
    rep     -   generic OptGuard report;  more  detailed  reports  can  be
                retrieved with other functions.

NOTE: false negatives (nonsmooth problems are not identified as  nonsmooth
      ones) are possible although unlikely.

      The reason  is  that  you  need  to  make several evaluations around
      nonsmoothness  in  order  to  accumulate  enough  information  about
      function curvature. Say, if you start right from the nonsmooth point,
      optimizer simply won't get enough data to understand what  is  going
      wrong before it terminates due to abrupt changes in the  derivative.
      It is also  possible  that  "unlucky"  step  will  move  us  to  the
      termination too quickly.

      Our current approach is to have less than 0.1%  false  negatives  in
      our test examples  (measured  with  multiple  restarts  from  random
      points), and to have exactly 0% false positives.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minnlcoptguardresults(minnlcstate &state, optguardreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minnlcoptguardsmoothness` function

/*************************************************************************
This  function  activates/deactivates nonsmoothness monitoring  option  of
the  OptGuard  integrity  checker. Smoothness  monitor  silently  observes
solution process and tries to detect ill-posed problems, i.e. ones with:
a) discontinuous target function (non-C0) and/or constraints
b) nonsmooth     target function (non-C1) and/or constraints

Smoothness monitoring does NOT interrupt optimization  even if it suspects
that your problem is nonsmooth. It just sets corresponding  flags  in  the
OptGuard report which can be retrieved after optimization is over.

Smoothness monitoring is a moderate overhead option which often adds  less
than 1% to the optimizer running time. Thus, you can use it even for large
scale problems.

NOTE: OptGuard does  NOT  guarantee  that  it  will  always  detect  C0/C1
      continuity violations.

      First, minor errors are hard to  catch - say, a 0.0001 difference in
      the model values at two sides of the gap may be due to discontinuity
      of the model - or simply because the model has changed.

      Second, C1-violations  are  especially  difficult  to  detect  in  a
      noninvasive way. The optimizer usually  performs  very  short  steps
      near the nonsmoothness, and differentiation  usually   introduces  a
      lot of numerical noise.  It  is  hard  to  tell  whether  some  tiny
      discontinuity in the slope is due to real nonsmoothness or just  due
      to numerical noise alone.

      Our top priority was to avoid false positives, so in some rare cases
      minor errors may went unnoticed (however, in most cases they can  be
      spotted with restart from different initial point).

INPUT PARAMETERS:
    state   -   algorithm state
    level   -   monitoring level:
                * 0 - monitoring is disabled
                * 1 - noninvasive low-overhead monitoring; function values
                      and/or gradients are recorded, but OptGuard does not
                      try to perform additional evaluations  in  order  to
                      get more information about suspicious locations.
                      This kind of monitoring does not work well with  SQP
                      because SQP solver needs just 1-2 function evaluations
                      per step, which is not enough for OptGuard  to  make
                      any conclusions.

=== EXPLANATION ==========================================================

One major source of headache during optimization  is  the  possibility  of
the coding errors in the target function/constraints (or their gradients).
Such  errors   most   often   manifest   themselves  as  discontinuity  or
nonsmoothness of the target/constraints.

Another frequent situation is when you try to optimize something involving
lots of min() and max() operations, i.e. nonsmooth target. Although not  a
coding error, it is nonsmoothness anyway - and smooth  optimizers  usually
stop right after encountering nonsmoothness, well before reaching solution.

OptGuard integrity checker helps you to catch such situations: it monitors
function values/gradients being passed  to  the  optimizer  and  tries  to
errors. Upon discovering suspicious pair of points it  raises  appropriate
flag (and allows you to continue optimization). When optimization is done,
you can study OptGuard result.

  -- ALGLIB --
     Copyright 21.11.2018 by Bochkanov Sergey
*************************************************************************/
void minnlcoptguardsmoothness(minnlcstate &state, const ae_int_t level, const xparams _xparams = alglib::xdefault);
void minnlcoptguardsmoothness(minnlcstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`minnlcoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    sjac    -   callback which calculates function vector fi[]
                and sparse Jacobian sjac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


CALLBACK PARALLELISM

The MINNLC optimizer supports parallel parallel  numerical differentiation
('callback parallelism'). This feature, which  is  present  in  commercial
ALGLIB  editions,  greatly   accelerates   optimization   with   numerical
differentiation of an expensive target functions.

Callback parallelism is usually  beneficial  when  computing  a  numerical
gradient requires more than several  milliseconds.  In this case  the  job
of computing individual gradient components can be split between  multiple
threads. Even inexpensive targets can benefit  from  parallelism,  if  you
have many variables.

ALGLIB Reference Manual, 'Working with commercial  version' section, tells
how to activate callback parallelism for your programming language.

NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied Jacobian, and one which uses  only  function
   vector and numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   you should choose appropriate variant of MinNLCOptimize() -  one  which
   accepts function AND Jacobian or one which accepts ONLY function.

   Be careful to choose variant of MinNLCOptimize()  which  corresponds to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to MinNLCOptimize()   and  specific
   function used to create optimizer.


                     |         USER PASSED TO MinNLCOptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   MinNLCCreateF()   |     works               FAILS
   MinNLCCreate()    |     FAILS               works

   Here "FAILS" denotes inappropriate combinations  of  optimizer creation
   function  and  MinNLCOptimize()  version.   Attemps   to    use    such
   combination will lead to exception. Either  you  did  not pass gradient
   when it WAS needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcoptimize(minnlcstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minnlcoptimize(minnlcstate &state,
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minnlcoptimize(minnlcstate &state,
    void (*sjac)(const real_1d_array &x, real_1d_array &fi, sparsematrix &s, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcrequesttermination(minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlcrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinNLCCreate call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minnlcrestartfrom(minnlcstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`minnlcresults` function

/*************************************************************************
MinNLC results:  the  solution  found,  completion  codes  and  additional
information.

If you activated OptGuard integrity checking functionality and want to get
OptGuard report, it can be retrieved with:
* minnlcoptguardresults() - for a primary report about (a) suspected C0/C1
  continuity violations and (b) errors in the analytic gradient.
* minnlcoptguardnonc1test0results() - for C1 continuity violation test #0,
  detailed line search log
* minnlcoptguardnonc1test1results() - for C1 continuity violation test #1,
  detailed line search log

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report, contains information about completion
                code, constraint violation at the solution and so on.

                rep.f contains objective value at the solution.

                You   should   check   rep.terminationtype  in  order   to
                distinguish successful termination from unsuccessful one:

                === FAILURE CODES ===
                * -8    internal  integrity control  detected  infinite or
                        NAN  values  in  function/gradient,  recovery  was
                        impossible. Abnormal termination signalled.
                * -3    box  constraints are infeasible.
                        Note: infeasibility of  non-box  constraints  does
                              NOT trigger emergency completion;  you  have
                              to examine rep.bcerr/rep.lcerr/rep.nlcerr to
                              detect possibly inconsistent constraints.

                === SUCCESS CODES ===
                *  2   scaled step is no more than EpsX.
                *  5   MaxIts steps were taken.
                *  8   user   requested    algorithm    termination    via
                       minnlcrequesttermination(), last accepted point  is
                       returned.

                === ADDITIONAL CODES ===
                * +800      if   during  algorithm  execution  the  solver
                            encountered NAN/INF values in  the  target  or
                            constraints but managed to recover by reducing
                            trust region radius, the  solver  returns  one
                            of SUCCESS codes but adds +800 to the code.

                Some solvers (as of ALGLIB 4.02, only SQP) return Lagrange
                multipliers in rep.lagbc/lagbcnz, laglc, lagnlc fields.

                More information about fields of this  structure  can  be
                found in the comments on the minnlcreport datatype.

  -- ALGLIB --
     Copyright 18.01.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcresults(const minnlcstate &state, real_1d_array &x, minnlcreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcresultsbuf` function

/*************************************************************************
NLC results

Buffered implementation of MinNLCResults() which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minnlcresultsbuf(const minnlcstate &state, real_1d_array &x, minnlcreport &rep, const xparams _xparams = alglib::xdefault);

`minnlcsetalgoaul2` function

/*************************************************************************
This function tells MinNLC unit to use the large-scale augmented Lagrangian
algorithm for nonlinearly constrained optimization.

This  algorithm  is  a  significant  refactoring  of  one  described in "A
Modified Barrier-Augmented  Lagrangian Method for Constrained Minimization
(1999)" by D.GOLDFARB,  R.POLYAK,  K. SCHEINBERG,  I.YUZEFOVICH  with  the
following additions:
* improved sparsity support
* improved handling of large-scale problems with the low rank  LBFGS-based
  sparse preconditioner
* automatic selection of the penalty parameter Rho

AUL solver can be significantly faster than SQP on easy  problems  due  to
cheaper iterations, although it needs more function evaluations. On large-
scale sparse problems one iteration of the AUL solver usually  costs  tens
times less than one iteration of the SQP solver.

However, the SQP solver is more robust than the AUL. In particular, it  is
much better at constraint enforcement and will  never escape feasible area
after constraints were successfully enforced.  It  also  needs  much  less
target function evaluations.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    MaxOuterIts-upper limit on outer iterations count:
                * MaxOuterIts=0 means that the solver  will  automatically
                  choose an upper limit. Recommended value.
                * MaxOuterIts>1 means that the AUL solver will performs at
                  most specified number of outer iterations

  -- ALGLIB --
     Copyright 22.09.2023 by Bochkanov Sergey
*************************************************************************/
void minnlcsetalgoaul2(minnlcstate &state, const ae_int_t maxouterits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcsetalgoorbit` function

/*************************************************************************
This function selects ORBIT solver, a model-based  derivative-free  solver
for minimization of expensive derivative-free functions.

The ORBIT algorithm by Wild and Shoemaker (2013) is an algorithm that uses
objective function values to build a smooth  RBF  model  (f=r^3)  that  is
minimized over a trust region in order to  identify  step  direction.  The
algorithm saves and reuses function values at all previously known points.

ALGLIB added to the original algorithm the following modifications:
* box, linear and nonlinear constraints
* improved tolerance to noise in the objective/constraints

Its  intended  area  of  application  is  a  low-accuracy  minimization of
expensive objectives with no  gradient  available.  It  is  expected  that
additional overhead of building and minimizing an RBF model is  negligible
when compared with the objective evaluation cost. Iteration overhead grows
as O(N^3), so this solver is recommended for problems with N below 100.

This algorithm has the following nice properties:
* no parameters to tune
* no convexity requirements for target function or constraints
* the initial point can be infeasible
* the algorithm respects box constraints in all  intermediate  points  (it
  does not even evaluate the target outside of the box constrained area)
* once linear and nonlinear constraints are enforced, the  algorithm  will
  try to respect them as much as possible.

When compared with SQP solver, ORBIT:
* is much faster than the finite-difference  based  serial  SQP  at  early
  stages of optimization, being able to achieve 0.1-0.01 relative accuracy
  about 4x-10x faster than SQP solver
* has slower asymptotic convergence on ill-conditioned problems, sometimes
  being unable to reduce error in objective or constraints below  1E-5  in
  a reasonable amount of time
* has  no  obvious  benefits  over  SQP  with  analytic gradient or highly
  parallelized (more than 10 cores) finite-difference SQP

NOTE: whilst technically this algorithm supports callback parallelism,  in
      practice it can't efficiently utilize parallel resources because  it
      issues requests for objective/constraints in  an  inherently  serial
      manner.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

    Rad0    -   initial  sampling   radius  (multiplied  by   per-variable
                scales),  Rad0>=0,  zero  value  means  automatic   radius
                selection.
                An ideal  value  is  large  enough  to  allow  significant
                progress by making a Rad0-sized step, but  not  too  large
                (so  that  initial  linear  model  well  approximates  the
                objective).
                Recommended values: 0.1 or 1  (assuming  properly   chosen
                variable scales.
                The solver can tolerate inappropriately  chosen  Rad0,  at
                the expense of additional function evaluations  needed  to
                adjust it.

    MaxNFEV -   MaxNFEV>=0,  with   zero  value  meaning  no  limit.  This
                parameter allows to control computational budget (measured
                in  function  evaluations).

                It provides somewhat finer control than  MaxIts  parameter
                of minnlcsetcond(), which controls the maximum  amount  of
                iterations performed by the algorithm, with  one iteration
                usually needing more than one function evaluation.

                The  solver  does  not  stop  immediately  after  reaching
                MaxNFEV evaluations, but  will  stop  shortly  after  that
                (usually within N+1 evaluations, often within 1-2).

  -- ALGLIB --
     Copyright 02.10.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcsetalgoorbit(minnlcstate &state, const double rad0, const ae_int_t maxnfev, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minnlcsetalgosl1qp` function

/*************************************************************************
This function selects a legacy solver: an L1 merit function based SQP with
the sparse l-BFGS update.

It is recommended to use either SQP or SQP-BFGS solvers  instead  of  this
one.  These  solvers  use  filters  to  provide  much  faster  and  robust
convergence.
>

  -- ALGLIB --
     Copyright 02.12.2019 by Bochkanov Sergey
*************************************************************************/
void minnlcsetalgosl1qp(minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlcsetalgosl1qpbfgs` function

/*************************************************************************
This function selects a legacy solver: an L1 merit function based SQP with
the dense BFGS update.

It is recommended to use either SQP or SQP-BFGS solvers  instead  of  this
one.  These  solvers  use  filters  to  provide  much  faster  and  robust
convergence.

  -- ALGLIB --
     Copyright 02.12.2019 by Bochkanov Sergey
*************************************************************************/
void minnlcsetalgosl1qpbfgs(minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlcsetalgosqp` function

/*************************************************************************
This function selects large-scale  sparse  filter-based  SQP  solver,  the
most robust solver in ALGLIB, a recommended option.

This algorithm is scalable to problems with tens of thousands of variables
and can efficiently handle sparsity of constraints.

The convergence is proved for the following case:
* function and constraints are continuously differentiable (C1 class)

This algorithm has the following nice properties:
* no parameters to tune
* no convexity requirements for target function or constraints
* the initial point can be infeasible
* the algorithm respects box constraints in all  intermediate  points  (it
  does not even evaluate the target outside of the box constrained area)
* once linear constraints are enforced, the algorithm will not violate them
* no such guarantees can be provided for nonlinear constraints,  but  once
  nonlinear  constraints  are  enforced,  the algorithm will try to respect
  them as much as possible
* numerical differentiation does  not  violate  box  constraints  (although
  general linear and nonlinear ones can be violated during differentiation)

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

===== TRACING SQP SOLVER =================================================

SQP solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'SQP'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
                  It also prints OptGuard  integrity  checker  report when
                  nonsmoothness of target/constraints is suspected.
* 'SQP.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'SQP'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format
* 'SQP.PROBING' - to let algorithm insert additional function  evaluations
                  before line search  in  order  to  build  human-readable
                  chart of the raw  Lagrangian  (~40  additional  function
                  evaluations is performed for  each  line  search).  This
                  symbol  also  implicitly  defines  'SQP'  and  activates
                  OptGuard integrity checker which detects continuity  and
                  smoothness violations. An OptGuard log is printed at the
                  end of the file.

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related   overhead.  Specifying  'SQP.PROBING'  adds   even  larger
overhead due to additional function evaluations being performed.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("SQP,SQP.PROBING,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 02.12.2023 by Bochkanov Sergey
*************************************************************************/
void minnlcsetalgosqp(minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlcsetalgosqpbfgs` function

/*************************************************************************
This function selects a special solver for low-dimensional  problems  with
expensive target function - the dense filer-based SQP-BFGS solver.

This algorithm uses a dense quadratic model of the  target  and  solves  a
dense QP subproblem at each step. Thus, it has difficulties scaling beyond
several hundreds of variables.  However,  it  usually  needs  the smallest
number of the target evaluations - sometimes  up  to  30%  less  than  the
sparse large-scale filter-based SQP.

The convergence is proved for the following case:
* function and constraints are continuously differentiable (C1 class)

This algorithm has the following nice properties:
* no parameters to tune
* no convexity requirements for target function or constraints
* the initial point can be infeasible
* the algorithm respects box constraints in all  intermediate  points  (it
  does not even evaluate the target outside of the box constrained area)
* once linear constraints are enforced, the algorithm will not violate them
* no such guarantees can be provided for nonlinear constraints,  but  once
  nonlinear  constraints  are  enforced,  the algorithm will try to respect
  them as much as possible
* numerical differentiation does  not  violate  box  constraints  (although
  general linear and nonlinear ones can be violated during differentiation)

INPUT PARAMETERS:
    State   -   structure which stores algorithm state

===== TRACING SQP SOLVER =================================================

SQP solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'SQP'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
                  It also prints OptGuard  integrity  checker  report when
                  nonsmoothness of target/constraints is suspected.
* 'SQP.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'SQP'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format
* 'SQP.PROBING' - to let algorithm insert additional function  evaluations
                  before line search  in  order  to  build  human-readable
                  chart of the raw  Lagrangian  (~40  additional  function
                  evaluations is performed for  each  line  search).  This
                  symbol  also  implicitly  defines  'SQP'  and  activates
                  OptGuard integrity checker which detects continuity  and
                  smoothness violations. An OptGuard log is printed at the
                  end of the file.

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related   overhead.  Specifying  'SQP.PROBING'  adds   even  larger
overhead due to additional function evaluations being performed.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("SQP,SQP.PROBING,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 02.12.2023 by Bochkanov Sergey
*************************************************************************/
void minnlcsetalgosqpbfgs(minnlcstate &state, const xparams _xparams = alglib::xdefault);

`minnlcsetbc` function

/*************************************************************************
This function sets boundary constraints for NLC optimizer.

Boundary constraints are inactive by  default  (after  initial  creation).
They are preserved after algorithm restart with  MinNLCRestartFrom().

You may combine boundary constraints with  general  linear ones - and with
nonlinear ones! Boundary constraints are  handled  more  efficiently  than
other types.  Thus,  if  your  problem  has  mixed  constraints,  you  may
explicitly specify some of them as boundary and save some time/space.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1:  it is possible to specify  BndL[i]=BndU[i].  In  this  case  I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2:  when you solve your problem  with  augmented  Lagrangian  solver,
         boundary constraints are  satisfied  only  approximately!  It  is
         possible   that  algorithm  will  evaluate  function  outside  of
         feasible area!

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcsetbc(minnlcstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`minnlcsetcond` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

This  function allows to set  iterations  limit  and  step-based  stopping
conditions. If you want the solver to stop upon having a small  change  in
the target, use minnlcsetcond3() function.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinNLCSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic
selection of the stopping condition.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcsetcond(minnlcstate &state, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcsetcond3` function

/*************************************************************************
This function sets stopping conditions for the optimizer.

This function allows to set three types of stopping conditions:
* iterations limit
* stopping upon performing a short step (depending on the specific  solver
  being used  it may stop as soon as the first short step was made, or
  only after performing several sequential short steps)
* stopping upon having a small change in  the  target  (depending  on  the
  specific solver being used it may stop as soon as the  first  step  with
  small change in the target was made, or only  after  performing  several
  sequential steps)

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsF    -   >=0
                The optimizer will stop as soon as the following condition
                is met:

                    |f_scl(k+1)-f_scl(k)| <= max(|f_scl(k+1)|,|f_scl(k)|,1)

                where f_scl is an internally used by the optimizer rescaled
                target (ALGLIB optimizers usually apply rescaling in order
                to normalize target and constraints).
    EpsX    -   >=0
                The subroutine finishes its work if  on  k+1-th  iteration
                the condition |v|<=EpsX is fulfilled, where:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinNLCSetScale()
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsF, EpsX=0 and MaxIts=0 (simultaneously) will lead to the
automatic selection of the stopping condition.

  -- ALGLIB --
     Copyright 21.09.2023 by Bochkanov Sergey
*************************************************************************/
void minnlcsetcond3(minnlcstate &state, const double epsf, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`minnlcsetlc` function

/*************************************************************************
This function sets linear constraints for MinNLC optimizer.

Linear constraints are inactive by default (after initial creation).  They
are preserved after algorithm restart with MinNLCRestartFrom().

You may combine linear constraints with boundary ones - and with nonlinear
ones! If your problem has mixed constraints, you  may  explicitly  specify
some of them as linear. It  may  help  optimizer   to   handle  them  more
efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinNLCCreate call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE 1: when you solve your problem  with  augmented  Lagrangian   solver,
        linear constraints are  satisfied  only   approximately!   It   is
        possible   that  algorithm  will  evaluate  function  outside   of
        feasible area!

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcsetlc(minnlcstate &state, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minnlcsetlc(minnlcstate &state, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

`minnlcsetlc2` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
a sparse constraining matrix A. Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

INPUT PARAMETERS:
    State   -   structure previously allocated with minnlccreate() call.
    A       -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 15.04.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcsetlc2(minnlcstate &state, const sparsematrix &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`minnlcsetlc2dense` function

/*************************************************************************
This function sets two-sided linear constraints AL <= A*x <= AU with dense
constraint matrix A.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnlccreate() call.
    A       -   linear constraints, array[K,N]. Each row of  A  represents
                one  constraint. One-sided  inequality   constraints, two-
                sided inequality  constraints,  equality  constraints  are
                supported (see below)
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number of equality/inequality constraints,  K>=0;  if  not
                given, inferred from sizes of A, AL, AU.

  -- ALGLIB --
     Copyright 15.04.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcsetlc2dense(minnlcstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minnlcsetlc2dense(minnlcstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

`minnlcsetlc2mixed` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
a mixed constraining matrix A including a sparse part (first SparseK rows)
and a dense part (last DenseK rows). Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

This function may be useful if constraint matrix includes large number  of
both types of rows - dense and sparse. If you have just a few sparse rows,
you  may  represent  them  in  dense  format  without losing  performance.
Similarly, if you have just a few dense rows, you may store them in sparse
format with almost same performance.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnlccreate() call.
    SparseA -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    SparseK -   number of sparse constraints, SparseK>=0
    DenseA  -   linear constraints, array[K,N], set of dense constraints.
                Each row of A represents one general linear constraint.
    DenseK  -   number of dense constraints, DenseK>=0
    AL, AU  -   lower and upper bounds, array[SparseK+DenseK], with former
                SparseK elements corresponding to sparse constraints,  and
                latter DenseK elements corresponding to dense constraints;
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 15.04.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcsetlc2mixed(minnlcstate &state, const sparsematrix &sparsea, const ae_int_t ksparse, const real_2d_array &densea, const ae_int_t kdense, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

`minnlcsetnlc` function

/*************************************************************************
This function sets nonlinear constraints for MinNLC optimizer.

It sets constraints of the form

    Ci(x)=0 for i=0..NLEC-1
    Ci(x)<=0 for i=NLEC..NLEC+NLIC-1

See MinNLCSetNLC2() for a modern function which allows greater flexibility
in the constraint specification.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcsetnlc(minnlcstate &state, const ae_int_t nlec, const ae_int_t nlic, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcsetnlc2` function

/*************************************************************************
This function sets two-sided nonlinear constraints for MinNLC optimizer.

In fact, this function sets  only  constraints  COUNT  and  their  BOUNDS.
Constraints  themselves  (constraint  functions)   are   passed   to   the
MinNLCOptimize() method as callbacks.

MinNLCOptimize() method accepts a user-defined vector function F[] and its
Jacobian J[], where:
* first element of F[] and first row of J[] correspond to the target
* subsequent NNLC components of F[] (and rows of J[]) correspond  to  two-
  sided nonlinear constraints NL<=C(x)<=NU, where
  * NL[i]=NU[i] => I-th row is an equality constraint Ci(x)=NL
  * NL[i]<NU[i] => I-th tow is a  two-sided constraint NL[i]<=Ci(x)<=NU[i]
  * NL[i]=-INF  => I-th row is an one-sided constraint Ci(x)<=NU[i]
  * NU[i]=+INF  => I-th row is an one-sided constraint NL[i]<=Ci(x)
  * NL[i]=-INF, NU[i]=+INF => constraint is ignored

NOTE: you may combine nonlinear constraints with linear/boundary ones.  If
      your problem has mixed constraints, you  may explicitly specify some
      of them as linear or box ones.
      It helps optimizer to handle them more efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinNLCCreate call.
    NL      -   array[NNLC], lower bounds, can contain -INF
    NU      -   array[NNLC], lower bounds, can contain +INF
    NNLC    -   constraints count, NNLC>=0

NOTE 1: nonlinear constraints are satisfied only  approximately!   It   is
        possible that the algorithm will evaluate the function  outside of
        the feasible area!

NOTE 2: algorithm scales variables  according  to the scale  specified by
        MinNLCSetScale()  function,  so it can handle problems with badly
        scaled variables (as long as we KNOW their scales).

        However,  there  is  no  way  to  automatically  scale   nonlinear
        constraints. Inappropriate scaling  of nonlinear  constraints  may
        ruin convergence. Solving problem with  constraint  "1000*G0(x)=0"
        is NOT the same as solving it with constraint "0.001*G0(x)=0".

        It means that YOU are  the  one who is responsible for the correct
        scaling of the nonlinear constraints Gi(x) and Hi(x). We recommend
        you to scale nonlinear constraints in such a way that the Jacobian
        rows have approximately unit magnitude  (for  problems  with  unit
        scale) or have magnitude approximately equal to 1/S[i] (where S is
        a scale set by MinNLCSetScale() function).

  -- ALGLIB --
     Copyright 23.09.2023 by Bochkanov Sergey
*************************************************************************/
void minnlcsetnlc2(minnlcstate &state, const real_1d_array &nl, const real_1d_array &nu, const ae_int_t nnlc, const xparams _xparams = alglib::xdefault);
void minnlcsetnlc2(minnlcstate &state, const real_1d_array &nl, const real_1d_array &nu, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcsetnumdiff` function

/*************************************************************************
This function sets specific finite  difference  formula  to  be  used  for
numerical differentiation.

It works only for optimizers created  with  minnlccreatef()  function;  in
other cases it has no effect.

INPUT PARAMETERS:
    State       -   structure previously allocated with MinNLCCreateF call.
    FormulaType -   formula type:
                    * 5 for a 5-point formula (actually, only 4 values per
                      variable  are  used, ones at x+h, x+h/2,  x-h/2  and
                      x-h; the central  one has  zero  multiplier  due  to
                      symmetry). The most precise and the  most  expensive
                      option, chosen by default
                    * 3 for a 3-point formula, which is also  known  as  a
                      symmetric difference quotient (the formula  actually
                      uses only two function values per variable:  at  x+h
                      and x-h). A   good  compromise  for  medium-accuracy
                      setups
                    * 2 for a forward (or backward, depending  on variable
                      bounds)  finite   difference  (f(x+h)-f(x))/h.  This
                      formula has the lowest accuracy. However, it  is  4x
                      faster than the 5-point formula and 2x  faster  than
                      the 3-point one because, in addition to the  central
                      value f(x), it needs only  one  additional  function
                      evaluation per variable.


  -- ALGLIB --
     Copyright 03.12.2024 by Bochkanov Sergey
*************************************************************************/
void minnlcsetnumdiff(minnlcstate &state, const ae_int_t formulatype, const xparams _xparams = alglib::xdefault);

`minnlcsetscale` function

/*************************************************************************
This function sets scaling coefficients for NLC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison  with  tolerances).  Scales
are also used by the finite difference variant of the optimizer - the step
along I-th axis is equal to DiffStep*S[I]. Finally,  variable  scales  are
used for preconditioning (i.e. to speed up the solver).

The scale of the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 06.06.2014 by Bochkanov Sergey
*************************************************************************/
void minnlcsetscale(minnlcstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minnlcsetstpmax` function

/*************************************************************************
This function sets maximum step length (after scaling of step vector  with
respect to variable scales specified by minnlcsetscale() call).

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0 (default),  if
                you don't want to limit step length.

Use this subroutine when you optimize target function which contains exp()
or  other  fast  growing  functions,  and optimization algorithm makes too
large  steps  which  leads  to overflow. This function allows us to reject
steps  that  are  too  large  (and  therefore  expose  us  to the possible
overflow) without actually calculating function value at the x+stp*d.

NOTE: different solvers employed by MinNLC  optimizer  may  use  different
      norms for the step.

  -- ALGLIB --
     Copyright 02.04.2010 by Bochkanov Sergey
*************************************************************************/
void minnlcsetstpmax(minnlcstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`minnlcsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to MinNLCOptimize().

NOTE: algorithm passes two parameters to rep() callback  -  current  point
      and penalized function value at current point. Important -  function
      value which is returned is NOT function being minimized. It  is  sum
      of the value of the function being minimized - and penalty term.

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minnlcsetxrep(minnlcstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minnlc_d_equality example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = -x0+x1
    //     f1(x0,x1) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = -x[0]+x[1];
    fi[1] = x[0]*x[0] + x[1]*x[1] - 1.0;
    jac[0][0] = -1.0;
    jac[0][1] = +1.0;
    jac[1][0] = 2*x[0];
    jac[1][1] = 2*x[1];
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = -x0+x1
        //
        // subject to nonlinear equality constraint
        //
        //    x0^2 + x1^2 - 1 = 0
        //
        // IMPORTANT: the   MINNLC   optimizer    supports    parallel   numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minnlcoptimize()  function for
        //            more information.
        //
        real_1d_array x0 = "[1,1]";
        real_1d_array s = "[1,1]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minnlcstate state;

        //
        // Create optimizer object and tune its settings:
        // * epsx=0.000001  stopping condition for inner iterations
        // * s=[1,1]        all variables have unit scale
        //
        minnlccreate(2, x0, state);
        minnlcsetcond(state, epsx, maxits);
        minnlcsetscale(state, s);

        //
        // Choose  one  of  nonlinear  programming  solvers  supported  by  MINNLC
        // optimizer.
        //
        // As of ALGLIB 4.01, the default (and recommended)  option  is to  use  a
        // large-scale filter-based SQP solver, which can utilize sparsity of  the
        // problem and uses a limited-memory BFGS update in order to  be  able  to
        // deal with thousands of variables.
        //
        // Other options include:
        // * SQP-BFGS (the same filter SQP solver relying on a dense BFGS  update,
        //   not intended for anything beyond 100 variables)
        // * ORBIT solver, a derivative-free solver  for optimization of expensive
        //   functions that are smooth, but have no gradient available
        // * AUL2 solver (a large-scale augmented  Lagrangian  solver for problems
        //   with  cheap  target  functions)
        // * SL1QP and SL1QP-BFGS legacy solvers which are similar to filter-based
        //   SQP/SQP-BFGS, but use a less  robust  L1  merit  function  to  handle
        //   constraints
        //
        minnlcsetalgosqp(state);

        //
        // Set constraints:
        //
        // Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        // nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        // elements being possibly infinite (means that this specific bound  is
        // ignored). It includes equality constraints,  upper/lower  inequality
        // constraints, range constraints. In particular, the constraint
        //
        //        x0^2 + x1^2 - 1 = 0
        //
        // can be specified by passing NL=[0], NU=[0] to minnlcsetnlc2().
        //
        // Constraining functions themselves are passed as part  of  a  problem
        // Jacobian (see below).
        //
        real_1d_array nl = "[0]";
        real_1d_array nu = "[0]";
        minnlcsetnlc2(state, nl, nu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear equality and inequality constraints.
        //
        // So, our vector function has form
        //
        //     {f0,f1} = { -x0+x1 , x0^2+x1^2-1 }
        //
        // with Jacobian
        //
        //         [  -1    +1  ]
        //     J = [            ]
        //         [ 2*x0  2*x1 ]
        //
        // with f0 being target function, f1 being constraining function. Number
        // of equality/inequality constraints is specified by minnlcsetnlc2().
        //
        minnlcreport rep;
        real_1d_array x1;
        alglib::minnlcoptimize(state, nlcfunc1_jac);
        minnlcresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.70710,-0.70710]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minnlc_d_inequality example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = -x0+x1
    //     f1(x0,x1) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = -x[0]+x[1];
    fi[1] = x[0]*x[0] + x[1]*x[1] - 1.0;
    jac[0][0] = -1.0;
    jac[0][1] = +1.0;
    jac[1][0] = 2*x[0];
    jac[1][1] = 2*x[1];
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = -x0+x1
        //
        // subject to box constraints
        //
        //    x0>=0, x1>=0
        //
        // and a nonlinear inequality constraint
        //
        //    x0^2 + x1^2 - 1 <= 0
        //
        // IMPORTANT: the   MINNLC   optimizer    supports    parallel   numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minnlcoptimize()  function for
        //            more information.
        //
        real_1d_array x0 = "[0,0]";
        real_1d_array s = "[1,1]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minnlcstate state;

        //
        // Create optimizer object and tune its settings:
        // * epsx=0.000001  stopping condition for inner iterations
        // * s=[1,1]        all variables have unit scale; it is important to
        //                  tell optimizer about scales of your variables - it
        //                  greatly accelerates convergence and helps to perform
        //                  some important integrity checks.
        //
        minnlccreate(2, x0, state);
        minnlcsetcond(state, epsx, maxits);
        minnlcsetscale(state, s);

        //
        // Choose  one  of  nonlinear  programming  solvers  supported  by  MINNLC
        // optimizer.
        //
        // As of ALGLIB 4.01, the default (and recommended)  option  is to  use  a
        // large-scale filter-based SQP solver, which can utilize sparsity of  the
        // problem and uses a limited-memory BFGS update in order to  be  able  to
        // deal with thousands of variables.
        //
        // Other options include:
        // * SQP-BFGS (the same filter SQP solver relying on a dense BFGS  update,
        //   not intended for anything beyond 100 variables)
        // * ORBIT solver, a derivative-free solver  for optimization of expensive
        //   functions that are smooth, but have no gradient available
        // * AUL2 solver (a large-scale augmented  Lagrangian  solver for problems
        //   with  cheap  target  functions)
        // * SL1QP and SL1QP-BFGS legacy solvers which are similar to filter-based
        //   SQP/SQP-BFGS, but use a less  robust  L1  merit  function  to  handle
        //   constraints
        //
        minnlcsetalgosqp(state);

        //
        // Set constraints:
        //
        // 1. box constraints are passed with minnlcsetbc() call. The  solver also
        //    supports linear constraints with minnlcsetlc().
        //
        // 2. nonlinear constraints are more tricky - you can not "pack" a general
        //    nonlinear  function  into  a  double  precision  array.  That's  why
        //    minnlcsetnlc2() does not accept constraints itself - only constraint
        //    bounds are passed.
        //
        //    Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        //    nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        //    elements being possibly infinite (means that this specific bound  is
        //    ignored). It includes equality constraints,  upper/lower  inequality
        //    constraints, range constraints. In particular, the constraint
        //
        //        x0^2 + x1^2 - 1 <= 0
        //
        //    can be specified by passing NL=[-INF], NU=[0] to minnlcsetnlc2().
        //
        //    Constraining functions themselves are passed as part  of  a  problem
        //    Jacobian (see below).
        //
        real_1d_array bndl = "[0,0]";
        real_1d_array bndu = "[+inf,+inf]";
        real_1d_array nl = "[-inf]";
        real_1d_array nu = "[0]";
        minnlcsetbc(state, bndl, bndu);
        minnlcsetnlc2(state, nl, nu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear constraints.
        //
        // So, our vector function has form
        //
        //     {f0,f1} = { -x0+x1 , x0^2+x1^2-1 }
        //
        // with Jacobian
        //
        //         [  -1    +1  ]
        //     J = [            ]
        //         [ 2*x0  2*x1 ]
        //
        // with f0 being target function, f1 being constraining function. Number
        // of equality/inequality constraints is specified by minnlcsetnlc2().
        //
        minnlcreport rep;
        real_1d_array x1;
        alglib::minnlcoptimize(state, nlcfunc1_jac);
        minnlcresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minnlc_d_mixed example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1,x2) = x0+x1
    //     f1(x0,x1,x2) = x2-exp(x0)
    //     f2(x0,x1,x2) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj]
    //
    fi[0] = x[0]+x[1];
    fi[1] = x[2]-exp(x[0]);
    fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
    jac[0][0] = 1.0;
    jac[0][1] = 1.0;
    jac[0][2] = 0.0;
    jac[1][0] = -exp(x[0]);
    jac[1][1] = 0.0;
    jac[1][2] = 1.0;
    jac[2][0] = 2*x[0];
    jac[2][1] = 2*x[1];
    jac[2][2] = 0.0;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = x0+x1
        //
        // subject to nonlinear inequality constraint
        //
        //    x0^2 + x1^2 - 1 <= 0
        //
        // and nonlinear equality constraint
        //
        //    x2-exp(x0) = 0
        //
        // IMPORTANT: the   MINNLC   optimizer    supports    parallel   numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minnlcoptimize() function  for
        //            more information.
        //
        real_1d_array x0 = "[0,0,0]";
        real_1d_array s = "[1,1,1]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minnlcstate state;
        minnlcreport rep;
        real_1d_array x1;

        //
        // Create optimizer object and tune its settings:
        // * epsx=0.000001  stopping condition for inner iterations
        // * s=[1,1]        all variables have unit scale
        // * upper limit on step length is specified (to avoid probing locations where exp() is large)
        //
        minnlccreate(3, x0, state);
        minnlcsetcond(state, epsx, maxits);
        minnlcsetscale(state, s);
        minnlcsetstpmax(state, 10.0);

        //
        // Choose  one  of  nonlinear  programming  solvers  supported  by  MINNLC
        // optimizer.
        //
        // As of ALGLIB 4.01, the default (and recommended)  option  is to  use  a
        // large-scale filter-based SQP solver, which can utilize sparsity of  the
        // problem and uses a limited-memory BFGS update in order to  be  able  to
        // deal with thousands of variables.
        //
        // Other options include:
        // * SQP-BFGS (the same filter SQP solver relying on a dense BFGS  update,
        //   not intended for anything beyond 100 variables)
        // * ORBIT solver, a derivative-free solver  for optimization of expensive
        //   functions that are smooth, but have no gradient available
        // * AUL2 solver (a large-scale augmented  Lagrangian  solver for problems
        //   with  cheap  target  functions)
        // * SL1QP and SL1QP-BFGS legacy solvers which are similar to filter-based
        //   SQP/SQP-BFGS, but use a less  robust  L1  merit  function  to  handle
        //   constraints
        //
        minnlcsetalgosqp(state);

        //
        // Set constraints:
        //
        // Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        // nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        // elements being possibly infinite (means that this specific bound  is
        // ignored). It includes equality constraints,  upper/lower  inequality
        // constraints, range constraints. In particular, a pair of constraints
        //
        //        x2-exp(x0)       = 0
        //        x0^2 + x1^2 - 1 <= 0
        //
        // can be specified by passing NL=[0,-INF], NU=[0,0] to minnlcsetnlc2().
        //
        // Constraining functions themselves are passed as part  of  a  problem
        // Jacobian (see below).
        //
        real_1d_array nl = "[0,-inf]";
        real_1d_array nu = "[0,0]";
        minnlcsetnlc2(state, nl, nu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear equality and inequality constraints.
        //
        // So, our vector function has form
        //
        //     {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
        //
        // with Jacobian
        //
        //         [  +1      +1       0 ]
        //     J = [-exp(x0)  0        1 ]
        //         [ 2*x0    2*x1      0 ]
        //
        // with f0 being target function, f1 being equality constraint "f1=0",
        // f2 being inequality constraint "f2<=0".
        //
        alglib::minnlcoptimize(state, nlcfunc2_jac);
        minnlcresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [-0.70710,-0.70710,0.49306]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minnlc_d_numdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc2_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1,x2) = x0+x1
    //     f1(x0,x1,x2) = x2-exp(x0)
    //     f2(x0,x1,x2) = x0^2+x1^2-1
    //
    fi[0] = x[0]+x[1];
    fi[1] = x[2]-exp(x[0]);
    fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = x0+x1
        //
        // subject to box constraints
        //
        //    0<=x0<+inf, -inf<x1<+int, -inf<x2<+int
        //
        // nonlinear inequality constraint
        //
        //    x0^2 + x1^2 - 1 <= 0
        //
        // and nonlinear equality constraint
        //
        //    x2-exp(x0) = 0
        //
        // using numerical differentiation.
        //
        // IMPORTANT: the   MINNLC   optimizer    supports    parallel   numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minnlcoptimize()  function  for
        //            more information.
        //
        real_1d_array x0 = "[0,0,0]";
        real_1d_array s = "[1,1,1]";
        double epsx = 0.000001;
        double diffstep = 0.000001;
        ae_int_t maxits = 0;
        minnlcstate state;
        minnlcreport rep;
        real_1d_array x1;

        //
        // Create optimizer object and tune its settings:
        // * epsx=0.000001      stopping condition for inner iterations
        // * diffstep=0.000001  numerical differentiation step (times variable scales)
        // * s=[1,1]            all variables have unit scale
        //
        minnlccreatef(3, x0, diffstep, state);
        minnlcsetcond(state, epsx, maxits);
        minnlcsetscale(state, s);
        minnlcsetstpmax(state, 10.0);

        //
        // Choose  one  of  nonlinear  programming  solvers  supported  by  MINNLC
        // optimizer.
        //
        // As of ALGLIB 4.01, the default (and recommended)  option  is to  use  a
        // large-scale filter-based SQP solver, which can utilize sparsity of  the
        // problem and uses a limited-memory BFGS update in order to  be  able  to
        // deal with thousands of variables.
        //
        // Alternatively, ORBIT solver can be used. This solver uses RBF surrogate
        // models to minimize expensive objectives with  no  gradient  information
        // available.
        //
        minnlcsetalgosqp(state);

        //
        // Set box constraints. ALGLIB  respects  box  constraints  and  does  not
        // evaluate  target  outside  of  a  box-constrained  area,  even   during
        // numerical differentiation. The finite difference  formula  is  modified
        // according to the current box constraints, if necessary.
        //
        real_1d_array bndl = "[0,-inf,-inf]";
        real_1d_array bndu = "[+inf,+inf,+inf]";
        minnlcsetbc(state, bndl, bndu);

        //
        // Set nonlinear constraints:
        //
        // Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        // nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        // elements being possibly infinite (means that this specific bound  is
        // ignored). It includes equality constraints,  upper/lower  inequality
        // constraints, range constraints. In particular, a pair of constraints
        //
        //        x2-exp(x0)       = 0
        //        x0^2 + x1^2 - 1 <= 0
        //
        // can be specified by passing NL=[0,-INF], NU=[0,0] to minnlcsetnlc2().
        //
        // Constraining functions themselves are passed as part  of  a  problem
        // function vector (see below).
        //
        real_1d_array nl = "[0,-inf]";
        real_1d_array nu = "[0,0]";
        minnlcsetnlc2(state, nl, nu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function but not its Jacobian, with
        // numerical differentiation used  to  compute  Jacobian  values.  The
        // first component of the function vector is a  target  function,  and
        // the next components are nonlinear constraints.
        //
        // So, our vector function has the form
        //
        //     {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
        //
        // with f0 being target function, f1 being equality constraint "f1=0",
        // f2 being inequality constraint "f2<=0".
        //
        alglib::minnlcoptimize(state, nlcfunc2_fvec);
        minnlcresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0,-1,1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minnlc_d_sparse example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc2_sjac(const real_1d_array &x, real_1d_array &fi, sparsematrix &sjac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1,x2) = x0+x1
    //     f1(x0,x1,x2) = x2-exp(x0)
    //     f2(x0,x1,x2) = x0^2+x1^2-1
    //
    // and Jacobian matrix J = [dfi/dxj].
    //
    // This callback returns Jacobian as a sparse CRS-based matrix. This format is intended
    // for large-scale problems, it allows to solve otherwise intractable tasks with hundreds
    // of thousands of variables. It will also work for our toy problem with just three variables,
    // though.
    //
    //
    // First, we calculate function vector fi[].
    //
    fi[0] = x[0]+x[1];
    fi[1] = x[2]-exp(x[0]);
    fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
    
    //
    // After that we initialize sparse Jacobian. On entry to this function sjac is a sparse
    // CRS matrix in a special initial state with N columns but no rows (such matrices can
    // be created with the sparsecreatecrsempty() function ).
    //
    // Such matrices can be used only for sequential addition of rows and nonzero elements.
    // You should add all rows that are expected (one for an objective and one per each
    // nonlinear constraint). Insufficient or excessive rows will be treated as an error.
    // Row elements must be added from left to right, i.e. column indexes must monotonically
    // increase.
    //
    // NOTE: you should NOT reinitialize sjac with sparsecreate() or any other function. It
    //       is important that you append rows/cols to the matrix, but do not create a new
    //       instance of the matrix object. Doing so may cause hard-to-detect errors in
    //       the present or future ALGLIB versions.
    //
    sparseappendemptyrow(sjac);
    sparseappendelement(sjac, 0, 1.0);
    sparseappendelement(sjac, 1, 1.0);
    sparseappendemptyrow(sjac);
    sparseappendelement(sjac, 0, -exp(x[0]));
    sparseappendelement(sjac, 2, 1.0);
    sparseappendemptyrow(sjac);
    sparseappendelement(sjac, 0, 2.0*x[0]);
    sparseappendelement(sjac, 1, 2.0*x[1]);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = x0+x1
        //
        // subject to nonlinear inequality constraint
        //
        //    x0^2 + x1^2 - 1 <= 0
        //
        // and nonlinear equality constraint
        //
        //    x2-exp(x0) = 0
        //
        // with their Jacobian being a sparse matrix.
        //
        // IMPORTANT: the   MINNLC   optimizer    supports    parallel   numerical
        //            differentiation  ('callback   parallelism').  This  feature,
        //            which  is present  in  commercial  ALGLIB  editions, greatly
        //            accelerates optimization with numerical  differentiation  of
        //            an expensive target functions.
        //
        //            Callback parallelism is usually  beneficial when computing a
        //            numerical gradient requires more than several  milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on   minnlcoptimize() function  for
        //            more information.
        //
        real_1d_array x0 = "[0,0,0]";
        real_1d_array s = "[1,1,1]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minnlcstate state;
        minnlcreport rep;
        real_1d_array x1;

        //
        // Create optimizer object and tune its settings:
        // * epsx=0.000001  stopping condition for inner iterations
        // * s=[1,1]        all variables have unit scale
        // * upper limit on step length is specified (to avoid probing locations where exp() is large)
        //
        minnlccreate(3, x0, state);
        minnlcsetcond(state, epsx, maxits);
        minnlcsetscale(state, s);
        minnlcsetstpmax(state, 10.0);

        //
        // Choose  one  of  nonlinear  programming  solvers  supported  by  MINNLC
        // optimizer.
        //
        // As of ALGLIB 4.02, the only solver which is fully  sparse-capable  is a
        // large-scale filter-based SQP solver, which can utilize sparsity of  the
        // problem and uses a limited-memory BFGS update in order to  be  able  to
        // deal with thousands of variables.
        //
        minnlcsetalgosqp(state);

        //
        // Set constraints:
        //
        // Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        // nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        // elements being possibly infinite (means that this specific bound  is
        // ignored). It includes equality constraints,  upper/lower  inequality
        // constraints, range constraints. In particular, a pair of constraints
        //
        //        x2-exp(x0)       = 0
        //        x0^2 + x1^2 - 1 <= 0
        //
        // can be specified by passing NL=[0,-INF], NU=[0,0] to minnlcsetnlc2().
        //
        // Constraining functions themselves are passed as part  of  a  problem
        // Jacobian (see below).
        //
        real_1d_array nl = "[0,-inf]";
        real_1d_array nu = "[0,0]";
        minnlcsetnlc2(state, nl, nu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear equality and inequality constraints.
        //
        // So, our vector function has form
        //
        //     {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
        //
        // with Jacobian
        //
        //         [  +1      +1       0 ]
        //     J = [-exp(x0)  0        1 ]
        //         [ 2*x0    2*x1      0 ]
        //
        // with f0 being target function, f1 being equality constraint "f1=0",
        // f2 being inequality constraint "f2<=0". The Jacobian is store as a
        // sparse matrix. See comments on the callback for  more  information
        // about working with sparse Jacobians.
        //
        alglib::minnlcoptimize(state, nlcfunc2_sjac);
        minnlcresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [-0.70710,-0.70710,0.49306]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minnlc_modelbased example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nlcfunc2_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1,x2) = x0+x1
    //     f1(x0,x1,x2) = x2-exp(x0)
    //     f2(x0,x1,x2) = x0^2+x1^2-1
    //
    fi[0] = x[0]+x[1];
    fi[1] = x[2]-exp(x[0]);
    fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = x0+x1
        //
        // subject to nonlinears constraints
        //
        //    x0^2 + x1^2 - 1 <= 0
        //    x2-exp(x0)       = 0
        //
        // using surrogate RBF models.
        //
        // This optimization mode is intended for expensive objectives/constraints
        // that  lack   derivative   information  (e.g.  obtained  from  numerical
        // simulation or by observing some physical  system).  It  provides  rapid
        // convergence to medium-quality approximate solutions (error in objective
        // being as low as 1% or 0.1%).
        //
        real_1d_array x0 = "[0,0,0]";
        real_1d_array s = "[1,1,1]";
        double epsx = 0.000001;
        ae_int_t maxits = 0;
        minnlcstate state;
        minnlcreport rep;
        real_1d_array x1;

        //
        // Create optimizer object and tune its settings:
        // * epsx=0.000001  stopping condition for inner iterations
        // * s=[1,1]        all variables have unit scale
        // * set ORBIT solver
        //
        minnlccreate(3, x0, state);
        minnlcsetcond(state, epsx, maxits);
        minnlcsetscale(state, s);
        minnlcsetalgoorbit(state, 0, 0);

        //
        // Set constraints:
        //
        // Since  version  4.01,  ALGLIB  supports  the  most  general  form of
        // nonlinear constraints: two-sided   constraints  NL<=C(x)<=NU,   with
        // elements being possibly infinite (means that this specific bound  is
        // ignored). It includes equality constraints,  upper/lower  inequality
        // constraints, range constraints. In particular, a pair of constraints
        //
        //        x2-exp(x0)       = 0
        //        x0^2 + x1^2 - 1 <= 0
        //
        // can be specified by passing NL=[0,-INF], NU=[0,0] to minnlcsetnlc2().
        //
        // Constraining functions themselves are passed  as  part  of  a  function
        // vector (see below).
        //
        real_1d_array nl = "[0,-inf]";
        real_1d_array nu = "[0,0]";
        minnlcsetnlc2(state, nl, nu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function (no derivative information  is
        // needed), with first component being objective  and  next  2  components
        // being nonlinear constraints.
        //
        // So, our vector function has form
        //
        //     {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
        //
        // with f0 being objective function, f1 being equality constraint "f1=0",
        // f2 being inequality constraint "f2<=0".
        //
        alglib::minnlcoptimize(state, nlcfunc2_fvec);
        minnlcresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [-0.70710,-0.70710,0.49306]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minns` subpackage

Classes

minnsreport
minnsstate

Functions

minnscreate
minnscreatef
minnsiteration
minnsoptimize
minnsrequesttermination
minnsrestartfrom
minnsresults
minnsresultsbuf
minnssetalgoags
minnssetbc
minnssetcond
minnssetlc
minnssetnlc
minnssetscale
minnssetxrep

Examples

minns_d_bc		Nonsmooth box constrained optimization
minns_d_diff		Nonsmooth unconstrained optimization with numerical differentiation
minns_d_nlc		Nonsmooth nonlinearly constrained optimization
minns_d_unconstrained		Nonsmooth unconstrained optimization

`minnsreport` class

/*************************************************************************
This structure stores optimization report:
* IterationsCount           total number of inner iterations
* NFEV                      number of gradient evaluations
* TerminationType           termination type (see below)
* CErr                      maximum violation of all types of constraints
* LCErr                     maximum violation of linear constraints
* NLCErr                    maximum violation of nonlinear constraints

TERMINATION CODES

TerminationType field contains completion code, which can be:
  -8    internal integrity control detected  infinite  or  NAN  values  in
        function/gradient. Abnormal termination signalled.
  -3    box constraints are inconsistent
  -1    inconsistent parameters were passed:
        * penalty parameter for minnssetalgoags() is zero,
          but we have nonlinear constraints set by minnssetnlc()
   2    sampling radius decreased below epsx
   5    MaxIts steps was taken
   7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.
   8    User requested termination via MinNSRequestTermination()

Other fields of this structure are not documented and should not be used!
*************************************************************************/
class minnsreport
{
public:
    minnsreport();
    minnsreport(const minnsreport &rhs);
    minnsreport& operator=(const minnsreport &rhs);
    virtual ~minnsreport();
    ae_int_t iterationscount;
    ae_int_t nfev;
    double cerr;
    double lcerr;
    double nlcerr;
    ae_int_t terminationtype;
    ae_int_t varidx;
    ae_int_t funcidx;
};

`minnsstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinNS subpackage to work  with  this
object
*************************************************************************/
class minnsstate
{
public:
    minnsstate();
    minnsstate(const minnsstate &rhs);
    minnsstate& operator=(const minnsstate &rhs);
    virtual ~minnsstate();
};

`minnscreate` function

/*************************************************************************
                  NONSMOOTH NONCONVEX OPTIMIZATION
            SUBJECT TO BOX/LINEAR/NONLINEAR-NONSMOOTH CONSTRAINTS

DESCRIPTION:

The  subroutine  minimizes  function   F(x)  of N arguments subject to any
combination of:
* bound constraints
* linear inequality constraints
* linear equality constraints
* nonlinear equality constraints Gi(x)=0
* nonlinear inequality constraints Hi(x)<=0

IMPORTANT: see MinNSSetAlgoAGS for important  information  on  performance
           restrictions of AGS solver.

REQUIREMENTS:
* starting point X0 must be feasible or not too far away from the feasible
  set
* F(), G(), H() are continuous, locally Lipschitz  and  continuously  (but
  not necessarily twice) differentiable in an open dense  subset  of  R^N.
  Functions F(), G() and H() may be nonsmooth and non-convex.
  Informally speaking, it means  that  functions  are  composed  of  large
  differentiable "patches" with nonsmoothness having  place  only  at  the
  boundaries between these "patches".
  Most real-life nonsmooth  functions  satisfy  these  requirements.  Say,
  anything which involves finite number of abs(), min() and max() is  very
  likely to pass the test.
  Say, it is possible to optimize anything of the following:
  * f=abs(x0)+2*abs(x1)
  * f=max(x0,x1)
  * f=sin(max(x0,x1)+abs(x2))
* for nonlinearly constrained problems: F()  must  be  bounded from  below
  without nonlinear constraints (this requirement is due to the fact that,
  contrary to box and linear constraints, nonlinear ones  require  special
  handling).
* user must provide function value and gradient for F(), H(), G()  at  all
  points where function/gradient can be calculated. If optimizer  requires
  value exactly at the boundary between "patches" (say, at x=0 for f=abs(x)),
  where gradient is not defined, user may resolve tie arbitrarily (in  our
  case - return +1 or -1 at its discretion).
* NS solver supports numerical differentiation, i.e. it may  differentiate
  your function for you,  but  it  results  in  2N  increase  of  function
  evaluations. Not recommended unless you solve really small problems. See
  minnscreatef() for more information on this functionality.

USAGE:

1. User initializes algorithm state with MinNSCreate() call  and   chooses
   what NLC solver to use. There is some solver which is used by  default,
   with default settings, but you should NOT rely on  default  choice.  It
   may change in future releases of ALGLIB without notice, and no one  can
   guarantee that new solver will be  able  to  solve  your  problem  with
   default settings.

   From the other side, if you choose solver explicitly, you can be pretty
   sure that it will work with new ALGLIB releases.

   In the current release following solvers can be used:
   * AGS solver (activated with MinNSSetAlgoAGS() function)

2. User adds boundary and/or linear and/or nonlinear constraints by  means
   of calling one of the following functions:
   a) MinNSSetBC() for boundary constraints
   b) MinNSSetLC() for linear constraints
   c) MinNSSetNLC() for nonlinear constraints
   You may combine (a), (b) and (c) in one optimization problem.

3. User sets scale of the variables with MinNSSetScale() function. It   is
   VERY important to set  scale  of  the  variables,  because  nonlinearly
   constrained problems are hard to solve when variables are badly scaled.

4. User sets stopping conditions with MinNSSetCond().

5. Finally, user calls MinNSOptimize()  function  which  takes   algorithm
   state and pointer (delegate, etc) to callback function which calculates
   F/G/H.

7. User calls MinNSResults() to get solution

8. Optionally user may call MinNSRestartFrom() to solve   another  problem
   with same N but another starting point. MinNSRestartFrom()  allows   to
   reuse already initialized structure.


INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

NOTE: minnscreatef() function may be used if  you  do  not  have  analytic
      gradient.   This   function  creates  solver  which  uses  numerical
      differentiation with user-specified step.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnscreate(const ae_int_t n, const real_1d_array &x, minnsstate &state, const xparams _xparams = alglib::xdefault);
void minnscreate(const real_1d_array &x, minnsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`minnscreatef` function

/*************************************************************************
Version of minnscreatef() which uses numerical differentiation. I.e.,  you
do not have to calculate derivatives yourself. However, this version needs
2N times more function evaluations.

2-point differentiation formula is  used,  because  more  precise  4-point
formula is unstable when used on non-smooth functions.

INPUT PARAMETERS:
    N       -   problem dimension, N>0:
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    X       -   starting point, array[N]:
                * it is better to set X to a feasible point
                * but X can be infeasible, in which case algorithm will try
                  to find feasible point first, using X as initial
                  approximation.
    DiffStep-   differentiation  step,  DiffStep>0.   Algorithm   performs
                numerical differentiation  with  step  for  I-th  variable
                being equal to DiffStep*S[I] (here S[] is a  scale vector,
                set by minnssetscale() function).
                Do not use  too  small  steps,  because  it  may  lead  to
                catastrophic cancellation during intermediate calculations.

OUTPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnscreatef(const ae_int_t n, const real_1d_array &x, const double diffstep, minnsstate &state, const xparams _xparams = alglib::xdefault);
void minnscreatef(const real_1d_array &x, const double diffstep, minnsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minnsiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool minnsiteration(minnsstate &state, const xparams _xparams = alglib::xdefault);

`minnsoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


NOTES:

1. This function has two different implementations: one which  uses  exact
   (analytical) user-supplied Jacobian, and one which uses  only  function
   vector and numerically  differentiates  function  in  order  to  obtain
   gradient.

   Depending  on  the  specific  function  used to create optimizer object
   you should choose appropriate variant of  minnsoptimize() -  one  which
   accepts function AND Jacobian or one which accepts ONLY function.

   Be careful to choose variant of minnsoptimize()  which  corresponds  to
   your optimization scheme! Table below lists different  combinations  of
   callback (function/gradient) passed to minnsoptimize()    and  specific
   function used to create optimizer.


                     |         USER PASSED TO minnsoptimize()
   CREATED WITH      |  function only   |  function and gradient
   ------------------------------------------------------------
   minnscreatef()    |     works               FAILS
   minnscreate()     |     FAILS               works

   Here "FAILS" denotes inappropriate combinations  of  optimizer creation
   function  and  minnsoptimize()  version.   Attemps   to    use     such
   combination will lead to exception. Either  you  did  not pass gradient
   when it WAS needed or you passed gradient when it was NOT needed.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnsoptimize(minnsstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void minnsoptimize(minnsstate &state,
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnsrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnsrequesttermination(minnsstate &state, const xparams _xparams = alglib::xdefault);

`minnsrestartfrom` function

/*************************************************************************
This subroutine restarts algorithm from new point.
All optimization parameters (including constraints) are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have  same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnscreate() call.
    X       -   new starting point.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnsrestartfrom(minnsstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`minnsresults` function

/*************************************************************************
MinNS results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report. You should check Rep.TerminationType
                in  order  to  distinguish  successful  termination  from
                unsuccessful one:
                * -8   internal integrity control  detected  infinite  or
                       NAN   values   in   function/gradient.    Abnormal
                       termination signalled.
                * -3   box constraints are inconsistent
                * -1   inconsistent parameters were passed:
                       * penalty parameter for minnssetalgoags() is zero,
                         but we have nonlinear constraints set by minnssetnlc()
                *  2   sampling radius decreased below epsx
                *  7    stopping conditions are too stringent,
                        further improvement is impossible,
                        X contains best point found so far.
                *  8    User requested termination via minnsrequesttermination()

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnsresults(const minnsstate &state, real_1d_array &x, minnsreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnsresultsbuf` function

/*************************************************************************

Buffered implementation of minnsresults() which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnsresultsbuf(const minnsstate &state, real_1d_array &x, minnsreport &rep, const xparams _xparams = alglib::xdefault);

`minnssetalgoags` function

/*************************************************************************
This function tells MinNS unit to use  AGS  (adaptive  gradient  sampling)
algorithm for nonsmooth constrained  optimization.  This  algorithm  is  a
slight modification of one described in  "An  Adaptive  Gradient  Sampling
Algorithm for Nonsmooth Optimization" by Frank E. Curtisy and Xiaocun Quez.

This optimizer has following benefits and drawbacks:
+ robustness; it can be used with nonsmooth and nonconvex functions.
+ relatively easy tuning; most of the metaparameters are easy to select.
- it has convergence of steepest descent, slower than CG/LBFGS.
- each iteration involves evaluation of ~2N gradient values  and  solution
  of 2Nx2N quadratic programming problem, which  limits  applicability  of
  algorithm by small-scale problems (up to 50-100).

IMPORTANT: this  algorithm  has  convergence  guarantees,   i.e.  it  will
           steadily move towards some stationary point of the function.

           However, "stationary point" does not  always  mean  "solution".
           Nonsmooth problems often have "flat spots",  i.e.  areas  where
           function do not change at all. Such "flat spots" are stationary
           points by definition, and algorithm may be caught here.

           Nonsmooth CONVEX tasks are not prone to  this  problem. Say, if
           your function has form f()=MAX(f0,f1,...), and f_i are  convex,
           then f() is convex too and you have guaranteed  convergence  to
           solution.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Radius  -   initial sampling radius, >=0.

                Internally multiplied  by  vector of  per-variable  scales
                specified by minnssetscale()).

                You should select relatively large sampling radius, roughly
                proportional to scaled length of the first  steps  of  the
                algorithm. Something close to 0.1 in magnitude  should  be
                good for most problems.

                AGS solver can automatically decrease radius, so too large
                radius is  not a problem (assuming that you  won't  choose
                so large radius that algorithm  will  sample  function  in
                too far away points, where gradient value is irrelevant).

                Too small radius won't cause algorithm to fail, but it may
                slow down algorithm (it may  have  to  perform  too  short
                steps).
    Penalty -   penalty coefficient for nonlinear constraints:
                * for problem with nonlinear constraints  should  be  some
                  problem-specific  positive   value,  large  enough  that
                  penalty term changes shape of the function.
                  Starting  from  some  problem-specific   value   penalty
                  coefficient becomes  large  enough  to  exactly  enforce
                  nonlinear constraints;  larger  values  do  not  improve
                  precision.
                  Increasing it too much may slow down convergence, so you
                  should choose it carefully.
                * can be zero for problems WITHOUT  nonlinear  constraints
                  (i.e. for unconstrained ones or ones with  just  box  or
                  linear constraints)
                * if you specify zero value for problem with at least  one
                  nonlinear  constraint,  algorithm  will  terminate  with
                  error code -1.

ALGORITHM OUTLINE

The very basic outline of unconstrained AGS algorithm is given below:

0. If sampling radius is below EpsX  or  we  performed  more  then  MaxIts
   iterations - STOP.
1. sample O(N) gradient values at random locations  around  current point;
   informally speaking, this sample is an implicit piecewise  linear model
   of the function, although algorithm formulation does  not  mention that
   explicitly
2. solve quadratic programming problem in order to find descent direction
3. if QP solver tells us that we  are  near  solution,  decrease  sampling
   radius and move to (0)
4. perform backtracking line search
5. after moving to new point, goto (0)

Constraint handling details:
* box constraints are handled exactly by algorithm
* linear/nonlinear constraints are handled by adding L1  penalty.  Because
  our solver can handle nonsmoothness, we can  use  L1  penalty  function,
  which is an exact one  (i.e.  exact  solution  is  returned  under  such
  penalty).
* penalty coefficient for  linear  constraints  is  chosen  automatically;
  however, penalty coefficient for nonlinear constraints must be specified
  by user.

===== TRACING AGS SOLVER =================================================

AGS solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'AGS'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'AGS.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'AGS'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format
* 'AGS.DETAILED.SAMPLE'-
                  for output of points being visited ,  search  directions
                  and gradient sample. May take a LOT of  space ,  do  not
                  use it on problems with more that several tens of vars.
                  This  symbol   also    implicitly   defines   'AGS'  and
                  'AGS.DETAILED'.

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("AGS,PREC.F6", "path/to/trace.log")
>


  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnssetalgoags(minnsstate &state, const double radius, const double penalty, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnssetbc` function

/*************************************************************************
This function sets boundary constraints.

Boundary constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with minnsrestartfrom().

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF.
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF.

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: AGS solver has following useful properties:
* bound constraints are always satisfied exactly
* function is evaluated only INSIDE area specified by  bound  constraints,
  even  when  numerical  differentiation is used (algorithm adjusts  nodes
  according to boundary constraints)

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnssetbc(minnsstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minnssetcond` function

/*************************************************************************
This function sets stopping conditions for iterations of optimizer.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0
                The AGS solver finishes its work if  on  k+1-th  iteration
                sampling radius decreases below EpsX.
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to  automatic
stopping criterion selection. We do not recommend you to rely  on  default
choice in production code.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnssetcond(minnsstate &state, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnssetlc` function

/*************************************************************************
This function sets linear constraints.

Linear constraints are inactive by default (after initial creation).
They are preserved after algorithm restart with minnsrestartfrom().

INPUT PARAMETERS:
    State   -   structure previously allocated with minnscreate() call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE: linear (non-bound) constraints are satisfied only approximately:

* there always exists some minor violation (about current sampling  radius
  in magnitude during optimization, about EpsX in the solution) due to use
  of penalty method to handle constraints.
* numerical differentiation, if used, may  lead  to  function  evaluations
  outside  of the feasible  area,   because   algorithm  does  NOT  change
  numerical differentiation formula according to linear constraints.

If you want constraints to be  satisfied  exactly, try to reformulate your
problem  in  such  manner  that  all constraints will become boundary ones
(this kind of constraints is always satisfied exactly, both in  the  final
solution and in all intermediate points).

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnssetlc(minnsstate &state, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minnssetlc(minnsstate &state, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

`minnssetnlc` function

/*************************************************************************
This function sets nonlinear constraints.

In fact, this function sets NUMBER of nonlinear  constraints.  Constraints
itself (constraint functions) are passed to minnsoptimize() method.   This
method requires user-defined vector function F[]  and  its  Jacobian  J[],
where:
* first component of F[] and first row  of  Jacobian  J[]  correspond   to
  function being minimized
* next NLEC components of F[] (and rows  of  J)  correspond  to  nonlinear
  equality constraints G_i(x)=0
* next NLIC components of F[] (and rows  of  J)  correspond  to  nonlinear
  inequality constraints H_i(x)<=0

NOTE: you may combine nonlinear constraints with linear/boundary ones.  If
      your problem has mixed constraints, you  may explicitly specify some
      of them as linear ones. It may help optimizer to  handle  them  more
      efficiently.

INPUT PARAMETERS:
    State   -   structure previously allocated with minnscreate() call.
    NLEC    -   number of Non-Linear Equality Constraints (NLEC), >=0
    NLIC    -   number of Non-Linear Inquality Constraints (NLIC), >=0

NOTE 1: nonlinear constraints are satisfied only  approximately!   It   is
        possible   that  algorithm  will  evaluate  function  outside   of
        the feasible area!

NOTE 2: algorithm scales variables  according  to   scale   specified   by
        minnssetscale()  function,  so  it can handle problems with  badly
        scaled variables (as long as we KNOW their scales).

        However,  there  is  no  way  to  automatically  scale   nonlinear
        constraints Gi(x) and Hi(x). Inappropriate scaling  of  Gi/Hi  may
        ruin convergence. Solving problem with  constraint  "1000*G0(x)=0"
        is NOT same as solving it with constraint "0.001*G0(x)=0".

        It  means  that  YOU  are  the  one who is responsible for correct
        scaling of nonlinear constraints Gi(x) and Hi(x). We recommend you
        to scale nonlinear constraints in such way that I-th component  of
        dG/dX (or dH/dx) has approximately unit  magnitude  (for  problems
        with unit scale)  or  has  magnitude approximately equal to 1/S[i]
        (where S is a scale set by minnssetscale() function).

NOTE 3: nonlinear constraints are always hard to handle,  no  matter  what
        algorithm you try to use. Even basic box/linear constraints modify
        function  curvature   by  adding   valleys  and  ridges.  However,
        nonlinear constraints add valleys which are very  hard  to  follow
        due to their "curved" nature.

        It means that optimization with single nonlinear constraint may be
        significantly slower than optimization with multiple linear  ones.
        It is normal situation, and we recommend you to  carefully  choose
        Rho parameter of minnssetalgoags(), because too  large  value  may
        slow down convergence.


  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnssetnlc(minnsstate &state, const ae_int_t nlec, const ae_int_t nlic, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minnssetscale` function

/*************************************************************************
This function sets scaling coefficients for NLC optimizer.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Scaling is also used by finite difference variant of the optimizer  - step
along I-th axis is equal to DiffStep*S[I].

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 18.05.2015 by Bochkanov Sergey
*************************************************************************/
void minnssetscale(minnsstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`minnssetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to minnsoptimize().

  -- ALGLIB --
     Copyright 28.11.2010 by Bochkanov Sergey
*************************************************************************/
void minnssetxrep(minnsstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

minns_d_bc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //
    // and Jacobian matrix J = [df0/dx0 df0/dx1]
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
    jac[0][0] = 2*alglib::sign(x[0]);
    jac[0][1] = alglib::sign(x[1]);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = 2*|x0|+|x1|
        //
        // subject to box constraints
        //
        //        1 <= x0 < +INF
        //     -INF <= x1 < +INF
        //
        // using nonsmooth nonlinear optimizer.
        //
        real_1d_array x0 = "[1,1]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[1,-inf]";
        real_1d_array bndu = "[+inf,+inf]";
        double epsx = 0.00001;
        double radius = 0.1;
        double rho = 0.0;
        ae_int_t maxits = 0;
        minnsstate state;
        minnsreport rep;
        real_1d_array x1;

        //
        // Create optimizer object, choose AGS algorithm and tune its settings:
        // * radius=0.1     good initial value; will be automatically decreased later.
        // * rho=0.0        penalty coefficient for nonlinear constraints; can be zero
        //                  because we do not have such constraints
        // * epsx=0.000001  stopping conditions
        // * s=[1,1]        all variables have unit scale
        //
        minnscreate(2, x0, state);
        minnssetalgoags(state, radius, rho);
        minnssetcond(state, epsx, maxits);
        minnssetscale(state, s);

        //
        // Set box constraints.
        //
        // General linear constraints are set in similar way (see comments on
        // minnssetlc() function for more information).
        //
        // You may combine box, linear and nonlinear constraints in one optimization
        // problem.
        //
        minnssetbc(state, bndl, bndu);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear equality and inequality constraints
        // (box/linear ones are passed separately by means of minnssetbc() and
        // minnssetlc() calls).
        //
        // If you do not have nonlinear constraints (exactly our situation), then
        // you will have one-component function vector and 1xN Jacobian matrix.
        //
        // So, our vector function has form
        //
        //     {f0} = { 2*|x0|+|x1| }
        //
        // with Jacobian
        //
        //         [                       ]
        //     J = [ 2*sign(x0)   sign(x1) ]
        //         [                       ]
        //
        // NOTE: nonsmooth optimizer requires considerably more function
        //       evaluations than smooth solver - about 2N times more. Using
        //       numerical differentiation introduces additional (multiplicative)
        //       2N speedup.
        //
        //       It means that if smooth optimizer WITH user-supplied gradient
        //       needs 100 function evaluations to solve 50-dimensional problem,
        //       then AGS solver with user-supplied gradient will need about 10.000
        //       function evaluations, and with numerical gradient about 1.000.000
        //       function evaluations will be performed.
        //
        // NOTE: AGS solver used by us can handle nonsmooth and nonconvex
        //       optimization problems. It has convergence guarantees, i.e. it will
        //       converge to stationary point of the function after running for some
        //       time.
        //
        //       However, it is important to remember that "stationary point" is not
        //       equal to "solution". If your problem is convex, everything is OK.
        //       But nonconvex optimization problems may have "flat spots" - large
        //       areas where gradient is exactly zero, but function value is far away
        //       from optimal. Such areas are stationary points too, and optimizer
        //       may be trapped here.
        //
        //       "Flat spots" are nonsmooth equivalent of the saddle points, but with
        //       orders of magnitude worse properties - they may be quite large and
        //       hard to avoid. All nonsmooth optimizers are prone to this kind of the
        //       problem, because it is impossible to automatically distinguish "flat
        //       spot" from true solution.
        //
        //       This note is here to warn you that you should be very careful when
        //       you solve nonsmooth optimization problems. Visual inspection of
        //       results is essential.
        //
        //
        alglib::minnsoptimize(state, nsfunc1_jac);
        minnsresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minns_d_diff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = 2*|x0|+|x1|
        //
        // using nonsmooth nonlinear optimizer with numerical
        // differentiation provided by ALGLIB.
        //
        // NOTE: nonsmooth optimizer requires considerably more function
        //       evaluations than smooth solver - about 2N times more. Using
        //       numerical differentiation introduces additional (multiplicative)
        //       2N speedup.
        //
        //       It means that if smooth optimizer WITH user-supplied gradient
        //       needs 100 function evaluations to solve 50-dimensional problem,
        //       then AGS solver with user-supplied gradient will need about 10.000
        //       function evaluations, and with numerical gradient about 1.000.000
        //       function evaluations will be performed.
        //
        real_1d_array x0 = "[1,1]";
        real_1d_array s = "[1,1]";
        double epsx = 0.00001;
        double diffstep = 0.000001;
        double radius = 0.1;
        double rho = 0.0;
        ae_int_t maxits = 0;
        minnsstate state;
        minnsreport rep;
        real_1d_array x1;

        //
        // Create optimizer object, choose AGS algorithm and tune its settings:
        // * radius=0.1     good initial value; will be automatically decreased later.
        // * rho=0.0        penalty coefficient for nonlinear constraints; can be zero
        //                  because we do not have such constraints
        // * epsx=0.000001  stopping conditions
        // * s=[1,1]        all variables have unit scale
        //
        minnscreatef(2, x0, diffstep, state);
        minnssetalgoags(state, radius, rho);
        minnssetcond(state, epsx, maxits);
        minnssetscale(state, s);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function, with first component
        // being target function, and next components being nonlinear equality
        // and inequality constraints (box/linear ones are passed separately
        // by means of minnssetbc() and minnssetlc() calls).
        //
        // If you do not have nonlinear constraints (exactly our situation), then
        // you will have one-component function vector.
        //
        // So, our vector function has form
        //
        //     {f0} = { 2*|x0|+|x1| }
        //
        alglib::minnsoptimize(state, nsfunc1_fvec);
        minnsresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.0000,0.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minns_d_nlc example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates function vector
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //     f1(x0,x1) = x0-1
    //     f2(x0,x1) = -x1-1
    //
    // and Jacobian matrix J
    //
    //         [ df0/dx0   df0/dx1 ]
    //     J = [ df1/dx0   df1/dx1 ]
    //         [ df2/dx0   df2/dx1 ]
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
    jac[0][0] = 2*alglib::sign(x[0]);
    jac[0][1] = alglib::sign(x[1]);
    fi[1] = x[0]-1;
    jac[1][0] = 1;
    jac[1][1] = 0;
    fi[2] = -x[1]-1;
    jac[2][0] = 0;
    jac[2][1] = -1;
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = 2*|x0|+|x1|
        //
        // subject to combination of equality and inequality constraints
        //
        //      x0  =  1
        //      x1 >= -1
        //
        // using nonsmooth nonlinear optimizer. Although these constraints
        // are linear, we treat them as general nonlinear ones in order to
        // demonstrate nonlinearly constrained optimization setup.
        //
        real_1d_array x0 = "[1,1]";
        real_1d_array s = "[1,1]";
        double epsx = 0.00001;
        double radius = 0.1;
        double rho = 50.0;
        ae_int_t maxits = 0;
        minnsstate state;
        minnsreport rep;
        real_1d_array x1;

        //
        // Create optimizer object, choose AGS algorithm and tune its settings:
        // * radius=0.1     good initial value; will be automatically decreased later.
        // * rho=50.0       penalty coefficient for nonlinear constraints. It is your
        //                  responsibility to choose good one - large enough that it
        //                  enforces constraints, but small enough in order to avoid
        //                  extreme slowdown due to ill-conditioning.
        // * epsx=0.000001  stopping conditions
        // * s=[1,1]        all variables have unit scale
        //
        minnscreate(2, x0, state);
        minnssetalgoags(state, radius, rho);
        minnssetcond(state, epsx, maxits);
        minnssetscale(state, s);

        //
        // Set general nonlinear constraints.
        //
        // This part is more tricky than working with box/linear constraints - you
        // can not "pack" general nonlinear function into double precision array.
        // That's why minnssetnlc() does not accept constraints itself - only
        // constraint COUNTS are passed: first parameter is number of equality
        // constraints, second one is number of inequality constraints.
        //
        // As for constraining functions - these functions are passed as part
        // of problem Jacobian (see below).
        //
        // NOTE: MinNS optimizer supports arbitrary combination of boundary, general
        //       linear and general nonlinear constraints. This example does not
        //       show how to work with general linear constraints, but you can
        //       easily find it in documentation on minnlcsetlc() function.
        //
        minnssetnlc(state, 1, 1);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear equality and inequality constraints
        // (box/linear ones are passed separately by means of minnssetbc() and
        // minnssetlc() calls).
        //
        // Nonlinear equality constraints have form Gi(x)=0, inequality ones
        // have form Hi(x)<=0, so we may have to "normalize" constraints prior
        // to passing them to optimizer (right side is zero, constraints are
        // sorted, multiplied by -1 when needed).
        //
        // So, our vector function has form
        //
        //     {f0,f1,f2} = { 2*|x0|+|x1|,  x0-1, -x1-1 }
        //
        // with Jacobian
        //
        //         [ 2*sign(x0)   sign(x1) ]
        //     J = [     1           0     ]
        //         [     0          -1     ]
        //
        // which means that we have optimization problem
        //
        //     min{f0} subject to f1=0, f2<=0
        //
        // which is essentially same as
        //
        //     min { 2*|x0|+|x1| } subject to x0=1, x1>=-1
        //
        // NOTE: AGS solver used by us can handle nonsmooth and nonconvex
        //       optimization problems. It has convergence guarantees, i.e. it will
        //       converge to stationary point of the function after running for some
        //       time.
        //
        //       However, it is important to remember that "stationary point" is not
        //       equal to "solution". If your problem is convex, everything is OK.
        //       But nonconvex optimization problems may have "flat spots" - large
        //       areas where gradient is exactly zero, but function value is far away
        //       from optimal. Such areas are stationary points too, and optimizer
        //       may be trapped here.
        //
        //       "Flat spots" are nonsmooth equivalent of the saddle points, but with
        //       orders of magnitude worse properties - they may be quite large and
        //       hard to avoid. All nonsmooth optimizers are prone to this kind of the
        //       problem, because it is impossible to automatically distinguish "flat
        //       spot" from true solution.
        //
        //       This note is here to warn you that you should be very careful when
        //       you solve nonsmooth optimization problems. Visual inspection of
        //       results is essential.
        //
        alglib::minnsoptimize(state, nsfunc2_jac);
        minnsresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minns_d_unconstrained example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  nsfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
    //
    // this callback calculates
    //
    //     f0(x0,x1) = 2*|x0|+x1
    //
    // and Jacobian matrix J = [df0/dx0 df0/dx1]
    //
    fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
    jac[0][0] = 2*alglib::sign(x[0]);
    jac[0][1] = alglib::sign(x[1]);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of
        //
        //     f(x0,x1) = 2*|x0|+|x1|
        //
        // using nonsmooth nonlinear optimizer.
        //
        real_1d_array x0 = "[1,1]";
        real_1d_array s = "[1,1]";
        double epsx = 0.00001;
        double radius = 0.1;
        double rho = 0.0;
        ae_int_t maxits = 0;
        minnsstate state;
        minnsreport rep;
        real_1d_array x1;

        //
        // Create optimizer object, choose AGS algorithm and tune its settings:
        // * radius=0.1     good initial value; will be automatically decreased later.
        // * rho=0.0        penalty coefficient for nonlinear constraints; can be zero
        //                  because we do not have such constraints
        // * epsx=0.000001  stopping conditions
        // * s=[1,1]        all variables have unit scale
        //
        minnscreate(2, x0, state);
        minnssetalgoags(state, radius, rho);
        minnssetcond(state, epsx, maxits);
        minnssetscale(state, s);

        //
        // Optimize and test results.
        //
        // Optimizer object accepts vector function and its Jacobian, with first
        // component (Jacobian row) being target function, and next components
        // (Jacobian rows) being nonlinear equality and inequality constraints
        // (box/linear ones are passed separately by means of minnssetbc() and
        // minnssetlc() calls).
        //
        // If you do not have nonlinear constraints (exactly our situation), then
        // you will have one-component function vector and 1xN Jacobian matrix.
        //
        // So, our vector function has form
        //
        //     {f0} = { 2*|x0|+|x1| }
        //
        // with Jacobian
        //
        //         [                       ]
        //     J = [ 2*sign(x0)   sign(x1) ]
        //         [                       ]
        //
        // NOTE: nonsmooth optimizer requires considerably more function
        //       evaluations than smooth solver - about 2N times more. Using
        //       numerical differentiation introduces additional (multiplicative)
        //       2N speedup.
        //
        //       It means that if smooth optimizer WITH user-supplied gradient
        //       needs 100 function evaluations to solve 50-dimensional problem,
        //       then AGS solver with user-supplied gradient will need about 10.000
        //       function evaluations, and with numerical gradient about 1.000.000
        //       function evaluations will be performed.
        //
        // NOTE: AGS solver used by us can handle nonsmooth and nonconvex
        //       optimization problems. It has convergence guarantees, i.e. it will
        //       converge to stationary point of the function after running for some
        //       time.
        //
        //       However, it is important to remember that "stationary point" is not
        //       equal to "solution". If your problem is convex, everything is OK.
        //       But nonconvex optimization problems may have "flat spots" - large
        //       areas where gradient is exactly zero, but function value is far away
        //       from optimal. Such areas are stationary points too, and optimizer
        //       may be trapped here.
        //
        //       "Flat spots" are nonsmooth equivalent of the saddle points, but with
        //       orders of magnitude worse properties - they may be quite large and
        //       hard to avoid. All nonsmooth optimizers are prone to this kind of the
        //       problem, because it is impossible to automatically distinguish "flat
        //       spot" from true solution.
        //
        //       This note is here to warn you that you should be very careful when
        //       you solve nonsmooth optimization problems. Visual inspection of
        //       results is essential.
        //
        alglib::minnsoptimize(state, nsfunc1_jac);
        minnsresults(state, x1, rep);
        printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.0000,0.0000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`minqp` subpackage

Classes

minqpreport
minqpstate

Functions

minqpaddlc2
minqpaddlc2dense
minqpaddlc2sparsefromdense
minqpaddpowccorthogonal
minqpaddpowccprimitive
minqpaddqc2
minqpaddqc2dense
minqpaddqc2list
minqpaddsoccorthogonal
minqpaddsoccprimitive
minqpclearcc
minqpclearqc
minqpcreate
minqpexport
minqpimport
minqpoptimize
minqpresults
minqpresultsbuf
minqpsetalgodenseaul
minqpsetalgodensegenipm
minqpsetalgodenseipm
minqpsetalgoquickqp
minqpsetalgosparseecqp
minqpsetalgosparsegenipm
minqpsetalgosparseipm
minqpsetbc
minqpsetbcall
minqpsetbci
minqpsetlc
minqpsetlc2
minqpsetlc2dense
minqpsetlc2mixed
minqpsetlcmixed
minqpsetlcmixedlegacy
minqpsetlcsparse
minqpsetlinearterm
minqpsetorigin
minqpsetquadraticterm
minqpsetquadratictermsparse
minqpsetscale
minqpsetscaleautodiag
minqpsetstartingpoint

Examples

minqp_conic		L1-penalized quadratic programming using conic constraints
minqp_d_bc1		Box constrained dense quadratic programming
minqp_d_lc1		Linearly constrained dense quadratic programming
minqp_d_nonconvex		Nonconvex quadratic programming
minqp_d_u1		Unconstrained dense quadratic programming
minqp_d_u2		Unconstrained sparse quadratic programming

`minqpreport` class

/*************************************************************************
This structure stores optimization report:
* InnerIterationsCount      number of inner iterations
* OuterIterationsCount      number of outer iterations
* NCholesky                 number of Cholesky decomposition
* NMV                       number of matrix-vector products
                            (only products calculated as part of iterative
                            process are counted)
* TerminationType           completion code (see below)
* F                         for positive terminationtype stores quadratic
                            model value at the solution
* LagBC                     Lagrange multipliers for box constraints,
                            array[N]
* LagLC                     Lagrange multipliers for linear constraints,
                            array[MSparse+MDense]
* LagQC                     Lagrange multipliers for quadratic constraints

=== COMPLETION CODES =====================================================

Completion codes:
* -9    failure of the automatic scale evaluation:  one  of  the  diagonal
        elements of the quadratic term is non-positive.  Specify  variable
        scales manually!
* -5    inappropriate solver was used:
        * QuickQP solver for a problem with general linear constraints (dense/sparse)
        * QuickQP/DENSE-AUL/DENSE-IPM/SPARSE-IPM for a problem with
          quadratic/conic constraints
        * ECQP for a problem with inequality or nonlinear equality constraints
* -4    the problem is highly likely to be unbounded; either one of the solvers
        found an unconstrained direction of negative curvature, or objective
        simply decreased for too much (more than 1E50).
* -3    inconsistent constraints (or, maybe, feasible point is
        too hard to find). If you are sure that constraints are feasible,
        try to restart optimizer with better initial approximation.
* -2    IPM solver has difficulty finding primal/dual feasible point.
        It is likely that the problem is either infeasible or unbounded,
        but it is difficult to determine exact reason for termination.
        X contains best point found so far.
*  1..4 successful completion
*  5    MaxIts steps was taken
*  7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.

=== LAGRANGE MULTIPLIERS =================================================

Some  optimizers  report  values of  Lagrange  multipliers  on  successful
completion (positive completion code):
* dense and sparse IPM/GENIPM return very precise Lagrange  multipliers as
  determined during solution process.
* DENSE-AUL-QP returns approximate Lagrange multipliers  (which  are  very
  close to "true"  Lagrange  multipliers  except  for  overconstrained  or
  degenerate problems)

Three arrays of multipliers are returned:
* LagBC is array[N] which is loaded with multipliers from box constraints;
  LagBC[i]>0 means that I-th constraint is at the  upper bound, LagBC[I]<0
  means that I-th constraint is at the lower bound, LagBC[I]=0 means  that
  I-th box constraint is inactive.
* LagLC is array[MSparse+MDense] which is  loaded  with  multipliers  from
  general  linear  constraints  (former  MSparse  elements  corresponds to
  sparse part of the constraint matrix, latter MDense are  for  the  dense
  constraints, as was specified by user).
  LagLC[i]>0 means that I-th constraint at  the  upper  bound,  LagLC[i]<0
  means that I-th constraint is at the lower bound, LagLC[i]=0 means  that
  I-th linear constraint is inactive.
* LagQC is array[MQC]  which stores multipliers for quadratic constraints.
  LagQC[i]>0 means that I-th constraint at  the  upper  bound,  LagQC[i]<0
  means that I-th constraint is at the lower bound, LagQC[i]=0 means  that
  I-th linear constraint is inactive.

On failure (or when optimizer does not support Lagrange multipliers) these
arrays are zero-filled.

It is expected that at solution the dual feasibility condition holds:

    C+H*(Xs-X0) + SUM(Ei*LagBC[i],i=0..n-1) + SUM(Ai*LagLC[i],i=0..m-1) + ... ~ 0

where
* C is a linear term
* H is a quadratic term
* Xs is a solution, and X0 is an origin term (zero by default)
* Ei is a vector with 1.0 at position I and 0 in other positions
* Ai is an I-th row of linear constraint matrix

NOTE: methods  from  IPM  family  may  also  return  meaningful   Lagrange
      multipliers  on  completion   with   code   -2   (infeasibility   or
      unboundedness  detected).
*************************************************************************/
class minqpreport
{
public:
    minqpreport();
    minqpreport(const minqpreport &rhs);
    minqpreport& operator=(const minqpreport &rhs);
    virtual ~minqpreport();
    ae_int_t inneriterationscount;
    ae_int_t outeriterationscount;
    ae_int_t nmv;
    ae_int_t ncholesky;
    ae_int_t terminationtype;
    double f;
    real_1d_array lagbc;
    real_1d_array laglc;
    real_1d_array lagqc;
};

`minqpstate` class

/*************************************************************************
This object stores nonlinear optimizer state.
You should use functions provided by MinQP subpackage to work with this
object
*************************************************************************/
class minqpstate
{
public:
    minqpstate();
    minqpstate(const minqpstate &rhs);
    minqpstate& operator=(const minqpstate &rhs);
    virtual ~minqpstate();
};

`minqpaddlc2` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present sparse constraints.

Constraint is passed in compressed format: as list of non-zero entries  of
coefficient vector A. Such approach is more efficient than  dense  storage
for highly sparse constraint vectors.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    IdxA    -   array[NNZ], indexes of non-zero elements of A:
                * can be unsorted
                * can include duplicate indexes (corresponding entries  of
                  ValA[] will be summed)
    ValA    -   array[NNZ], values of non-zero elements of A
    NNZ     -   number of non-zero coefficients in A
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minqpaddlc2(minqpstate &state, const integer_1d_array &idxa, const real_1d_array &vala, const ae_int_t nnz, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minqpaddlc2dense` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
matrix of currently present dense constraints.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    A       -   linear constraint coefficient, array[N], right side is NOT
                included.
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint Ai*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint Ai*x<=AU
                * AU=+INF  => one-sided constraint AL<=Ai*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minqpaddlc2dense(minqpstate &state, const real_1d_array &a, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minqpaddlc2sparsefromdense` function

/*************************************************************************
This function appends two-sided linear constraint  AL <= A*x <= AU  to the
list of currently present sparse constraints.

Constraint vector A is  passed  as  a  dense  array  which  is  internally
sparsified by this function.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    DA      -   array[N], constraint vector
    AL, AU  -   lower and upper bounds;
                * AL=AU    => equality constraint A*x
                * AL<AU    => two-sided constraint AL<=A*x<=AU
                * AL=-INF  => one-sided constraint A*x<=AU
                * AU=+INF  => one-sided constraint AL<=A*x
                * AL=-INF, AU=+INF => constraint is ignored

  -- ALGLIB --
     Copyright 19.07.2018 by Bochkanov Sergey
*************************************************************************/
void minqpaddlc2sparsefromdense(minqpstate &state, const real_1d_array &da, const double al, const double au, const xparams _xparams = alglib::xdefault);

`minqpaddpowccorthogonal` function

/*************************************************************************
This function appends an axis-orthogonal power cone constraint of the form

         (          k-kp-1      )     k-1
    sqrt ( theta^2 + SUM y[i]^2 ) <=  MUL y[i]^alpha[i]
         (           i=0        )    i=k-kp

where

    y[i] = a[i]*x[idx[i]]+c[i], y[i]>=0

    0<alpha[i]<1, with SUM(alpha[i])<=1

    1<=kp<=k, with kp=k meaning that we have MUL(y[i]^alpha[i])>=|theta|

Alternatively, if ApplyOrigin parameter  is  True,  x[i]  is  replaced  by
x[i]-origin[i] (applies to all variables).

Unlike many other conic solvers, ALGLIB provides a flexible conic API that
allows alpha[] to sum up to any positive value less than  or  equal  to  1
(e.g. it is possible to formulate |x|<z^0.33 without  using  slack  vars).
Furthermore, ALGLIB allows conic constraints to overlap, i.e. it allows  a
variable to be a part of multiple conic constraints, or to appear multiple
times in the same constraint.

INPUT PARAMETERS:
    State   -   structure previously allocated  with  minqpcreate()  call.
    Idx     -   array[K] (or larger, only leading  K  elements  are  used)
                storing variable indexes. Indexes can be  unsorted  and/or
                non-distinct.
    A       -   array[K]  (or larger, only leading  K  elements are used),
                variable multipliers. Can contain zero values.
    C       -   array[K] (or larger, only leading  K  elements  are used),
                variable shifts.
    K       -   cone dimensionality, K>=1. It is possible to have K>N.
    Theta   -   additional constant term, can be zero
    AlphaV  -   array[KPow], power coefficients:
                * 0<AlphaV[i]<=1
                * 0<SUM(AlphaV[])<=1
    KPow    -   1<=KPow<=K, with KPow=K being correctly handled.

RESULT:
    constraint index in a conic constraints list, starting from 0

NOTE: power cone constraints are always  convex,  so having them preserves
      convexity of the QP problem.

NOTE: A starting point that is strictly feasible with respect to both  box
      and conic constraints greatly helps the solver to power up; however,
      it will work even without such a point,  albeit  at  somewhat  lower
      performance.

NOTE: a power cone with alpha<1 is sensitive to numerical errors near  the
      origin.  Suppose,  for  example,  that  we  have a constraint of the
      form |y|<=x^alpha with alpha=0.25 and that x is zero at the solution
      Furthermore, suppose that we perturbed x by as little  as  eps=1E-8.
      Because of alpha=0.25 the constraint is  perturbed  by  as  much  as
      eps^(1/4)=0.01!

      Such great sensitivity is  explained  by  the  non-differentiability
      of x^alpha for alpha<1 near x=0.  It  does  not  prevent  the  conic
      solver  from  converging  to   the precise solution, it merely makes
      constraints  extremely  sensitive  to  small   errors,   e.g.   ones
      introduced during the presolve/postsolve, like discussed below.

      The conic solver sometimes has to insert slack variables, most often
      because of constraints referring the same right-hand  side  variable
      twice.   For   example,  for  sqrt(x0^2+x1^2)<=(y0-1)^0.5*(y0+1)^0.5
      it will automatically add t=y0+1 (a linear equality constraint)  and
      will rewrite the  constraint  as  sqrt(x0^2+x1^2)<=(y0-1)^0.5*t^0.5.
      The solver will have no  difficulty  solving  the  problem,  even if
      the optimal t is close to zero. However, the equality y0=t-1 will be
      satisfied only approximately, and even a tiny error will be  greatly
      magnified when evaluating the constraint violation.

      Thus,   one  has  to  be  very  careful  when  evaluating constraint
      violation errors  for  power  cones.  Having  high  error  does  not
      necessarily mean that the solver has failed.

  -- ALGLIB --
     Copyright 09.09.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddpowccorthogonal(minqpstate &state, const integer_1d_array &idx, const real_1d_array &a, const real_1d_array &c, const ae_int_t k, const double theta, const real_1d_array &alphav, const ae_int_t kpow, const bool applyorigin, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpaddpowccprimitive` function

/*************************************************************************
This function appends a primitive power cone constraint of the form

        (                                                  )
    sqrt(x[range0]^2 + x[range0+1]^2 + ... + x[range1-1]^2 ) <= x[axisidx]^alpha
        (                                                  )

or, written in another form,

    (    (                                                  ))^(1/alpha)
    (sqrt(x[range0]^2 + x[range0+1]^2 + ... + x[range1-1]^2 ))           <= x[axisidx]
    (    (                                                  ))

where

    0<alpha<=1

with 'primitive' meaning that there are no per-variable  scales  and  that
variables under the square root have sequential indexes. More general form
of power cone constraints can be specified with minqpaddpowccorthogonal().

Alternatively, if ApplyOrigin parameter  is  True,  x[i]  is  replaced  by
x[i]-origin[i] (applies to all variables).

Unlike many other conic solvers, ALGLIB provides a flexible conic API that
allows alpha to be any positive value less than or equal to 1 (e.g.  it is
possible to formulate |x|<z^0.33 without using  slack vars that are  fixed
at 1.0). Furthermore, ALGLIB allows conic constraints to overlap, i.e.  it
allows a variable to be a part of multiple conic constraints.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    Range0,
    Range1  -   0<=range0<=range1<=N, variable range for the LHS;
                * squared variables x[range0]...x[range1-1] are summed up
                  under the square root.
                * range0=range1 means that the constraint is interpreted
                  as x[AxisIdx]>=0.
    AxisIdx -   RHS variable index:
                * 0<=AxisIdx<N
                * either AxisIdx<range0 or AxisAdx>=Range1.
    Alpha   -   power parameter, 0<alpha<=1

RESULT:
    constraint index in a conic constraints list, starting from 0

NOTE: power cone constraints are always  convex,  so having them preserves
      convexity of the QP problem.

NOTE: A starting point that is strictly feasible with respect to both  box
      and conic constraints greatly helps the solver to power up; however,
      it will work even without such a point,  albeit  at  somewhat  lower
      performance.

NOTE: a power cone with alpha<1 is sensitive to numerical errors near  the
      origin.  Suppose,  for  example,  that  we  have a constraint of the
      form |y|<=x^alpha with alpha=0.25 and that x is zero at the solution
      Furthermore, suppose that we perturbed x by as little  as  eps=1E-8.
      Because of alpha=0.25 the constraint is  perturbed  by  as  much  as
      eps^(1/4)=0.01!

      Such great sensitivity is  explained  by  the  non-differentiability
      of x^alpha for alpha<1 near x=0.  It  does  not  prevent  the  conic
      solver  from  converging  to   the precise solution, it merely makes
      constraints  extremely  sensitive  to  small   errors,   e.g.   ones
      introduced during the presolve/postsolve, like discussed below.

      The conic solver sometimes has to insert slack variables, most often
      because of constraints referring the same right-hand  side  variable
      twice.   For   example,  for  sqrt(x0^2+x1^2)<=(y0-1)^0.5*(y0+1)^0.5
      it will automatically add t=y0+1 (a linear equality constraint)  and
      will rewrite the  constraint  as  sqrt(x0^2+x1^2)<=(y0-1)^0.5*t^0.5.
      The solver will have no  difficulty  solving  the  problem,  even if
      the optimal t is close to zero. However, the equality y0=t-1 will be
      satisfied only approximately, and even a tiny error will be  greatly
      magnified when evaluating the constraint violation.

      Thus,   one  has  to  be  very  careful  when  evaluating constraint
      violation errors  for  power  cones.  Having  high  error  does  not
      necessarily mean that the solver has failed.

  -- ALGLIB --
     Copyright 19.11.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddpowccprimitive(minqpstate &state, const ae_int_t range0, const ae_int_t range1, const ae_int_t axisidx, const double alpha, const bool applyorigin, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpaddqc2` function

/*************************************************************************
This function appends a two-sided quadratic constraint of the form

    CL <= b'x + 0.5*x'*Q*x <= CU

or (depending on the ApplyOrigin parameter)

    CL <= b'(x-origin) + 0.5*(x-origin)'*Q*(x-origin) <= CU

to the set of currently present constraints. The linear term is  given  by
a dense array, the quadratic term is given by a sparse array.

Here CL can be finite or -INF (absense of constraint), CU can be finite or
+INF (absense of constraint), CL<=CU, with  CL=CU  denoting   an  equality
constraint. Q is an arbitrary (including indefinite) symmetric matrix.

The  function  has  O(max(N,NNZ)) memory and time requirements  because  a
dense array is used to store linear term and because  most  sparse  matrix
storage formats supported by ALGLIB need at least O(N) memory even for  an
empty quadratic constraint matrix.

Use minqpaddqc2list() if you have to add many constraints with  much  less
than N nonzero elements.

IMPORTANT: ALGLIB  supports  arbitrary  quadratic  constraints,  including
           nonconvex ones. However, only convex constraints (combined with
           the convex objective) result in guaranteed convergence  to  the
           global minimizer. In all other cases, only local convergence to
           a local minimum is guaranteed.

           A convex constraint is a  constraint  of  the  following  form:
           b'*(x-origin) + 0.5(x-origin)'*Q*(x-origin) <=CU, with Q  being
           a semidefinite matrix. All other modifications  are  nonconvex:
           * -x0^2<=1 is nonconvex
           *  x0^2>=1 is nonconvex (despite Q=1 being positive definite)
           *  x0^2 =1 is nonconvex

           The latter case is notable because it effectively converts a QP
           problem into a mixed integer QP program. Smooth interior  point
           solver can not efficiently handle such programs, converging  to
           a randomly chosen x0 (either +1 or -1) and  keeping  its  value
           fixed during the optimization.

           It is also  notable  that  larger  equality  constraints  (e.g.
           x0^2+x1^2=1) are much less difficult  to  handle  because  they
           form large connected regions within the parameters space.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    Q       -   symmetric matrix Q in a sparse matrix storage format:
                * if IsUpper=True, then the upper triangle  is  given, and
                  the lower triangle is ignored
                * if IsUpper=False, then the lower triangle is  given, and
                  the upper triangle is ignored
                * any sparse matrix storage format present in ALGLIB is
                  supported
                * the matrix must be exactly NxN
    IsUpper -   whether upper or lower triangle of Q is used
    B       -   array[N], linear term
    CL, CU  -   lower and upper bounds:
                * CL can be finite or -INF (absence of a bound)
                * CU can be finite or +INF (absence of a bound)
                * CL<=CU, with CL=CU meaning an equality constraint
                * CL=-INF, CU=+INF => constraint is ignored
    ApplyOrigin-whether origin (as specified by minqpsetorigin) is applied
                to the constraint or not. If no origin was specified, this
                parameter has no effect.

RESULT:
    constraint index, starting from 0

  -- ALGLIB --
     Copyright 19.07.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddqc2(minqpstate &state, const sparsematrix &q, const bool isupper, const real_1d_array &b, const double cl, const double cu, const bool applyorigin, const xparams _xparams = alglib::xdefault);

`minqpaddqc2dense` function

/*************************************************************************
This function appends a two-sided quadratic constraint of the form

    CL <= b'x + 0.5*x'*Q*x <= CU

or (depending on the ApplyOrigin parameter)

    CL <= b'(x-origin) + 0.5*(x-origin)'*Q*(x-origin) <= CU

to the set of currently present constraints. The linear and quadratic terms
are given by dense arrays.

Here CL can be finite or -INF (absense of constraint), CU can be finite or
+INF (absense of constraint), CL<=CU, with  CL=CU  denoting   an  equality
constraint. Q is an arbitrary (including indefinite) symmetric matrix.

This function trades convenience of using dense arrays for the efficiency.
Because dense NxN storage is used, merely calling this function has O(N^2)
complexity, no matter how sparse the Q is.

Use minqpaddqc2() or minqpaddqc2list() if you have sparse  Q  and/or  many
constraints to handle.

IMPORTANT: ALGLIB  supports  arbitrary  quadratic  constraints,  including
           nonconvex ones. However, only convex constraints (combined with
           the convex objective) result in guaranteed convergence  to  the
           global minimizer. In all other cases, only local convergence to
           a local minimum is guaranteed.

           A convex constraint is a  constraint  of  the  following  form:
           b'*(x-origin) + 0.5(x-origin)'*Q*(x-origin) <=CU, with Q  being
           a semidefinite matrix. All other modifications  are  nonconvex:
           * -x0^2<=1 is nonconvex
           *  x0^2>=1 is nonconvex (despite Q=1 being positive definite)
           *  x0^2 =1 is nonconvex

           The latter case is notable because it effectively converts a QP
           problem into a mixed integer QP program. Smooth interior  point
           solver can not efficiently handle such programs, converging  to
           a randomly chosen x0 (either +1 or -1) and  keeping  its  value
           fixed during the optimization.

           It is also  notable  that  larger  equality  constraints  (e.g.
           x0^2+x1^2=1) are much less difficult  to  handle  because  they
           form large connected regions within the parameters space.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    Q       -   array[N,N], symmetric matrix Q:
                * if IsUpper=True, then the upper triangle  is  given, and
                  the lower triangle is ignored
                * if IsUpper=False, then the lower triangle is  given, and
                  the upper triangle is ignored
                * if more than N rows/cols are present, only leading N
                  elements are used
    IsUpper -   whether upper or lower triangle of Q is used
    B       -   array[N], linear term
    CL, CU  -   lower and upper bounds:
                * CL can be finite or -INF (absence of a bound)
                * CU can be finite or +INF (absence of a bound)
                * CL<=CU, with CL=CU meaning an equality constraint
                * CL=-INF, CU=+INF => constraint is ignored
    ApplyOrigin-whether origin (as specified by minqpsetorigin) is applied
                to the constraint or not. If no origin was specified, this
                parameter has no effect.

RESULT:
    constraint index, starting from 0

  -- ALGLIB --
     Copyright 19.06.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddqc2dense(minqpstate &state, const real_2d_array &q, const bool isupper, const real_1d_array &b, const double cl, const double cu, const bool applyorigin, const xparams _xparams = alglib::xdefault);

`minqpaddqc2list` function

/*************************************************************************
This function appends a two-sided quadratic constraint of the form

    CL <= b'x + 0.5*x'*Q*x <= CU

or (depending on the ApplyOrigin parameter)

    CL <= b'(x-origin) + 0.5*(x-origin)'*Q*(x-origin) <= CU

to the set of currently present constraints.  Both  linear  and  quadratic
terms are given as lists of non-zero entries.

Here CL can be finite or -INF (absense of constraint), CU can be finite or
+INF (absense of constraint), CL<=CU, with  CL=CU  denoting   an  equality
constraint. Q is an arbitrary (including indefinite) symmetric matrix.

The function needs O(NNZ) memory for temporaries and  O(NNZ*logNNZ)  time,
where NNZ is a  total  number  of  non-zeros  in  both  lists.  For  small
constraints it can be orders of magnitude faster than  minqpaddqc2()  with
its  O(max(N,NNZ)) temporary memory or minqpaddqc2dense() with its  O(N^2)
temporaries. Thus, it is recommended if you have many small constraints.

NOTE: in  the  end,  all  quadratic  constraints  are  stored  in the same
      memory-efficient compressed format. However, you have to allocate an
      NxN  temporary  dense  matrix  when  you  pass  a  constraint  using
      minqpaddqc2dense(). Similarly, data structures used as a part of the
      API  provided  by   minqpaddqc2()   have   O(N)   temporary   memory
      requirements.

IMPORTANT: ALGLIB  supports  arbitrary  quadratic  constraints,  including
           nonconvex ones. However, only convex constraints (combined with
           the convex objective) result in guaranteed convergence  to  the
           global minimizer. In all other cases, only local convergence to
           a local minimum is guaranteed.

           A convex constraint is a  constraint  of  the  following  form:
           b'*(x-origin) + 0.5(x-origin)'*Q*(x-origin) <=CU, with Q  being
           a semidefinite matrix. All other modifications  are  nonconvex:
           * -x0^2<=1 is nonconvex
           *  x0^2>=1 is nonconvex (despite Q=1 being positive definite)
           *  x0^2 =1 is nonconvex

           The latter case is notable because it effectively converts a QP
           problem into a mixed integer QP program. Smooth interior  point
           solver can not efficiently handle such programs, converging  to
           a randomly chosen x0 (either +1 or -1) and  keeping  its  value
           fixed during the optimization.

           It is also  notable  that  larger  equality  constraints  (e.g.
           x0^2+x1^2=1) are much less difficult  to  handle  because  they
           form large connected regions within the parameters space.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    QRIdx   -   array[QNNZ], stores row indexes of QNNZ  nonzero  elements
                of a symmetric matrix Q
    QCIdx   -   array[QNNZ], stores col indexes of QNNZ  nonzero  elements
                of a symmetric matrix Q
    QVals   -   array[QNNZ], stores values of QNNZ  nonzero  elements of a
                symmetric matrix Q
    QNNZ    -   number of non-zero elements in Q, QNNZ>=0
    IsUpper -   whether upper or lower triangle of Q is used:
                * if IsUpper=True, then only elements with QRIdx[I]<=QCIdx[I]
                  are used and the rest is ignored
                * if IsUpper=False, then only elements with QRIdx[I]>=QCIdx[I]
                  are used and the rest is ignored
    BIdx    -   array[BNNZ], indexes of BNNZ nonzero elements of a linear term
    BVals   -   array[BNNZ], values of BNNZ nonzero elements of a linear term
    BNNZ    -   number of nonzero elements in B, BNNZ>=0
    CL, CU  -   lower and upper bounds:
                * CL can be finite or -INF (absence of a bound)
                * CU can be finite or +INF (absence of a bound)
                * CL<=CU, with CL=CU meaning an equality constraint
                * CL=-INF, CU=+INF => constraint is ignored
    ApplyOrigin-whether origin (as specified by minqpsetorigin) is applied
                to the constraint or not. If no origin was specified, this
                parameter has no effect.

RESULT:
    constraint index, starting from 0

  -- ALGLIB --
     Copyright 19.07.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddqc2list(minqpstate &state, const integer_1d_array &qridx, const integer_1d_array &qcidx, const real_1d_array &qvals, const ae_int_t qnnz, const bool isupper, const integer_1d_array &bidx, const real_1d_array &bvals, const ae_int_t bnnz, const double cl, const double cu, const bool applyorigin, const xparams _xparams = alglib::xdefault);

`minqpaddsoccorthogonal` function

/*************************************************************************
This function appends an axis-orthogonal second-order  conic constraint of
the form

         ( k-2 (                     )^2          )
    sqrt ( SUM ( a[i]*x[idx[i]]+c[i] )  + theta^2 ) <= a[k-1]*x[idx[k-1]]+c[k-1]
         ( i=0 (                     )            )

Alternatively, if ApplyOrigin parameter  is  True,  x[i]  is  replaced  by
x[i]-origin[i] (applies to all variables).

Unlike many other conic solvers, ALGLIB provides a flexible conic API that
allows a[] to have zero elements at arbitrary positions (e.g.,  |x|<=const
can be handled just as easy as |x|<=y). Furthermore, ALGLIB  allows  conic
constraints to overlap, i.e. it allows a variable to be a part of multiple
conic constraints, or to appear multiple times in the same constraint.

NOTE: second-order conic constraints are always  convex,  so  having  them
      preserves convexity of the QP problem.

NOTE: A starting point that is strictly feasible with respect to both  box
      and conic constraints greatly helps the solver to power up; however,
      it will work even without such a point,  albeit  at  somewhat  lower
      performance.

INPUT PARAMETERS:
    State   -   structure previously allocated  with  minqpcreate()  call.
    Idx     -   array[K] (or larger, only leading  K  elements  are  used)
                storing variable indexes. Indexes can be  unsorted  and/or
                non-distinct.
    A       -   array[K]  (or larger, only leading  K  elements are used),
                variable multipliers. Can contain zero values.
    C       -   array[K] (or larger, only leading  K  elements  are used),
                variable shifts.
    K       -   cone dimensionality, K>=1. It is possible to have K>N.
    Theta   -   additional constant term, can be zero

RESULT:
    constraint index in a conic constraints list, starting from 0

  -- ALGLIB --
     Copyright 09.09.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddsoccorthogonal(minqpstate &state, const integer_1d_array &idx, const real_1d_array &a, const real_1d_array &c, const ae_int_t k, const double theta, const bool applyorigin, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpaddsoccprimitive` function

/*************************************************************************
This function appends a primitive second-order  conic  constraint  of  the
form

        (                                                  )
    sqrt(x[range0]^2 + x[range0+1]^2 + ... + x[range1-1]^2 ) <= x[axisidx]
        (                                                  )

with 'primitive' meaning that there are no per-variable  scales  and  that
variables under the square root have sequential indexes. More general form
of conic constraints can be specified with minqpaddsoccorthogonal().

Alternatively, if ApplyOrigin parameter  is  True,  x[i]  is  replaced  by
x[i]-origin[i] (applies to all variables).

Unlike  many  other  conic  solvers,  ALGLIB  allows  conic constraints to
overlap, i.e. it allows  a  variable  to  be  a  part  of  multiple  conic
constraints.

NOTE: second-order conic constraints are always  convex,  so  having  them
      preserves convexity of the QP problem.

NOTE: A starting point that is strictly feasible with respect to both  box
      and conic constraints greatly helps the solver to power up; however,
      it will work even without such a point,  albeit  at  somewhat  lower
      performance.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    Range0,
    Range1  -   0<=range0<=range1<=N, variable range for the LHS;
                * squared variables x[range0]...x[range1-1] are summed up
                  under the square root.
                * range0=range1 means that the constraint is interpreted
                  as x[AxisIdx]>=0.
    AxisIdx -   RHS variable index:
                * 0<=AxisIdx<N
                * either AxisIdx<range0 or AxisAdx>=Range1.

RESULT:
    constraint index in a conic constraints list, starting from 0

  -- ALGLIB --
     Copyright 09.09.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t minqpaddsoccprimitive(minqpstate &state, const ae_int_t range0, const ae_int_t range1, const ae_int_t axisidx, const bool applyorigin, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpclearcc` function

/*************************************************************************
This function clears the list of conic constraints. Other  constraints are
not modified.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.

  -- ALGLIB --
     Copyright 19.06.2024 by Bochkanov Sergey
*************************************************************************/
void minqpclearcc(minqpstate &state, const xparams _xparams = alglib::xdefault);

`minqpclearqc` function

/*************************************************************************
This function clears the list of quadratic constraints. Other  constraints
are not modified.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.

  -- ALGLIB --
     Copyright 19.06.2024 by Bochkanov Sergey
*************************************************************************/
void minqpclearqc(minqpstate &state, const xparams _xparams = alglib::xdefault);

`minqpcreate` function

/*************************************************************************
                    CONSTRAINED QUADRATIC PROGRAMMING

The subroutine creates QP optimizer. After initial creation,  it  contains
default optimization problem with zero quadratic and linear terms  and  no
constraints.

In order to actually solve something you should:

specify objective:
* set linear term with minqpsetlinearterm()
* set quadratic term with minqpsetquadraticterm() or
  minqpsetquadratictermsparse()

specify constraints:
* set variable bounds with minqpsetbc() or minqpsetbcall()
* specify linear constraint matrix with one of the following functions:
  * modern API:
    * minqpsetlc2()       for sparse two-sided constraints AL <= A*x <= AU
    * minqpsetlc2dense()  for dense  two-sided constraints AL <= A*x <= AU
    * minqpsetlc2mixed()  for mixed  two-sided constraints AL <= A*x <= AU
    * minqpaddlc2dense()  to add one dense row to the dense constraint submatrix
    * minqpaddlc2()       to add one sparse row to the sparse constraint submatrix
    * minqpaddlc2sparsefromdense() to add one sparse row (passed as a dense array) to the sparse constraint submatrix
  * legacy API:
    * minqpsetlc()        for dense one-sided equality/inequality constraints
    * minqpsetlcsparse()  for sparse one-sided equality/inequality constraints
    * minqpsetlcmixed()   for mixed dense/sparse one-sided equality/inequality constraints
* add two-sided quadratic constraint(s) of the form CL <= b'x+0.5*x'Qx <= CU
  with one of the following functions:
  * minqpaddqc2()         for a quadratic constraint given by a sparse
                          matrix structure; has O(max(N,NNZ)) memory and
                          running time requirements.
  * minqpaddqc2dense()    for a quadratic constraint given by a dense
                          matrix; has O(N^2) memory and running time requirements.
  * minqpaddqc2list()     for a sparse quadratic constraint given by
                          a list of non-zero  entries;  has  O(NNZ) memory
                          and O(NNZ*logNNZ)  running   time  requirements,
                          ideal for constraints  with  much  less  than  N
                          nonzero elements.
* add second order cone constraints with:
  * minqpaddsoccprimitive()     for a primitive second order cone constraint
  * minqpaddsoccorthogonal()    for an axis-orthogonal second order cone constraint
* add power cone constraints with:
  * minqpaddpowccprimitive()    for a primitive power cone constraint
  * minqpaddpowccorthogonal()   for an axis-orthogonal power cone constraint

configure and run QP solver:
* choose appropriate QP solver and set it  and  its stopping  criteria  by
  means of minqpsetalgo??????() function
* call minqpoptimize() to run the solver and  minqpresults()  to  get  the
  solution vector and additional information.

Following solvers are recommended for  convex  and  semidefinite  problems
with box and linear constraints:
* QuickQP for dense problems with box-only constraints (or no constraints
  at all)
* DENSE-IPM-QP for  convex  or  semidefinite  problems  with   medium  (up
  to several thousands) variable count, dense/sparse  quadratic  term  and
  any number  (up  to  many  thousands)  of  dense/sparse  general  linear
  constraints
* SPARSE-IPM-QP for convex  or  semidefinite  problems  with   large (many
  thousands) variable count, sparse quadratic term AND linear constraints.
* SPARSE-ECQP for convex having only  linear  equality  constraints.  This
  specialized solver can be orders of magnitude faster than IPM.

If your problem happens to be nonconvex or has nonlinear constraints, then
you can use:
* DENSE-GENIPM or SPARSE-GENIPM  solver  which  supports  convex/nonconvex
  QP   problems  with  box,  linear,  quadratic  equality/inequality   and
  conic constraints.
* QuickQP for small dense nonconvex problems with box-only constraints

INPUT PARAMETERS:
    N       -   problem size

OUTPUT PARAMETERS:
    State   -   optimizer with zero quadratic/linear terms
                and no constraints

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpcreate(const ae_int_t n, minqpstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minqpexport` function

/*************************************************************************
Export current QP problem stored in the solver into  QPXProblem  instance.
This  instance  can  be  serialized  into   ALGLIB-specific   format   and
unserialized from several widely acnowledged formats.

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    P       -   QPXProblem instance storing current objective and
                constraints.

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void minqpexport(minqpstate &state, qpxproblem &p, const xparams _xparams = alglib::xdefault);

`minqpimport` function

/*************************************************************************
Imports QP problem, as defined by QPXProblem instance, creating a QP solver
with objective/constraints/scales/origin set to that stored in the instance.

INPUT PARAMETERS:
    P       -   QPXProblem instance storing current objective and
                constraints.

OUTPUT PARAMETERS:
    State   -   newly created solver

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void minqpimport(qpxproblem &p, minqpstate &s, const xparams _xparams = alglib::xdefault);

`minqpoptimize` function

/*************************************************************************
This function solves quadratic programming problem.

Prior to calling this function you should choose solver by means of one of
the following functions:

* minqpsetalgoquickqp()     - for QuickQP solver
* minqpsetalgodenseaul()    - for Dense-AUL-QP solver
* minqpsetalgodenseipm()    - for convex Dense-IPM-QP solver
* minqpsetalgosparseipm()   - for convex Sparse-IPM-QP solver
* minqpsetalgodensegenipm() - for convex/nonconvex Dense-IPM-QP solver with conic constraints
* minqpsetalgosparsegenipm()- for convex/nonconvex Sparse-IPM-QP solver with conic constraints

These functions also allow you to control stopping criteria of the solver.
If you did not set solver,  MinQP  subpackage  will  automatically  select
solver for your problem and will run it with default stopping criteria.

However, it is better to set explicitly solver and its stopping criteria.

INPUT PARAMETERS:
    State   -   algorithm state

You should use MinQPResults() function to access results after calls
to this function.

  -- ALGLIB --
     Copyright 2011-2024 by Bochkanov Sergey.
     Special thanks to Elvira Illarionova  for  important  suggestions  on
     the linearly constrained QP algorithm.
*************************************************************************/
void minqpoptimize(minqpstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minqpresults` function

/*************************************************************************
QP solver results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution (on failure - the best point found
                so far).
    Rep     -   optimization report, contains:
                * completion code in Rep.TerminationType (positive  values
                  denote some kind of success, negative - failures)
                * Lagrange multipliers - for QP solvers which support them
                * other statistics
                See comments on minqpreport structure for more information

Following completion codes are returned in Rep.TerminationType:
* -9    failure of the automatic scale evaluation:  one  of  the  diagonal
        elements of the quadratic term is non-positive.  Specify  variable
        scales manually!
* -5    inappropriate solver was used:
        * QuickQP solver for problem with general linear constraints
        * QuickQP/DENSE-AUL/DENSE-IPM/SPARSE-IPM for a problem with
          quadratic/conic constraints
* -4    the function is unbounded from below even under constraints,
        no meaningful minimum can be found.
* -3    inconsistent constraints (or, maybe, feasible point is too hard to
        find).
* -2    IPM solver has difficulty finding primal/dual feasible point.
        It is likely that the problem is either infeasible or unbounded,
        but it is difficult to determine exact reason for termination.
        X contains best point found so far.
*  >0   success
*  7    stopping conditions are too stringent,
        further improvement is impossible,
        X contains best point found so far.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpresults(const minqpstate &state, real_1d_array &x, minqpreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minqpresultsbuf` function

/*************************************************************************
QP results

Buffered implementation of MinQPResults() which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpresultsbuf(const minqpstate &state, real_1d_array &x, minqpreport &rep, const xparams _xparams = alglib::xdefault);

`minqpsetalgodenseaul` function

/*************************************************************************
This function tells QP solver to use DENSE-AUL algorithm and sets stopping
criteria for the algorithm.

This  algorithm  is  intended  for  non-convex problems with moderate  (up
to several thousands) variable count and arbitrary number  of  constraints
which are either (a) effectively convexified under constraints or (b) have
unique solution even with nonconvex target.

IMPORTANT: when DENSE-IPM solver is applicable, its performance is usually
           much better than that of DENSE-AUL.
           We recommend  you to use DENSE-AUL only when other solvers  can
           not be used.

ALGORITHM FEATURES:

* supports  box  and  dense/sparse  general   linear   equality/inequality
  constraints
* convergence is theoretically proved for positive-definite  (convex)   QP
  problems. Semidefinite and non-convex problems can be solved as long  as
  they  are   bounded  from  below  under  constraints,  although  without
  theoretical guarantees.

ALGORITHM OUTLINE:

* this  algorithm   is   an   augmented   Lagrangian   method  with  dense
  preconditioner (hence  its  name).
* it performs several outer iterations in order to refine  values  of  the
  Lagrange multipliers. Single outer  iteration  is  a  solution  of  some
  unconstrained optimization problem: first  it  performs  dense  Cholesky
  factorization of the Hessian in order to build preconditioner  (adaptive
  regularization is applied to enforce positive  definiteness),  and  then
  it uses L-BFGS optimizer to solve optimization problem.
* typically you need about 5-10 outer iterations to converge to solution

ALGORITHM LIMITATIONS:

* because dense Cholesky driver is used, this algorithm has O(N^2)  memory
  requirements and O(OuterIterations*N^3) minimum running time.  From  the
  practical  point  of  view,  it  limits  its  applicability  by  several
  thousands of variables.
  From  the  other  side,  variables  count  is  the most limiting factor,
  and dependence on constraint count is  much  more  lower. Assuming  that
  constraint matrix is sparse, it may handle tens of thousands  of general
  linear constraints.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   >=0, stopping criteria for inner optimizer.
                Inner  iterations  are  stopped  when  step  length  (with
                variable scaling being applied) is less than EpsX.
                See  minqpsetscale()  for  more  information  on  variable
                scaling.
    Rho     -   penalty coefficient, Rho>0:
                * large enough  that  algorithm  converges  with   desired
                  precision.
                * not TOO large to prevent ill-conditioning
                * recommended values are 100, 1000 or 10000
    ItsCnt  -   number of outer iterations:
                * recommended values: 10-15 (although  in  most  cases  it
                  converges within 5 iterations, you may need a  few  more
                  to be sure).
                * ItsCnt=0 means that small number of outer iterations  is
                  automatically chosen (10 iterations in current version).
                * ItsCnt=1 means that AUL algorithm performs just as usual
                  penalty method.
                * ItsCnt>1 means that  AUL  algorithm  performs  specified
                  number of outer iterations

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing  EpsX=0  will  lead  to  automatic  step  length  selection
      (specific step length chosen may change in the future  versions  of
      ALGLIB, so it is better to specify step length explicitly).

  -- ALGLIB --
     Copyright 20.08.2016 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgodenseaul(minqpstate &state, const double epsx, const double rho, const ae_int_t itscnt, const xparams _xparams = alglib::xdefault);

`minqpsetalgodensegenipm` function

/*************************************************************************
This function tells QP solver to  use  DENSE-GENIPM  QP algorithm and sets
stopping criteria for the algorithm.

This  algorithm  is  intended  for convex/nonconvex box/linearly/conically
constrained QP problems with moderate (up to several thousands)  variables
count and arbitrary number  of  constraints.  Use  SPARSE-GENIPM  if  your
problem is sparse.

The algorithm is a generalization of DENSE-IPM solver, capable of handling
more general constraints as well as nonconvexity of  the  target.  In  the
latter case, a local solution is found.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

ALGORITHM FEATURES:

* supports box, linear equality/inequality and conic constraints
* for convex problems returns the global (and the only) solution
* can handle non-convex  problem  (only  a  locally  optimal  solution  is
  returned in this case)

ALGORITHM LIMITATIONS:

* because a dense Cholesky driver is used, for  N-dimensional problem with
  M dense constaints this algorithm has O(N^2+N*M) memory requirements and
  O(N^3+M*N^2) running time.
  Having sparse constraints with Z nonzeros per row  relaxes  storage  and
  running time down to O(N^2+M*Z) and O(N^3+M*Z^2)
  From the practical  point  of  view,  it  limits  its  applicability  by
  several thousands of variables.
  From  the  other  side,  variables  count  is  the most limiting factor,
  and dependence on constraint count is  much  more  lower. Assuming  that
  the constraint matrix is sparse, it may  handle  tens  of  thousands  of
  general linear constraints.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Eps     -   >=0, stopping criteria. The algorithm stops  when   primal
                and dual infeasiblities as well as complementarity gap are
                less than Eps.

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing EpsX=0 will lead to automatic selection of small epsilon.

===== TRACING GENIPM SOLVER ==============================================

GENIPM solver supports advanced tracing capabilities. You can log algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'GENIPM'      - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'GENIPM.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'GENIPM'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("GENIPM,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 01.05.2024 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgodensegenipm(minqpstate &state, const double eps, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpsetalgodenseipm` function

/*************************************************************************
This function tells QP solver to  use  DENSE-IPM  QP  algorithm  and  sets
stopping criteria for the algorithm.

This  algorithm  is  intended for convex and semidefinite QP (but not QCQP
or conic) problems with moderate (up to several thousands) variable  count
and arbitrary number of linear constraints. Quadratic and conic constraints
are supported by another solver (DENSE-GENIPM).

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

IMPORTANT: this  algorithm  is  likely  to  fail  on  nonconvex  problems,
           furthermore, sometimes it fails without a notice. If you try to
           run DENSE-IPM on a problem  with  indefinite  matrix (a  matrix
           having  at least one negative eigenvalue) then depending on the
           circumstances it may either (a) stall at some  arbitrary point,
           or (b) throw an exception due to the failure  of  the  Cholesky
           decomposition.

           Use GENIPM algorithm if  your  problem  is  nonconvex  or has a
           potential of becoming  nonconvex.  The GENIPM solver  can  also
           handle problems with quadratic and conic constraints.

ALGORITHM FEATURES:

* supports  box  and  dense/sparse  general   linear   equality/inequality
  constraints

ALGORITHM OUTLINE:

* this  algorithm  is  our implementation  of  interior  point  method  as
  formulated by  R.J.Vanderbei, with minor modifications to the  algorithm
  (damped Newton directions are extensively used)
* like all interior point methods, this algorithm  tends  to  converge  in
  roughly same number of iterations (between 15 and 50) independently from
  the problem dimensionality

ALGORITHM LIMITATIONS:

* because a dense Cholesky driver is used, for  N-dimensional problem with
  M dense constaints this algorithm has O(N^2+N*M) memory requirements and
  O(N^3+M*N^2) running time.
  Having sparse constraints with Z nonzeros per row  relaxes  storage  and
  running time down to O(N^2+M*Z) and O(N^3+M*Z^2)
  From the practical  point  of  view,  it  limits  its  applicability  by
  several thousands of variables.
  From  the  other  side,  variables  count  is  the most limiting factor,
  and dependence on constraint count is  much  more  lower. Assuming  that
  the constraint matrix is sparse, it may  handle  tens  of  thousands  of
  general linear constraints.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Eps     -   >=0, stopping criteria. The algorithm stops  when   primal
                and dual infeasiblities as well as complementarity gap are
                less than Eps.

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing EpsX=0 will lead to automatic selection of small epsilon.

===== TRACING IPM SOLVER =================================================

IPM solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'IPM'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'IPM.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'IPM'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("IPM,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgodenseipm(minqpstate &state, const double eps, const xparams _xparams = alglib::xdefault);

`minqpsetalgoquickqp` function

/*************************************************************************
This function tells solver to use QuickQP  algorithm:  special  extra-fast
algorithm for problems with box-only constrants. It may  solve  non-convex
problems as long as they are bounded from below under constraints.

ALGORITHM FEATURES:
* several times faster than DENSE-IPM when running on box-only problem
* utilizes accelerated methods for activation of constraints.
* supports dense and sparse QP problems
* supports ONLY box constraints; general linear constraints are NOT
  supported by this solver
* can solve all types of problems  (convex,  semidefinite,  nonconvex)  as
  long as they are bounded from below under constraints.
  Say, it is possible to solve "min{-x^2} subject to -1<=x<=+1".
  In convex/semidefinite case global minimum  is  returned,  in  nonconvex
  case - algorithm returns one of the local minimums.

ALGORITHM OUTLINE:

* algorithm  performs  two kinds of iterations: constrained CG  iterations
  and constrained Newton iterations
* initially it performs small number of constrained CG  iterations,  which
  can efficiently activate/deactivate multiple constraints
* after CG phase algorithm tries to calculate Cholesky  decomposition  and
  to perform several constrained Newton steps. If  Cholesky  decomposition
  failed (matrix is indefinite even under constraints),  we  perform  more
  CG iterations until we converge to such set of constraints  that  system
  matrix becomes  positive  definite.  Constrained  Newton  steps  greatly
  increase convergence speed and precision.
* algorithm interleaves CG and Newton iterations which  allows  to  handle
  indefinite matrices (CG phase) and quickly converge after final  set  of
  constraints is found (Newton phase). Combination of CG and Newton phases
  is called "outer iteration".
* it is possible to turn off Newton  phase  (beneficial  for  semidefinite
  problems - Cholesky decomposition will fail too often)

ALGORITHM LIMITATIONS:

* algorithm does not support general  linear  constraints;  only  box ones
  are supported
* Cholesky decomposition for sparse problems  is  performed  with  Skyline
  Cholesky solver, which is intended for low-profile matrices. No profile-
  reducing reordering of variables is performed in this version of ALGLIB.
* problems with near-zero negative eigenvalues (or exacty zero  ones)  may
  experience about 2-3x performance penalty. The reason is  that  Cholesky
  decomposition can not be performed until we identify directions of  zero
  and negative curvature and activate corresponding boundary constraints -
  but we need a lot of trial and errors because these directions  are hard
  to notice in the matrix spectrum.
  In this case you may turn off Newton phase of algorithm.
  Large negative eigenvalues  are  not  an  issue,  so  highly  non-convex
  problems can be solved very efficiently.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsG    -   >=0
                The  subroutine  finishes  its  work   if   the  condition
                |v|<EpsG is satisfied, where:
                * |.| means Euclidian norm
                * v - scaled constrained gradient vector, v[i]=g[i]*s[i]
                * g - gradient
                * s - scaling coefficients set by MinQPSetScale()
    EpsF    -   >=0
                The  subroutine  finishes its work if exploratory steepest
                descent  step  on  k+1-th iteration  satisfies   following
                condition:  |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1}
    EpsX    -   >=0
                The  subroutine  finishes its work if exploratory steepest
                descent  step  on  k+1-th iteration  satisfies   following
                condition:
                * |.| means Euclidian norm
                * v - scaled step vector, v[i]=dx[i]/s[i]
                * dx - step vector, dx=X(k+1)-X(k)
                * s - scaling coefficients set by MinQPSetScale()
    MaxOuterIts-maximum number of OUTER iterations.  One  outer  iteration
                includes some amount of CG iterations (from 5 to  ~N)  and
                one or several (usually small amount) Newton steps.  Thus,
                one outer iteration has high cost, but can greatly  reduce
                funcation value.
                Use 0 if you do not want to limit number of outer iterations.
    UseNewton-  use Newton phase or not:
                * Newton phase improves performance of  positive  definite
                  dense problems (about 2 times improvement can be observed)
                * can result in some performance penalty  on  semidefinite
                  or slightly negative definite  problems  -  each  Newton
                  phase will bring no improvement (Cholesky failure),  but
                  still will require computational time.
                * if you doubt, you can turn off this  phase  -  optimizer
                  will retain its most of its high speed.

IT IS VERY IMPORTANT TO CALL MinQPSetScale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS STOPPING CRITERIA ARE SCALE-DEPENDENT!

Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead
to automatic stopping criterion selection (presently it is  small    step
length, but it may change in the future versions of ALGLIB).

  -- ALGLIB --
     Copyright 22.05.2014 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgoquickqp(minqpstate &state, const double epsg, const double epsf, const double epsx, const ae_int_t maxouterits, const bool usenewton, const xparams _xparams = alglib::xdefault);

`minqpsetalgosparseecqp` function

/*************************************************************************
This function tells QP solver to use an ECQP algorithm.

This  algorithm  is  intended  for sparse convex problems with only linear
equality constraints. It can handle millions of variables and constraints,
assuming that the problem is sufficiently  sparse.  However,  it  can  NOT
deal with nonlinear equality constraints or inequality constraints of  any
type (including box ones), nor it can deal with nonconvex problems.

When applicable, it outperforms SPARSE-IPM by  tens  of  times.  It  is  a
regularized direct linear algebra solver that performs several  rounds  of
iterative  refinement  in  order to improve a solution. Thus, due  to  its
direct nature, it does not need stopping criteria and performs much faster
than interior point methods.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern of quadratic term and constraints.
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

IMPORTANT: internally this solver performs large  and  sparse  (N+M)x(N+M)
           triangular factorization. So it expects both quadratic term and
           constraints to be highly sparse. However, its  running  time is
           influenced by BOTH fill factor and sparsity pattern.

           Generally we expect that no more than few nonzero  elements per
           row are present. However different sparsity patterns may result
           in completely different running  times  even  given  same  fill
           factor.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Eps     -   >=0, stopping criteria. The algorithm stops  when   primal
                and dual infeasiblities are less than Eps.

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing EpsX=0 will lead to automatic selection of small epsilon.

  -- ALGLIB --
     Copyright 01.07.2024 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgosparseecqp(minqpstate &state, const xparams _xparams = alglib::xdefault);

`minqpsetalgosparsegenipm` function

/*************************************************************************
This function tells QP solver to  use SPARSE-GENIPM  QP algorithm and sets
stopping criteria for the algorithm.

This   algorithm  is intended for convex/nonconvex box/linearly/conically/
quadratically constrained QP  problems  with  sparse  quadratic  term  and
constraints. It can handle millions of variables and constraints, assuming
that the problem is sufficiently sparse. If your problem is small (several
thousands vars at most) and dense, consider using DENSE-GENIPM  as  a more
efficient alternative.

The  algorithm  is  a  generalization of the SPARSE-IPM solver, capable of
handling more general constraints as well as nonconvexity of  the  target.
In the latter case, a local solution is found.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern of quadratic term and constraints.
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

IMPORTANT: internally this solver performs large  and  sparse  (N+M)x(N+M)
           triangular factorization. So it expects both quadratic term and
           constraints to be highly sparse. However, its  running  time is
           influenced by BOTH fill factor and sparsity pattern.

           Generally we expect that no more than few nonzero  elements per
           row are present. However different sparsity patterns may result
           in completely different running  times  even  given  same  fill
           factor.

           In many cases this algorithm outperforms DENSE-IPM by order  of
           magnitude. However, in some cases you may  get  better  results
           with DENSE-IPM even when solving sparse task.

ALGORITHM FEATURES:


* supports box, linear equality/inequality constraints
* for convex problems returns the global (and the only) solution
* can handle non-convex  problem  (only  a  locally  optimal  solution  is
  returned in this case)
* specializes on large-scale sparse problems

ALGORITHM LIMITATIONS:

* this algorithm may handle moderate number  of dense constraints, usually
  no more than a thousand of dense ones without losing its efficiency.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Eps     -   >=0, stopping criteria. The algorithm stops  when   primal
                and dual infeasiblities as well as complementarity gap are
                less than Eps.

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing EpsX=0 will lead to automatic selection of small epsilon.

===== TRACING GENIPM SOLVER ==============================================

GENIPM solver supports advanced tracing capabilities. You can log algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'GENIPM'      - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'GENIPM.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'IPM'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("GENIPM,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 01.05.2024 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgosparsegenipm(minqpstate &state, const double eps, const xparams _xparams = alglib::xdefault);

`minqpsetalgosparseipm` function

/*************************************************************************
This function tells QP solver to  use  SPARSE-IPM  QP algorithm  and  sets
stopping criteria for the algorithm.

This  algorithm  is  intended for convex and semidefinite QP (but not QCQP
or conic) problems with large  variable  and  constraint  count and sparse
quadratic term and sparse linear constraints. It was successfully used for
problems with millions of variables and constraints. Quadratic  and  conic
constraints are supported by another solver (SPARSE-GENIPM).

It is possible to have some limited set of dense linear constraints - they
will be handled separately  by  the  dense  BLAS  -  but  the  more  dense
constraints you have, the more time solver needs.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern of quadratic term and constraints.
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

IMPORTANT: internally this solver performs large  and  sparse  (N+M)x(N+M)
           triangular factorization. So it expects both quadratic term and
           constraints to be highly sparse. However, its  running  time is
           influenced by BOTH fill factor and sparsity pattern.

           Generally we expect that no more than few nonzero  elements per
           row are present. However different sparsity patterns may result
           in completely different running  times  even  given  same  fill
           factor.

           In many cases this algorithm outperforms DENSE-IPM by order  of
           magnitude. However, in some cases you may  get  better  results
           with DENSE-IPM even when solving sparse task.

IMPORTANT: this algorithm won't work for nonconvex problems. If you try to
           run SPARSE-IPM  on a problem with indefinite quadratic term  (a
           matrix having at least one negative eigenvalue) then  depending
           on the circumstances it may either (a) stall  at some arbitrary
           point, or (b) throw an exception due  to  the  failure  of  the
           Cholesky decomposition.

           Use GENIPM algorithm if  your  problem  is  nonconvex  or has a
           potential of becoming  nonconvex.  The  GENIPM  solver can also
           handle problems with quadratic and conic constraints.

ALGORITHM FEATURES:

* supports  box  and  dense/sparse  general   linear   equality/inequality
  constraints
* specializes on large-scale sparse problems

ALGORITHM OUTLINE:

* this  algorithm  is  our implementation  of  interior  point  method  as
  formulated by  R.J.Vanderbei, with minor modifications to the  algorithm
  (damped Newton directions are extensively used)
* like all interior point methods, this algorithm  tends  to  converge  in
  roughly same number of iterations (between 15 and 50) independently from
  the problem dimensionality

ALGORITHM LIMITATIONS:

* this algorithm may handle moderate number  of dense constraints, usually
  no more than a thousand of dense ones without losing its efficiency.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    Eps     -   >=0, stopping criteria. The algorithm stops  when   primal
                and dual infeasiblities as well as complementarity gap are
                less than Eps.

IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS  ALGORITHM
BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT!

NOTE: Passing EpsX=0 will lead to automatic selection of small epsilon.

===== TRACING IPM SOLVER =================================================

IPM solver supports advanced tracing capabilities. You can trace algorithm
output by specifying following trace symbols (case-insensitive)  by  means
of trace_file() call:
* 'IPM'         - for basic trace of algorithm  steps and decisions.  Only
                  short scalars (function values and deltas) are  printed.
                  N-dimensional quantities like search directions are  NOT
                  printed.
* 'IPM.DETAILED'- for output of points being visited and search directions
                  This  symbol  also  implicitly  defines  'IPM'. You  can
                  control output format by additionally specifying:
                  * nothing     to output in  6-digit exponential format
                  * 'PREC.E15'  to output in 15-digit exponential format
                  * 'PREC.F6'   to output in  6-digit fixed-point format

By default trace is disabled and adds  no  overhead  to  the  optimization
process. However, specifying any of the symbols adds some  formatting  and
output-related overhead.

You may specify multiple symbols by separating them with commas:
>
> alglib::trace_file("IPM,PREC.F6", "path/to/trace.log")
>

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minqpsetalgosparseipm(minqpstate &state, const double eps, const xparams _xparams = alglib::xdefault);

`minqpsetbc` function

/*************************************************************************
This function sets box constraints for QP solver

Box constraints are inactive by default (after  initial  creation).  After
being  set,  they are  preserved until explicitly overwritten with another
minqpsetbc()  or  minqpsetbcall()  call,  or  partially  overwritten  with
minqpsetbci() call.

Following types of constraints are supported:

    DESCRIPTION         CONSTRAINT              HOW TO SPECIFY
    fixed variable      x[i]=Bnd[i]             BndL[i]=BndU[i]
    lower bound         BndL[i]<=x[i]           BndU[i]=+INF
    upper bound         x[i]<=BndU[i]           BndL[i]=-INF
    range               BndL[i]<=x[i]<=BndU[i]  ...
    free variable       -                       BndL[I]=-INF, BndU[I]+INF

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (latter is recommended because
                it will allow solver to use better algorithm).
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (latter is recommended because
                it will allow solver to use better algorithm).

NOTE: infinite values can be specified by means of Double.PositiveInfinity
      and  Double.NegativeInfinity  (in  C#)  and  alglib::fp_posinf   and
      alglib::fp_neginf (in C++).

NOTE: you may replace infinities by very small/very large values,  but  it
      is not recommended because large numbers may introduce large numerical
      errors in the algorithm.

NOTE: if constraints for all variables are same you may use minqpsetbcall()
      which allows to specify constraints without using arrays.

NOTE: BndL>BndU will result in QP problem being recognized as infeasible.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetbc(minqpstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpsetbcall` function

/*************************************************************************
This function sets box constraints for QP solver (all variables  at  once,
same constraints for all variables)

Box constraints are inactive by default (after  initial  creation).  After
being  set,  they are  preserved until explicitly overwritten with another
minqpsetbc()  or  minqpsetbcall()  call,  or  partially  overwritten  with
minqpsetbci() call.

Following types of constraints are supported:

    DESCRIPTION         CONSTRAINT              HOW TO SPECIFY
    fixed variable      x[i]=Bnd                BndL=BndU
    lower bound         BndL<=x[i]              BndU=+INF
    upper bound         x[i]<=BndU              BndL=-INF
    range               BndL<=x[i]<=BndU        ...
    free variable       -                       BndL=-INF, BndU+INF

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bound, same for all variables
    BndU    -   upper bound, same for all variables

NOTE: infinite values can be specified by means of Double.PositiveInfinity
      and  Double.NegativeInfinity  (in  C#)  and  alglib::fp_posinf   and
      alglib::fp_neginf (in C++).

NOTE: you may replace infinities by very small/very large values,  but  it
      is not recommended because large numbers may introduce large numerical
      errors in the algorithm.

NOTE: BndL>BndU will result in QP problem being recognized as infeasible.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetbcall(minqpstate &state, const double bndl, const double bndu, const xparams _xparams = alglib::xdefault);

`minqpsetbci` function

/*************************************************************************
This function sets box constraints for I-th variable (other variables are
not modified).

Following types of constraints are supported:

    DESCRIPTION         CONSTRAINT              HOW TO SPECIFY
    fixed variable      x[i]=Bnd                BndL=BndU
    lower bound         BndL<=x[i]              BndU=+INF
    upper bound         x[i]<=BndU              BndL=-INF
    range               BndL<=x[i]<=BndU        ...
    free variable       -                       BndL=-INF, BndU+INF

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bound
    BndU    -   upper bound

NOTE: infinite values can be specified by means of Double.PositiveInfinity
      and  Double.NegativeInfinity  (in  C#)  and  alglib::fp_posinf   and
      alglib::fp_neginf (in C++).

NOTE: you may replace infinities by very small/very large values,  but  it
      is not recommended because large numbers may introduce large numerical
      errors in the algorithm.

NOTE: BndL>BndU will result in QP problem being recognized as infeasible.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetbci(minqpstate &state, const ae_int_t i, const double bndl, const double bndu, const xparams _xparams = alglib::xdefault);

`minqpsetlc` function

/*************************************************************************
This function sets dense linear constraints for QP optimizer.

This  function  overrides  results  of  previous  calls  to  minqpsetlc(),
minqpsetlcsparse() and minqpsetlcmixed().  After  call  to  this  function
all non-box constraints are dropped, and you have only  those  constraints
which were specified in the present call.

If you want  to  specify  mixed  (with  dense  and  sparse  terms)  linear
constraints, you should call minqpsetlcmixed().

INPUT PARAMETERS:
    State   -   structure previously allocated with MinQPCreate call.
    C       -   linear constraints, array[K,N+1].
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0:
                * if given, only leading K elements of C/CT are used
                * if not given, automatically determined from sizes of C/CT

NOTE 1: linear (non-bound) constraints are satisfied only approximately  -
        there always exists some violation due  to  numerical  errors  and
        algorithmic limitations.

  -- ALGLIB --
     Copyright 19.06.2012 by Bochkanov Sergey
*************************************************************************/
void minqpsetlc(minqpstate &state, const real_2d_array &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minqpsetlc(minqpstate &state, const real_2d_array &c, const integer_1d_array &ct, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpsetlc2` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
sparse constraining matrix A. Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    A       -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minqpsetlc2(minqpstate &state, const sparsematrix &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`minqpsetlc2dense` function

/*************************************************************************
This function sets two-sided linear constraints AL <= A*x <= AU with dense
constraint matrix A.

NOTE: knowing  that  constraint  matrix  is  dense  helps  some QP solvers
      (especially modern IPM method) to utilize efficient  dense  Level  3
      BLAS for dense parts of the problem. If your problem has both  dense
      and sparse constraints, you  can  use  minqpsetlc2mixed()  function,
      which will result in dense algebra being applied to dense terms, and
      sparse sparse linear algebra applied to sparse terms.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    A       -   linear constraints, array[K,N]. Each row of  A  represents
                one  constraint. One-sided  inequality   constraints, two-
                sided inequality  constraints,  equality  constraints  are
                supported (see below)
    AL, AU  -   lower and upper bounds, array[K];
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number of equality/inequality constraints,  K>=0;  if  not
                given, inferred from sizes of A, AL, AU.

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minqpsetlc2dense(minqpstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t k, const xparams _xparams = alglib::xdefault);
void minqpsetlc2dense(minqpstate &state, const real_2d_array &a, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

`minqpsetlc2mixed` function

/*************************************************************************
This  function  sets  two-sided linear  constraints  AL <= A*x <= AU  with
mixed constraining matrix A including sparse part (first SparseK rows) and
dense part (last DenseK rows). Recommended for large-scale problems.

This  function  overwrites  linear  (non-box)  constraints set by previous
calls (if such calls were made).

This function may be useful if constraint matrix includes large number  of
both types of rows - dense and sparse. If you have just a few sparse rows,
you  may  represent  them  in  dense  format  without losing  performance.
Similarly, if you have just a few dense rows, you may store them in sparse
format with almost same performance.

INPUT PARAMETERS:
    State   -   structure previously allocated with minqpcreate() call.
    SparseA -   sparse matrix with size [K,N] (exactly!).
                Each row of A represents one general linear constraint.
                A can be stored in any sparse storage format.
    SparseK -   number of sparse constraints, SparseK>=0
    DenseA  -   linear constraints, array[K,N], set of dense constraints.
                Each row of A represents one general linear constraint.
    DenseK  -   number of dense constraints, DenseK>=0
    AL, AU  -   lower and upper bounds, array[SparseK+DenseK], with former
                SparseK elements corresponding to sparse constraints,  and
                latter DenseK elements corresponding to dense constraints;
                * AL[i]=AU[i] => equality constraint Ai*x
                * AL[i]<AU[i] => two-sided constraint AL[i]<=Ai*x<=AU[i]
                * AL[i]=-INF  => one-sided constraint Ai*x<=AU[i]
                * AU[i]=+INF  => one-sided constraint AL[i]<=Ai*x
                * AL[i]=-INF, AU[i]=+INF => constraint is ignored
    K       -   number  of equality/inequality constraints, K>=0.  If  K=0
                is specified, A, AL, AU are ignored.

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minqpsetlc2mixed(minqpstate &state, const sparsematrix &sparsea, const ae_int_t ksparse, const real_2d_array &densea, const ae_int_t kdense, const real_1d_array &al, const real_1d_array &au, const xparams _xparams = alglib::xdefault);

`minqpsetlcmixed` function

/*************************************************************************
This function sets mixed linear constraints, which include a set of  dense
rows, and a set of sparse rows.

This  function  overrides  results  of  previous  calls  to  minqpsetlc(),
minqpsetlcsparse() and minqpsetlcmixed().

This function may be useful if constraint matrix includes large number  of
both types of rows - dense and sparse. If you have just a few sparse rows,
you  may  represent  them  in  dense  format  without losing  performance.
Similarly, if you have just a few dense rows, you may store them in sparse
format with almost same performance.

INPUT PARAMETERS:
    State   -   structure previously allocated with MinQPCreate call.
    SparseC -   linear constraints, sparse  matrix with dimensions EXACTLY
                EQUAL TO [SparseK,N+1].  Each  row  of  C  represents  one
                constraint, either equality or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    SparseCT-   type of sparse constraints, array[K]:
                * if SparseCT[i]>0, then I-th constraint is SparseC[i,*]*x >= SparseC[i,n+1]
                * if SparseCT[i]=0, then I-th constraint is SparseC[i,*]*x  = SparseC[i,n+1]
                * if SparseCT[i]<0, then I-th constraint is SparseC[i,*]*x <= SparseC[i,n+1]
    SparseK -   number of sparse equality/inequality constraints, K>=0
    DenseC  -   dense linear constraints, array[K,N+1].
                Each row of DenseC represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of DenseC (including right part) must be finite.
    DenseCT -   type of constraints, array[K]:
                * if DenseCT[i]>0, then I-th constraint is DenseC[i,*]*x >= DenseC[i,n+1]
                * if DenseCT[i]=0, then I-th constraint is DenseC[i,*]*x  = DenseC[i,n+1]
                * if DenseCT[i]<0, then I-th constraint is DenseC[i,*]*x <= DenseC[i,n+1]
    DenseK  -   number of equality/inequality constraints, DenseK>=0

NOTE 1: linear (non-box) constraints  are  satisfied only approximately  -
        there always exists some violation due  to  numerical  errors  and
        algorithmic limitations.

NOTE 2: due to backward compatibility reasons SparseC can be  larger  than
        [SparseK,N+1]. In this case only leading  [SparseK,N+1]  submatrix
        will be  used.  However,  the  rest  of  ALGLIB  has  more  strict
        requirements on the input size, so we recommend you to pass sparse
        term whose size exactly matches algorithm expectations.

  -- ALGLIB --
     Copyright 22.08.2016 by Bochkanov Sergey
*************************************************************************/
void minqpsetlcmixed(minqpstate &state, const sparsematrix &sparsec, const integer_1d_array &sparsect, const ae_int_t sparsek, const real_2d_array &densec, const integer_1d_array &densect, const ae_int_t densek, const xparams _xparams = alglib::xdefault);

`minqpsetlcmixedlegacy` function

/*************************************************************************
This function provides legacy API for specification of mixed  dense/sparse
linear constraints.

New conventions used by ALGLIB since release  3.16.0  state  that  set  of
sparse constraints comes first,  followed  by  set  of  dense  ones.  This
convention is essential when you talk about things like order of  Lagrange
multipliers.

However, legacy API accepted mixed  constraints  in  reverse  order.  This
function is here to simplify situation with code relying on legacy API. It
simply accepts constraints in one order (old) and passes them to new  API,
now in correct order.

  -- ALGLIB --
     Copyright 01.11.2019 by Bochkanov Sergey
*************************************************************************/
void minqpsetlcmixedlegacy(minqpstate &state, const real_2d_array &densec, const integer_1d_array &densect, const ae_int_t densek, const sparsematrix &sparsec, const integer_1d_array &sparsect, const ae_int_t sparsek, const xparams _xparams = alglib::xdefault);

`minqpsetlcsparse` function

/*************************************************************************
This function sets sparse linear constraints for QP optimizer.

This  function  overrides  results  of  previous  calls  to  minqpsetlc(),
minqpsetlcsparse() and minqpsetlcmixed().  After  call  to  this  function
all non-box constraints are dropped, and you have only  those  constraints
which were specified in the present call.

If you want  to  specify  mixed  (with  dense  and  sparse  terms)  linear
constraints, you should call minqpsetlcmixed().

INPUT PARAMETERS:
    State   -   structure previously allocated with MinQPCreate call.
    C       -   linear  constraints,  sparse  matrix  with  dimensions  at
                least [K,N+1]. If matrix has  larger  size,  only  leading
                Kx(N+1) rectangle is used.
                Each row of C represents one constraint, either equality
                or inequality (see below):
                * first N elements correspond to coefficients,
                * last element corresponds to the right part.
                All elements of C (including right part) must be finite.
    CT      -   type of constraints, array[K]:
                * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1]
                * if CT[i]=0, then I-th constraint is C[i,*]*x  = C[i,n+1]
                * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1]
    K       -   number of equality/inequality constraints, K>=0

NOTE 1: linear (non-bound) constraints are satisfied only approximately  -
        there always exists some violation due  to  numerical  errors  and
        algorithmic limitations.

  -- ALGLIB --
     Copyright 22.08.2016 by Bochkanov Sergey
*************************************************************************/
void minqpsetlcsparse(minqpstate &state, const sparsematrix &c, const integer_1d_array &ct, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`minqpsetlinearterm` function

/*************************************************************************
This function sets linear term for QP solver.

By default, linear term is zero.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    B       -   linear term, array[N].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetlinearterm(minqpstate &state, const real_1d_array &b, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`minqpsetorigin` function

/*************************************************************************
This  function sets origin for QP solver. By default, following QP program
is solved:

    min(0.5*x'*A*x+b'*x)

This function allows to solve a different problem:

    min(0.5*(x-x_origin)'*A*(x-x_origin)+b'*(x-x_origin))

Specification of non-zero origin  affects  function  being  minimized  and
quadratic/conic constraints, but not box and linear constraints which  are
still calculated without origin.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    XOrigin -   origin, array[N].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetorigin(minqpstate &state, const real_1d_array &xorigin, const xparams _xparams = alglib::xdefault);

`minqpsetquadraticterm` function

/*************************************************************************
This  function  sets  dense  quadratic  term  for  QP solver. By  default,
quadratic term is zero.

IMPORTANT:

This solver minimizes following  function:
    f(x) = 0.5*x'*A*x + b'*x.
Note that quadratic term has 0.5 before it. So if  you  want  to  minimize
    f(x) = x^2 + x
you should rewrite your problem as follows:
    f(x) = 0.5*(2*x^2) + x
and your matrix A will be equal to [[2.0]], not to [[1.0]]

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    A       -   matrix, array[N,N]
    IsUpper -   storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetquadraticterm(minqpstate &state, const real_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5]

`minqpsetquadratictermsparse` function

/*************************************************************************
This  function  sets  sparse  quadratic  term  for  QP solver. By default,
quadratic  term  is  zero.  This  function  overrides  previous  calls  to
minqpsetquadraticterm() or minqpsetquadratictermsparse().

NOTE: dense solvers like DENSE-AUL-QP or DENSE-IPM-QP  will  convert  this
      matrix to dense storage anyway.

IMPORTANT:

This solver minimizes following  function:
    f(x) = 0.5*x'*A*x + b'*x.
Note that quadratic term has 0.5 before it. So if  you  want  to  minimize
    f(x) = x^2 + x
you should rewrite your problem as follows:
    f(x) = 0.5*(2*x^2) + x
and your matrix A will be equal to [[2.0]], not to [[1.0]]

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    A       -   matrix, array[N,N]
    IsUpper -   (optional) storage type:
                * if True, symmetric matrix  A  is  given  by  its  upper
                  triangle, and the lower triangle isn't used
                * if False, symmetric matrix  A  is  given  by  its lower
                  triangle, and the upper triangle isn't used
                * if not given, both lower and upper  triangles  must  be
                  filled.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetquadratictermsparse(minqpstate &state, const sparsematrix &a, const bool isupper, const xparams _xparams = alglib::xdefault);

Examples: [1]

`minqpsetscale` function

/*************************************************************************
This function sets scaling coefficients.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison  with  tolerances)  and  as
preconditioner.

Scale of the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the
   function

If you do not know how to choose scales of your variables, you can:
* read www.alglib.net/optimization/scaling.php article
* use minqpsetscaleautodiag(), which calculates scale  using  diagonal  of
  the  quadratic  term:  S  is  set to 1/sqrt(diag(A)), which works well
  sometimes.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 14.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetscale(minqpstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`minqpsetscaleautodiag` function

/*************************************************************************
This function sets automatic evaluation of variable scaling.

IMPORTANT: this function works only for  matrices  with positive  diagonal
           elements! Zero or negative elements will  result  in  -9  error
           code  being  returned.  Specify  scale  vector  manually   with
           minqpsetscale() in such cases.

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison  with  tolerances)  and  as
preconditioner.

The  best  way  to  set  scaling  is  to manually specify variable scales.
However, sometimes you just need quick-and-dirty solution  -  either  when
you perform fast prototyping, or when you know your problem well  and  you
are 100% sure that this quick solution is robust enough in your case.

One such solution is to evaluate scale of I-th variable as 1/Sqrt(A[i,i]),
where A[i,i] is an I-th diagonal element of the quadratic term.

Such approach works well sometimes, but you have to be careful here.

INPUT PARAMETERS:
    State   -   structure stores algorithm state

  -- ALGLIB --
     Copyright 26.12.2017 by Bochkanov Sergey
*************************************************************************/
void minqpsetscaleautodiag(minqpstate &state, const xparams _xparams = alglib::xdefault);

`minqpsetstartingpoint` function

/*************************************************************************
This function sets starting point for QP solver. It is useful to have good
initial approximation to the solution, because it will increase  speed  of
convergence and identification of active constraints.

NOTE: interior point solvers ignore initial point provided by user.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    X       -   starting point, array[N].

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void minqpsetstartingpoint(minqpstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

minqp_conic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of a quadratic subject to L1 penalty,
        // i.e. minimization of 
        //
        //     F(x0,x1) = 0.5*x'Ax + b'x + penalty
        //              = 0.5*(2*x0^2 + 2*x1^2) - 6*x0 - 4*x1 + 5*|x0| + 5*|x1|.
        //
        // The exact solution is
        //
        //     [x0,x1] = [0.5,0]
        //
        // The L1 penalty can be modeled by adding two slack variables s0 and s1,
        // constraining them from below using second order or power cones as follows:
        //
        //     sqrt(x0^2) <= s0         this constraint is modelled as a second order cone
        //     sqrt(x1^2) <= s1         this constraint is modelled as a power cone
        //
        // and by adding 5*s0+5*s1 to the linear term. This trick increases variables
        // count by 2x, so in real life problems you may want to use SPARSE-GENIPM solver
        // to deal with the reformulated problem (this solver is more efficient when multiple
        // slacks are present).
        //
        real_2d_array a = "[[2,0,0,0],[0,2,0,0],[0,0,0,0],[0,0,0,0]]";
        real_1d_array b = "[-6,-4,5,5]";
        real_1d_array s = "[1,1,1,1]";
        bool isupper = true;
        real_1d_array x;
        minqpstate state;
        minqpreport rep;

        //
        // create the solver, set quadratic/linear terms
        //
        minqpcreate(4, state);
        minqpsetquadraticterm(state, a, isupper);
        minqpsetlinearterm(state, b);

        //
        // Set the scale of the parameters.
        //
        minqpsetscale(state, s);

        //
        // Specify constraints on slacks. The second order cone constraint is a bit more efficient,
        // but it does not allow to specify penalties other than the L1. The power cone constraint
        // takes a bit more time to handle, but it can be used to model |x|^alpha penalty for
        // any alpha>=1.
        //
        // Functions below add so called primitive conic constraints, ones that constrain a square
        // root of a squared sum of variables in [idx0,idx1-1] range by a variable with index axisIdx.
        // Below the second order cone constraint has idx0=0, idx1=1, axisIdx=2; the power cone
        // constraint has idx0=1, idx1=2, axisIdx=3.
        //
        // More general formulation can be specified with minqpaddsoccorthogonal() or minqpaddpowccorthogonal().
        //
        double alpha = 1.0;
        minqpaddsoccprimitive(state, 0, 1, 2, false);
        minqpaddpowccprimitive(state, 1, 2, 3, alpha, false);

        //
        // Solve problem with the sparse interior-point method (SPARSE-GENIPM) solver.
        //
        // This solver is intended for large-scale sparse problems with box, linear,
        // quadratic and conic constraints, but it will work on such a toy problem too.
        //
        // Commercial ALGLIB can parallelize sparse Cholesky factorization which is the
        // most time-consuming part of the algorithm. See the ALGLIB Reference Manual for
        // more information on how to activate parallelism support.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgosparsegenipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [0.5,0,0.5,0]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minqp_d_bc1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
        // subject to Box constraints 0<=x0<=2.5, 0<=x1<=2.5
        //
        // Exact solution is [x0,x1] = [2.5,2]
        //
        // IMPORTANT: this solver minimizes  following  function:
        //
        //     f(x) = 0.5*x'*A*x + b'*x.
        //
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        //
        // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
        //
        //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
        //
        // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
        //
        real_2d_array a = "[[2,0],[0,2]]";
        real_1d_array b = "[-6,-4]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[0.0,0.0]";
        real_1d_array bndu = "[2.5,2.5]";
        bool isupper = true;
        real_1d_array x;
        minqpstate state;
        minqpreport rep;

        // create solver, set quadratic/linear terms
        minqpcreate(2, state);
        minqpsetquadraticterm(state, a, isupper);
        minqpsetlinearterm(state, b);
        minqpsetbc(state, bndl, bndu);

        // Set scale of the parameters.
        // It is strongly recommended that you set scale of your variables.
        // Knowing their scales is essential for evaluation of stopping criteria
        // and for preconditioning of the algorithm steps.
        // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
        //
        // NOTE: for convex problems you may try using minqpsetscaleautodiag()
        //       which automatically determines variable scales.
        minqpsetscale(state, s);

        //
        // Solve problem with the sparse interior-point method (sparse IPM) solver.
        //
        // This solver is intended for large-scale sparse problems with box and linear
        // constraints, but it will work on such a toy problem too.
        //
        // Commercial ALGLIB can parallelize sparse Cholesky factorization which is the
        // most time-consuming part of the algorithm. See the ALGLIB Reference Manual for
        // more information on how to activate parallelism support.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgosparseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]

        //
        // Solve problem with dense IPM solver.
        //
        // This solver is optimized for problems with dense linear constraints and/or
        // dense quadratic term.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgodenseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minqp_d_lc1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
        // subject to linear constraint x0+x1<=2
        //
        // Exact solution is [x0,x1] = [1.5,0.5]
        //
        // IMPORTANT: this solver minimizes  following  function:
        //
        //     f(x) = 0.5*x'*A*x + b'*x.
        //
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        //
        // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
        //
        //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
        //
        // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
        //
        real_2d_array a = "[[2,0],[0,2]]";
        real_1d_array b = "[-6,-4]";
        real_1d_array s = "[1,1]";
        real_2d_array c = "[[1.0,1.0,2.0]]";
        integer_1d_array ct = "[-1]";
        bool isupper = true;
        real_1d_array x;
        minqpstate state;
        minqpreport rep;

        // create solver, set quadratic/linear terms
        minqpcreate(2, state);
        minqpsetquadraticterm(state, a, isupper);
        minqpsetlinearterm(state, b);
        minqpsetlc(state, c, ct);

        // Set scale of the parameters.
        // It is strongly recommended that you set scale of your variables.
        // Knowing their scales is essential for evaluation of stopping criteria
        // and for preconditioning of the algorithm steps.
        // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
        //
        // NOTE: for convex problems you may try using minqpsetscaleautodiag()
        //       which automatically determines variable scales.
        minqpsetscale(state, s);

        //
        // Solve problem with the sparse interior-point method (sparse IPM) solver.
        //
        // This solver is intended for large-scale sparse problems with box and linear
        // constraints, but it will work on such a toy problem too.
        //
        // Commercial ALGLIB can parallelize sparse Cholesky factorization which is the
        // most time-consuming part of the algorithm. See the ALGLIB Reference Manual for
        // more information on how to activate parallelism support.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgosparseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.5,0.5]

        //
        // Solve problem with dense IPM solver.
        //
        // This solver is optimized for problems with dense linear constraints and/or
        // dense quadratic term.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgodenseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.5,0.5]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minqp_d_nonconvex example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of nonconvex function
        //     F(x0,x1) = -(x0^2+x1^2)
        // subject to constraints x0,x1 in [1.0,2.0]
        // Exact solution is [x0,x1] = [2,2].
        //
        // Non-convex problems are harder to solve than convex ones, and they
        // may have more than one local minimum. However, ALGLIB solvers may deal
        // with such problems (although they do not guarantee convergence to
        // global minimum).
        //
        // IMPORTANT: this solver minimizes  following  function:
        //     f(x) = 0.5*x'*A*x + b'*x.
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        //
        // For example, our function is f(x)=-(x0^2+x1^2), but we rewrite it as 
        //     f(x) = 0.5*(-2*x0^2-2*x1^2)
        // and pass diag(-2,-2) as quadratic term - NOT diag(-1,-1)!
        //
        real_2d_array a = "[[-2,0],[0,-2]]";
        real_1d_array x0 = "[1,1]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[1.0,1.0]";
        real_1d_array bndu = "[2.0,2.0]";
        bool isupper = true;
        real_1d_array x;
        minqpstate state;
        minqpreport rep;

        // create solver, set quadratic/linear terms, constraints
        minqpcreate(2, state);
        minqpsetquadraticterm(state, a, isupper);
        minqpsetstartingpoint(state, x0);
        minqpsetbc(state, bndl, bndu);

        // Set scale of the parameters.
        // It is strongly recommended that you set scale of your variables.
        // Knowing their scales is essential for evaluation of stopping criteria
        // and for preconditioning of the algorithm steps.
        // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
        //
        // NOTE: there also exists minqpsetscaleautodiag() function
        //       which automatically determines variable scales; however,
        //       it does NOT work for non-convex problems.
        minqpsetscale(state, s);

        //
        // Solve problem with DENSE-GENIPM solver.
        //
        // This solver is optimized for nonconvex problems with up to several thousands of
        // variables and large amount of general linear constraints.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgodensegenipm(state, 1.0e-9);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,2]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minqp_d_u1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
        //
        // Exact solution is [x0,x1] = [3,2]
        //
        // IMPORTANT: this solver minimizes  following  function:
        //
        //     f(x) = 0.5*x'*A*x + b'*x.
        //
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        //
        // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
        //
        //     f(x) = 0.5*(2*x0^2+2*x1^2) + .... 
        //
        // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
        //
        real_2d_array a = "[[2,0],[0,2]]";
        real_1d_array b = "[-6,-4]";
        real_1d_array s = "[1,1]";
        bool isupper = true;
        real_1d_array x;
        minqpstate state;
        minqpreport rep;

        // create the solver, set quadratic/linear terms
        minqpcreate(2, state);
        minqpsetquadraticterm(state, a, isupper);
        minqpsetlinearterm(state, b);

        // Set the scale of the parameters.
        // It is strongly recommended that you set the scale of your variables.
        // Knowing their scales is essential for evaluation of stopping criteria
        // and for preconditioning of the algorithm steps.
        // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
        //
        // NOTE: for convex problems you may try using minqpsetscaleautodiag()
        //       which automatically determines variable scales.
        minqpsetscale(state, s);

        //
        // Solve problem with the sparse interior-point method (sparse IPM) solver.
        //
        // This solver is intended for large-scale sparse problems with box and linear
        // constraints, but it will work on such a toy problem too.
        //
        // Commercial ALGLIB can parallelize sparse Cholesky factorization which is the
        // most time-consuming part of the algorithm. See the ALGLIB Reference Manual for
        // more information on how to activate parallelism support.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgosparseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]

        //
        // Solve problem with dense IPM solver.
        //
        // This solver is optimized for problems with dense linear constraints and/or
        // dense quadratic term.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgodenseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

minqp_d_u2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1,
        // with quadratic term given by sparse matrix structure.
        //
        // Exact solution is [x0,x1] = [3,2]
        //
        // We provide algorithm with starting point, although in this case
        // (dense matrix, no constraints) it can work without such information.
        //
        // IMPORTANT: this solver minimizes  following  function:
        //     f(x) = 0.5*x'*A*x + b'*x.
        // Note that quadratic term has 0.5 before it. So if you want to minimize
        // quadratic function, you should rewrite it in such way that quadratic term
        // is multiplied by 0.5 too.
        //
        // For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as 
        //     f(x) = 0.5*(2*x0^2+2*x1^2) + ....
        // and pass diag(2,2) as quadratic term - NOT diag(1,1)!
        //
        sparsematrix a;
        real_1d_array b = "[-6,-4]";
        real_1d_array x0 = "[0,1]";
        real_1d_array s = "[1,1]";
        real_1d_array x;
        minqpstate state;
        minqpreport rep;

        // initialize sparsematrix structure
        sparsecreate(2, 2, 0, a);
        sparseset(a, 0, 0, 2.0);
        sparseset(a, 1, 1, 2.0);

        // create solver, set quadratic/linear terms
        minqpcreate(2, state);
        minqpsetquadratictermsparse(state, a, true);
        minqpsetlinearterm(state, b);
        minqpsetstartingpoint(state, x0);

        // Set scale of the parameters.
        // It is strongly recommended that you set scale of your variables.
        // Knowing their scales is essential for evaluation of stopping criteria
        // and for preconditioning of the algorithm steps.
        // You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
        //
        // NOTE: for convex problems you may try using minqpsetscaleautodiag()
        //       which automatically determines variable scales.
        minqpsetscale(state, s);

        //
        // Solve problem with the sparse interior-point method (sparse IPM) solver.
        //
        // This solver is intended for large-scale sparse problems with box and linear
        // constraints, but it will work on such a toy problem too.
        //
        // Commercial ALGLIB can parallelize sparse Cholesky factorization which is the
        // most time-consuming part of the algorithm. See the ALGLIB Reference Manual for
        // more information on how to activate parallelism support.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgosparseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]

        //
        // Solve problem with dense IPM solver.
        //
        // This solver is optimized for problems with dense linear constraints and/or
        // dense quadratic term.
        //
        // Default stopping criteria are used.
        //
        minqpsetalgodenseipm(state, 0.0);
        minqpoptimize(state);
        minqpresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`mlpbase` subpackage

Classes

modelerrors
multilayerperceptron

Functions

mlpactivationfunction
mlpallerrorssparsesubset
mlpallerrorssubset
mlpavgce
mlpavgcesparse
mlpavgerror
mlpavgerrorsparse
mlpavgrelerror
mlpavgrelerrorsparse
mlpclserror
mlpcopy
mlpcopytunableparameters
mlpcreate0
mlpcreate1
mlpcreate2
mlpcreateb0
mlpcreateb1
mlpcreateb2
mlpcreatec0
mlpcreatec1
mlpcreatec2
mlpcreater0
mlpcreater1
mlpcreater2
mlperror
mlperrorn
mlperrorsparse
mlperrorsparsesubset
mlperrorsubset
mlpgetinputscaling
mlpgetinputscount
mlpgetlayerscount
mlpgetlayersize
mlpgetneuroninfo
mlpgetoutputscaling
mlpgetoutputscount
mlpgetweight
mlpgetweightscount
mlpgrad
mlpgradbatch
mlpgradbatchsparse
mlpgradbatchsparsesubset
mlpgradbatchsubset
mlpgradn
mlpgradnbatch
mlphessianbatch
mlphessiannbatch
mlpinitpreprocessor
mlpissoftmax
mlpprocess
mlpprocessi
mlpproperties
mlprandomize
mlprandomizefull
mlprelclserror
mlprelclserrorsparse
mlprmserror
mlprmserrorsparse
mlpserialize
mlpsetinputscaling
mlpsetneuroninfo
mlpsetoutputscaling
mlpsetweight
mlpunserialize

Examples

`modelerrors` class

/*************************************************************************
Model's errors:
    * RelCLSError   -   fraction of misclassified cases.
    * AvgCE         -   acerage cross-entropy
    * RMSError      -   root-mean-square error
    * AvgError      -   average error
    * AvgRelError   -   average relative error

NOTE 1: RelCLSError/AvgCE are zero on regression problems.

NOTE 2: on classification problems  RMSError/AvgError/AvgRelError  contain
        errors in prediction of posterior probabilities
*************************************************************************/
class modelerrors
{
public:
    modelerrors();
    modelerrors(const modelerrors &rhs);
    modelerrors& operator=(const modelerrors &rhs);
    virtual ~modelerrors();
    double relclserror;
    double avgce;
    double rmserror;
    double avgerror;
    double avgrelerror;
};

`multilayerperceptron` class

/*************************************************************************

*************************************************************************/
class multilayerperceptron
{
public:
    multilayerperceptron();
    multilayerperceptron(const multilayerperceptron &rhs);
    multilayerperceptron& operator=(const multilayerperceptron &rhs);
    virtual ~multilayerperceptron();
};

`mlpactivationfunction` function

/*************************************************************************
Neural network activation function

INPUT PARAMETERS:
    NET         -   neuron input
    K           -   function index (zero for linear function)

OUTPUT PARAMETERS:
    F           -   function
    DF          -   its derivative
    D2F         -   its second derivative

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpactivationfunction(const double net, const ae_int_t k, double &f, double &df, double &d2f, const xparams _xparams = alglib::xdefault);

`mlpallerrorssparsesubset` function

/*************************************************************************
Calculation of all types of errors on subset of dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset given by sparse matrix;
                one sample = one row;
                first NIn columns contain inputs,
                next NOut columns - desired outputs.
    SetSize -   real size of XY, SetSize>=0;
    Subset  -   subset of SubsetSize elements, array[SubsetSize];
    SubsetSize- number of elements in Subset[] array:
                * if SubsetSize>0, rows of XY with indices Subset[0]...
                  ...Subset[SubsetSize-1] are processed
                * if SubsetSize=0, zeros are returned
                * if SubsetSize<0, entire dataset is  processed;  Subset[]
                  array is ignored in this case.

OUTPUT PARAMETERS:
    Rep     -   it contains all type of errors.


  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
void mlpallerrorssparsesubset(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t setsize, const integer_1d_array &subset, const ae_int_t subsetsize, modelerrors &rep, const xparams _xparams = alglib::xdefault);

`mlpallerrorssubset` function

/*************************************************************************
Calculation of all types of errors on subset of dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset; one sample = one row;
                first NIn columns contain inputs,
                next NOut columns - desired outputs.
    SetSize -   real size of XY, SetSize>=0;
    Subset  -   subset of SubsetSize elements, array[SubsetSize];
    SubsetSize- number of elements in Subset[] array:
                * if SubsetSize>0, rows of XY with indices Subset[0]...
                  ...Subset[SubsetSize-1] are processed
                * if SubsetSize=0, zeros are returned
                * if SubsetSize<0, entire dataset is  processed;  Subset[]
                  array is ignored in this case.

OUTPUT PARAMETERS:
    Rep     -   it contains all type of errors.

  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
void mlpallerrorssubset(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t setsize, const integer_1d_array &subset, const ae_int_t subsetsize, modelerrors &rep, const xparams _xparams = alglib::xdefault);

`mlpavgce` function

/*************************************************************************
Average cross-entropy  (in bits  per element) on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
CrossEntropy/(NPoints*LN(2)).
Zero if network solves regression task.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 08.01.2009 by Bochkanov Sergey
*************************************************************************/
double mlpavgce(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpavgcesparse` function

/*************************************************************************
Average  cross-entropy  (in bits  per element)  on the  test set  given by
sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
CrossEntropy/(NPoints*LN(2)).
Zero if network solves regression task.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 9.08.2012 by Bochkanov Sergey
*************************************************************************/
double mlpavgcesparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpavgerror` function

/*************************************************************************
Average absolute error on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means average error when estimating posterior probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 11.03.2008 by Bochkanov Sergey
*************************************************************************/
double mlpavgerror(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpavgerrorsparse` function

/*************************************************************************
Average absolute error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means average error when estimating posterior probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double mlpavgerrorsparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpavgrelerror` function

/*************************************************************************
Average relative error on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means  average  relative  error  when  estimating posterior probability of
belonging to the correct class.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 11.03.2008 by Bochkanov Sergey
*************************************************************************/
double mlpavgrelerror(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpavgrelerrorsparse` function

/*************************************************************************
Average relative error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
Its meaning for regression task is obvious. As for classification task, it
means  average  relative  error  when  estimating posterior probability of
belonging to the correct class.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double mlpavgrelerrorsparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpclserror` function

/*************************************************************************
Classification error of the neural network on dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
    classification error (number of misclassified cases)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
ae_int_t mlpclserror(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpcopy` function

/*************************************************************************
Copying of neural network

INPUT PARAMETERS:
    Network1 -   original

OUTPUT PARAMETERS:
    Network2 -   copy

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcopy(const multilayerperceptron &network1, multilayerperceptron &network2, const xparams _xparams = alglib::xdefault);

`mlpcopytunableparameters` function

/*************************************************************************
This function copies tunable  parameters (weights/means/sigmas)  from  one
network to another with same architecture. It  performs  some  rudimentary
checks that architectures are same, and throws exception if check fails.

It is intended for fast copying of states between two  network  which  are
known to have same geometry.

INPUT PARAMETERS:
    Network1 -   source, must be correctly initialized
    Network2 -   target, must have same architecture

OUTPUT PARAMETERS:
    Network2 -   network state is copied from source to target

  -- ALGLIB --
     Copyright 20.06.2013 by Bochkanov Sergey
*************************************************************************/
void mlpcopytunableparameters(const multilayerperceptron &network1, multilayerperceptron &network2, const xparams _xparams = alglib::xdefault);

`mlpcreate0` function

/*************************************************************************
Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
layers, with linear output layer. Network weights are  filled  with  small
random values.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcreate0(const ae_int_t nin, const ae_int_t nout, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreate1` function

/*************************************************************************
Same  as  MLPCreate0,  but  with  one  hidden  layer  (NHid  neurons) with
non-linear activation function. Output layer is linear.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcreate1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreate2` function

/*************************************************************************
Same as MLPCreate0, but with two hidden layers (NHid1 and  NHid2  neurons)
with non-linear activation function. Output layer is linear.
 $ALL

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcreate2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreateb0` function

/*************************************************************************
Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
layers with non-linear output layer. Network weights are filled with small
random values.

Activation function of the output layer takes values:

    (B, +INF), if D>=0

or

    (-INF, B), if D<0.


  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpcreateb0(const ae_int_t nin, const ae_int_t nout, const double b, const double d, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreateb1` function

/*************************************************************************
Same as MLPCreateB0 but with non-linear hidden layer.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpcreateb1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, const double b, const double d, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreateb2` function

/*************************************************************************
Same as MLPCreateB0 but with two non-linear hidden layers.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpcreateb2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, const double b, const double d, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreatec0` function

/*************************************************************************
Creates classifier network with NIn  inputs  and  NOut  possible  classes.
Network contains no hidden layers and linear output  layer  with  SOFTMAX-
normalization  (so  outputs  sums  up  to  1.0  and  converge to posterior
probabilities).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcreatec0(const ae_int_t nin, const ae_int_t nout, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreatec1` function

/*************************************************************************
Same as MLPCreateC0, but with one non-linear hidden layer.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcreatec1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreatec2` function

/*************************************************************************
Same as MLPCreateC0, but with two non-linear hidden layers.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpcreatec2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreater0` function

/*************************************************************************
Creates  neural  network  with  NIn  inputs,  NOut outputs, without hidden
layers with non-linear output layer. Network weights are filled with small
random values. Activation function of the output layer takes values [A,B].

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpcreater0(const ae_int_t nin, const ae_int_t nout, const double a, const double b, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreater1` function

/*************************************************************************
Same as MLPCreateR0, but with non-linear hidden layer.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpcreater1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, const double a, const double b, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreater2` function

/*************************************************************************
Same as MLPCreateR0, but with two non-linear hidden layers.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpcreater2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, const double a, const double b, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlperror` function

/*************************************************************************
Error of the neural network on dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
double mlperror(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlperrorn` function

/*************************************************************************
Natural error function for neural network, internal subroutine.

NOTE: this function is single-threaded. Unlike other  error  function,  it
receives no speed-up from being executed in SMP mode.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
double mlperrorn(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t ssize, const xparams _xparams = alglib::xdefault);

`mlperrorsparse` function

/*************************************************************************
Error of the neural network on dataset given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
double mlperrorsparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlperrorsparsesubset` function

/*************************************************************************
Error of the neural network on subset of sparse dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network   -     neural network;
    XY        -     training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    SetSize   -     real size of XY, SetSize>=0;
                    it is used when SubsetSize<0;
    Subset    -     subset of SubsetSize elements, array[SubsetSize];
    SubsetSize-     number of elements in Subset[] array:
                    * if SubsetSize>0, rows of XY with indices Subset[0]...
                      ...Subset[SubsetSize-1] are processed
                    * if SubsetSize=0, zeros are returned
                    * if SubsetSize<0, entire dataset is  processed;  Subset[]
                      array is ignored in this case.

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
double mlperrorsparsesubset(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t setsize, const integer_1d_array &subset, const ae_int_t subsetsize, const xparams _xparams = alglib::xdefault);

`mlperrorsubset` function

/*************************************************************************
Error of the neural network on subset of dataset.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network   -     neural network;
    XY        -     training  set,  see  below  for  information  on   the
                    training set format;
    SetSize   -     real size of XY, SetSize>=0;
    Subset    -     subset of SubsetSize elements, array[SubsetSize];
    SubsetSize-     number of elements in Subset[] array:
                    * if SubsetSize>0, rows of XY with indices Subset[0]...
                      ...Subset[SubsetSize-1] are processed
                    * if SubsetSize=0, zeros are returned
                    * if SubsetSize<0, entire dataset is  processed;  Subset[]
                      array is ignored in this case.

RESULT:
    sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2)

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.09.2012 by Bochkanov Sergey
*************************************************************************/
double mlperrorsubset(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t setsize, const integer_1d_array &subset, const ae_int_t subsetsize, const xparams _xparams = alglib::xdefault);

`mlpgetinputscaling` function

/*************************************************************************
This function returns offset/scaling coefficients for I-th input of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index

OUTPUT PARAMETERS:
    Mean        -   mean term
    Sigma       -   sigma term, guaranteed to be nonzero.

I-th input is passed through linear transformation
    IN[i] = (IN[i]-Mean)/Sigma
before feeding to the network

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpgetinputscaling(const multilayerperceptron &network, const ae_int_t i, double &mean, double &sigma, const xparams _xparams = alglib::xdefault);

`mlpgetinputscount` function

/*************************************************************************
Returns number of inputs.

  -- ALGLIB --
     Copyright 19.10.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t mlpgetinputscount(const multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpgetlayerscount` function

/*************************************************************************
This function returns total number of layers (including input, hidden and
output layers).

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t mlpgetlayerscount(const multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpgetlayersize` function

/*************************************************************************
This function returns size of K-th layer.

K=0 corresponds to input layer, K=CNT-1 corresponds to output layer.

Size of the output layer is always equal to the number of outputs, although
when we have softmax-normalized network, last neuron doesn't have any
connections - it is just zero.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t mlpgetlayersize(const multilayerperceptron &network, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`mlpgetneuroninfo` function

/*************************************************************************
This function returns information about Ith neuron of Kth layer

INPUT PARAMETERS:
    Network     -   network
    K           -   layer index
    I           -   neuron index (within layer)

OUTPUT PARAMETERS:
    FKind       -   activation function type (used by MLPActivationFunction())
                    this value is zero for input or linear neurons
    Threshold   -   also called offset, bias
                    zero for input neurons

NOTE: this function throws exception if layer or neuron with  given  index
do not exists.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpgetneuroninfo(multilayerperceptron &network, const ae_int_t k, const ae_int_t i, ae_int_t &fkind, double &threshold, const xparams _xparams = alglib::xdefault);

`mlpgetoutputscaling` function

/*************************************************************************
This function returns offset/scaling coefficients for I-th output of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index

OUTPUT PARAMETERS:
    Mean        -   mean term
    Sigma       -   sigma term, guaranteed to be nonzero.

I-th output is passed through linear transformation
    OUT[i] = OUT[i]*Sigma+Mean
before returning it to user. In case we have SOFTMAX-normalized network,
we return (Mean,Sigma)=(0.0,1.0).

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpgetoutputscaling(const multilayerperceptron &network, const ae_int_t i, double &mean, double &sigma, const xparams _xparams = alglib::xdefault);

`mlpgetoutputscount` function

/*************************************************************************
Returns number of outputs.

  -- ALGLIB --
     Copyright 19.10.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t mlpgetoutputscount(const multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpgetweight` function

/*************************************************************************
This function returns information about connection from I0-th neuron of
K0-th layer to I1-th neuron of K1-th layer.

INPUT PARAMETERS:
    Network     -   network
    K0          -   layer index
    I0          -   neuron index (within layer)
    K1          -   layer index
    I1          -   neuron index (within layer)

RESULT:
    connection weight (zero for non-existent connections)

This function:
1. throws exception if layer or neuron with given index do not exists.
2. returns zero if neurons exist, but there is no connection between them

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
double mlpgetweight(multilayerperceptron &network, const ae_int_t k0, const ae_int_t i0, const ae_int_t k1, const ae_int_t i1, const xparams _xparams = alglib::xdefault);

`mlpgetweightscount` function

/*************************************************************************
Returns number of weights.

  -- ALGLIB --
     Copyright 19.10.2011 by Bochkanov Sergey
*************************************************************************/
ae_int_t mlpgetweightscount(const multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpgrad` function

/*************************************************************************
Gradient calculation

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    X       -   input vector, length of array must be at least NIn
    DesiredY-   desired outputs, length of array must be at least NOut
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpgrad(multilayerperceptron &network, const real_1d_array &x, const real_1d_array &desiredy, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlpgradbatch` function

/*************************************************************************
Batch gradient calculation for a set of inputs/outputs

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in dense format; one sample = one row:
                * first NIn columns contain inputs,
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SSize   -   number of elements in XY
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpgradbatch(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t ssize, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlpgradbatchsparse` function

/*************************************************************************
Batch gradient calculation for a set  of inputs/outputs  given  by  sparse
matrices

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in sparse format; one sample = one row:
                * MATRIX MUST BE STORED IN CRS FORMAT
                * first NIn columns contain inputs.
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SSize   -   number of elements in XY
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 26.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpgradbatchsparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t ssize, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlpgradbatchsparsesubset` function

/*************************************************************************
Batch gradient calculation for a set of inputs/outputs  for  a  subset  of
dataset given by set of indexes.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in sparse format; one sample = one row:
                * MATRIX MUST BE STORED IN CRS FORMAT
                * first NIn columns contain inputs,
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SetSize -   real size of XY, SetSize>=0;
    Idx     -   subset of SubsetSize elements, array[SubsetSize]:
                * Idx[I] stores row index in the original dataset which is
                  given by XY. Gradient is calculated with respect to rows
                  whose indexes are stored in Idx[].
                * Idx[]  must store correct indexes; this function  throws
                  an  exception  in  case  incorrect index (less than 0 or
                  larger than rows(XY)) is given
                * Idx[]  may  store  indexes  in  any  order and even with
                  repetitions.
    SubsetSize- number of elements in Idx[] array:
                * positive value means that subset given by Idx[] is processed
                * zero value results in zero gradient
                * negative value means that full dataset is processed
    Grad      - possibly  preallocated array. If size of array is  smaller
                than WCount, it will be reallocated. It is  recommended to
                reuse  previously  allocated  array  to  reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad    -   gradient  of  E  with  respect   to  weights  of  network,
                array[WCount]

NOTE: when  SubsetSize<0 is used full dataset by call MLPGradBatchSparse
      function.

  -- ALGLIB --
     Copyright 26.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpgradbatchsparsesubset(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t setsize, const integer_1d_array &idx, const ae_int_t subsetsize, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlpgradbatchsubset` function

/*************************************************************************
Batch gradient calculation for a subset of dataset

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   original dataset in dense format; one sample = one row:
                * first NIn columns contain inputs,
                * for regression problem, next NOut columns store
                  desired outputs.
                * for classification problem, next column (just one!)
                  stores class number.
    SetSize -   real size of XY, SetSize>=0;
    Idx     -   subset of SubsetSize elements, array[SubsetSize]:
                * Idx[I] stores row index in the original dataset which is
                  given by XY. Gradient is calculated with respect to rows
                  whose indexes are stored in Idx[].
                * Idx[]  must store correct indexes; this function  throws
                  an  exception  in  case  incorrect index (less than 0 or
                  larger than rows(XY)) is given
                * Idx[]  may  store  indexes  in  any  order and even with
                  repetitions.
    SubsetSize- number of elements in Idx[] array:
                * positive value means that subset given by Idx[] is processed
                * zero value results in zero gradient
                * negative value means that full dataset is processed
    Grad      - possibly  preallocated array. If size of array is  smaller
                than WCount, it will be reallocated. It is  recommended to
                reuse  previously  allocated  array  to  reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E         - error function, SUM(sqr(y[i]-desiredy[i])/2,i)
    Grad      - gradient  of  E  with  respect   to  weights  of  network,
                array[WCount]

  -- ALGLIB --
     Copyright 26.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpgradbatchsubset(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t setsize, const integer_1d_array &idx, const ae_int_t subsetsize, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlpgradn` function

/*************************************************************************
Gradient calculation (natural error function is used)

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    X       -   input vector, length of array must be at least NIn
    DesiredY-   desired outputs, length of array must be at least NOut
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, sum-of-squares for regression networks,
                cross-entropy for classification networks.
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpgradn(multilayerperceptron &network, const real_1d_array &x, const real_1d_array &desiredy, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlpgradnbatch` function

/*************************************************************************
Batch gradient calculation for a set of inputs/outputs
(natural error function is used)

INPUT PARAMETERS:
    Network -   network initialized with one of the network creation funcs
    XY      -   set of inputs/outputs; one sample = one row;
                first NIn columns contain inputs,
                next NOut columns - desired outputs.
    SSize   -   number of elements in XY
    Grad    -   possibly preallocated array. If size of array is smaller
                than WCount, it will be reallocated. It is recommended to
                reuse previously allocated array to reduce allocation
                overhead.

OUTPUT PARAMETERS:
    E       -   error function, sum-of-squares for regression networks,
                cross-entropy for classification networks.
    Grad    -   gradient of E with respect to weights of network, array[WCount]

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpgradnbatch(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t ssize, double &e, real_1d_array &grad, const xparams _xparams = alglib::xdefault);

`mlphessianbatch` function

/*************************************************************************
Batch Hessian calculation using R-algorithm.
Internal subroutine.

  -- ALGLIB --
     Copyright 26.01.2008 by Bochkanov Sergey.

     Hessian calculation based on R-algorithm described in
     "Fast Exact Multiplication by the Hessian",
     B. A. Pearlmutter,
     Neural Computation, 1994.
*************************************************************************/
void mlphessianbatch(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t ssize, double &e, real_1d_array &grad, real_2d_array &h, const xparams _xparams = alglib::xdefault);

`mlphessiannbatch` function

/*************************************************************************
Batch Hessian calculation (natural error function) using R-algorithm.
Internal subroutine.

  -- ALGLIB --
     Copyright 26.01.2008 by Bochkanov Sergey.

     Hessian calculation based on R-algorithm described in
     "Fast Exact Multiplication by the Hessian",
     B. A. Pearlmutter,
     Neural Computation, 1994.
*************************************************************************/
void mlphessiannbatch(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t ssize, double &e, real_1d_array &grad, real_2d_array &h, const xparams _xparams = alglib::xdefault);

`mlpinitpreprocessor` function

/*************************************************************************
Internal subroutine.

  -- ALGLIB --
     Copyright 30.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlpinitpreprocessor(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t ssize, const xparams _xparams = alglib::xdefault);

`mlpissoftmax` function

/*************************************************************************
Tells whether network is SOFTMAX-normalized (i.e. classifier) or not.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
bool mlpissoftmax(const multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    Network -   neural network
    X       -   input vector,  array[0..NIn-1].

OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

See also MLPProcessI

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpprocess(multilayerperceptron &network, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`mlpprocessi` function

/*************************************************************************
'interactive'  variant  of  MLPProcess  for  languages  like  Python which
support constructs like "Y = MLPProcess(NN,X)" and interactive mode of the
interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 21.09.2010 by Bochkanov Sergey
*************************************************************************/
void mlpprocessi(multilayerperceptron &network, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`mlpproperties` function

/*************************************************************************
Returns information about initialized network: number of inputs, outputs,
weights.

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlpproperties(const multilayerperceptron &network, ae_int_t &nin, ae_int_t &nout, ae_int_t &wcount, const xparams _xparams = alglib::xdefault);

`mlprandomize` function

/*************************************************************************
Randomization of neural network weights

  -- ALGLIB --
     Copyright 06.11.2007 by Bochkanov Sergey
*************************************************************************/
void mlprandomize(multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlprandomizefull` function

/*************************************************************************
Randomization of neural network weights and standartisator

  -- ALGLIB --
     Copyright 10.03.2008 by Bochkanov Sergey
*************************************************************************/
void mlprandomizefull(multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlprelclserror` function

/*************************************************************************
Relative classification error on the test set.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Percent   of incorrectly   classified  cases.  Works  both  for classifier
networks and general purpose networks used as classifiers.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 25.12.2008 by Bochkanov Sergey
*************************************************************************/
double mlprelclserror(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlprelclserrorsparse` function

/*************************************************************************
Relative classification error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. Sparse matrix must use CRS format
                    for storage.
    NPoints     -   points count, >=0.

RESULT:
Percent   of incorrectly   classified  cases.  Works  both  for classifier
networks and general purpose networks used as classifiers.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double mlprelclserrorsparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlprmserror` function

/*************************************************************************
RMS error on the test set given.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format;
    NPoints     -   points count.

RESULT:
Root mean  square error. Its meaning for regression task is obvious. As for
classification  task,  RMS  error  means  error  when estimating  posterior
probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 04.11.2007 by Bochkanov Sergey
*************************************************************************/
double mlprmserror(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlprmserrorsparse` function

/*************************************************************************
RMS error on the test set given by sparse matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    Network     -   neural network;
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.  Sparse  matrix  must  use  CRS  format for
                    storage.
    NPoints     -   points count, >=0.

RESULT:
Root mean  square error. Its meaning for regression task is obvious. As for
classification  task,  RMS  error  means  error  when estimating  posterior
probabilities.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
dataset format is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 09.08.2012 by Bochkanov Sergey
*************************************************************************/
double mlprmserrorsparse(multilayerperceptron &network, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void mlpserialize(const multilayerperceptron &obj, std::string &s_out);
void mlpserialize(const multilayerperceptron &obj, std::ostream &s_out);

`mlpsetinputscaling` function

/*************************************************************************
This function sets offset/scaling coefficients for I-th input of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index
    Mean        -   mean term
    Sigma       -   sigma term (if zero, will be replaced by 1.0)

NTE: I-th input is passed through linear transformation
    IN[i] = (IN[i]-Mean)/Sigma
before feeding to the network. This function sets Mean and Sigma.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpsetinputscaling(multilayerperceptron &network, const ae_int_t i, const double mean, const double sigma, const xparams _xparams = alglib::xdefault);

`mlpsetneuroninfo` function

/*************************************************************************
This function modifies information about Ith neuron of Kth layer

INPUT PARAMETERS:
    Network     -   network
    K           -   layer index
    I           -   neuron index (within layer)
    FKind       -   activation function type (used by MLPActivationFunction())
                    this value must be zero for input neurons
                    (you can not set activation function for input neurons)
    Threshold   -   also called offset, bias
                    this value must be zero for input neurons
                    (you can not set threshold for input neurons)

NOTES:
1. this function throws exception if layer or neuron with given index do
   not exists.
2. this function also throws exception when you try to set non-linear
   activation function for input neurons (any kind of network) or for output
   neurons of classifier network.
3. this function throws exception when you try to set non-zero threshold for
   input neurons (any kind of network).

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpsetneuroninfo(multilayerperceptron &network, const ae_int_t k, const ae_int_t i, const ae_int_t fkind, const double threshold, const xparams _xparams = alglib::xdefault);

`mlpsetoutputscaling` function

/*************************************************************************
This function sets offset/scaling coefficients for I-th output of the
network.

INPUT PARAMETERS:
    Network     -   network
    I           -   input index
    Mean        -   mean term
    Sigma       -   sigma term (if zero, will be replaced by 1.0)

OUTPUT PARAMETERS:

NOTE: I-th output is passed through linear transformation
    OUT[i] = OUT[i]*Sigma+Mean
before returning it to user. This function sets Sigma/Mean. In case we
have SOFTMAX-normalized network, you can not set (Sigma,Mean) to anything
other than(0.0,1.0) - this function will throw exception.

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpsetoutputscaling(multilayerperceptron &network, const ae_int_t i, const double mean, const double sigma, const xparams _xparams = alglib::xdefault);

`mlpsetweight` function

/*************************************************************************
This function modifies information about connection from I0-th neuron of
K0-th layer to I1-th neuron of K1-th layer.

INPUT PARAMETERS:
    Network     -   network
    K0          -   layer index
    I0          -   neuron index (within layer)
    K1          -   layer index
    I1          -   neuron index (within layer)
    W           -   connection weight (must be zero for non-existent
                    connections)

This function:
1. throws exception if layer or neuron with given index do not exists.
2. throws exception if you try to set non-zero weight for non-existent
   connection

  -- ALGLIB --
     Copyright 25.03.2011 by Bochkanov Sergey
*************************************************************************/
void mlpsetweight(multilayerperceptron &network, const ae_int_t k0, const ae_int_t i0, const ae_int_t k1, const ae_int_t i1, const double w, const xparams _xparams = alglib::xdefault);

`mlpunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void mlpunserialize(const std::string &s_in, multilayerperceptron &obj);
void mlpunserialize(const std::istream &s_in, multilayerperceptron &obj);

`mlpe` subpackage

Classes

mlpensemble

Functions

mlpeavgce
mlpeavgerror
mlpeavgrelerror
mlpecreate0
mlpecreate1
mlpecreate2
mlpecreateb0
mlpecreateb1
mlpecreateb2
mlpecreatec0
mlpecreatec1
mlpecreatec2
mlpecreatefromnetwork
mlpecreater0
mlpecreater1
mlpecreater2
mlpeissoftmax
mlpeprocess
mlpeprocessi
mlpeproperties
mlperandomize
mlperelclserror
mlpermserror
mlpeserialize
mlpeunserialize

Examples

`mlpensemble` class

/*************************************************************************
Neural networks ensemble
*************************************************************************/
class mlpensemble
{
public:
    mlpensemble();
    mlpensemble(const mlpensemble &rhs);
    mlpensemble& operator=(const mlpensemble &rhs);
    virtual ~mlpensemble();
};

`mlpeavgce` function

/*************************************************************************
Average cross-entropy (in bits per element) on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    CrossEntropy/(NPoints*LN(2)).
    Zero if ensemble solves regression task.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double mlpeavgce(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpeavgerror` function

/*************************************************************************
Average error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for classification task
it means average error when estimating posterior probabilities.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double mlpeavgerror(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpeavgrelerror` function

/*************************************************************************
Average relative error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    Its meaning for regression task is obvious. As for classification task
it means average relative error when estimating posterior probabilities.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double mlpeavgrelerror(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpecreate0` function

/*************************************************************************
Like MLPCreate0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreate0(const ae_int_t nin, const ae_int_t nout, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreate1` function

/*************************************************************************
Like MLPCreate1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreate1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreate2` function

/*************************************************************************
Like MLPCreate2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreate2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreateb0` function

/*************************************************************************
Like MLPCreateB0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreateb0(const ae_int_t nin, const ae_int_t nout, const double b, const double d, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreateb1` function

/*************************************************************************
Like MLPCreateB1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreateb1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, const double b, const double d, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreateb2` function

/*************************************************************************
Like MLPCreateB2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreateb2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, const double b, const double d, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreatec0` function

/*************************************************************************
Like MLPCreateC0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreatec0(const ae_int_t nin, const ae_int_t nout, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreatec1` function

/*************************************************************************
Like MLPCreateC1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreatec1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreatec2` function

/*************************************************************************
Like MLPCreateC2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreatec2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreatefromnetwork` function

/*************************************************************************
Creates ensemble from network. Only network geometry is copied.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreatefromnetwork(const multilayerperceptron &network, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreater0` function

/*************************************************************************
Like MLPCreateR0, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreater0(const ae_int_t nin, const ae_int_t nout, const double a, const double b, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreater1` function

/*************************************************************************
Like MLPCreateR1, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreater1(const ae_int_t nin, const ae_int_t nhid, const ae_int_t nout, const double a, const double b, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpecreater2` function

/*************************************************************************
Like MLPCreateR2, but for ensembles.

  -- ALGLIB --
     Copyright 18.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpecreater2(const ae_int_t nin, const ae_int_t nhid1, const ae_int_t nhid2, const ae_int_t nout, const double a, const double b, const ae_int_t ensemblesize, mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpeissoftmax` function

/*************************************************************************
Return normalization type (whether ensemble is SOFTMAX-normalized or not).

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
bool mlpeissoftmax(const mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlpeprocess` function

/*************************************************************************
Procesing

INPUT PARAMETERS:
    Ensemble-   neural networks ensemble
    X       -   input vector,  array[0..NIn-1].
    Y       -   (possibly) preallocated buffer; if size of Y is less than
                NOut, it will be reallocated. If it is large enough, it
                is NOT reallocated, so we can save some time on reallocation.


OUTPUT PARAMETERS:
    Y       -   result. Regression estimate when solving regression  task,
                vector of posterior probabilities for classification task.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpeprocess(mlpensemble &ensemble, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`mlpeprocessi` function

/*************************************************************************
'interactive'  variant  of  MLPEProcess  for  languages  like Python which
support constructs like "Y = MLPEProcess(LM,X)" and interactive mode of the
interpreter

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpeprocessi(mlpensemble &ensemble, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`mlpeproperties` function

/*************************************************************************
Return ensemble properties (number of inputs and outputs).

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpeproperties(const mlpensemble &ensemble, ae_int_t &nin, ae_int_t &nout, const xparams _xparams = alglib::xdefault);

`mlperandomize` function

/*************************************************************************
Randomization of MLP ensemble

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlperandomize(mlpensemble &ensemble, const xparams _xparams = alglib::xdefault);

`mlperelclserror` function

/*************************************************************************
Relative classification error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    percent of incorrectly classified cases.
    Works both for classifier betwork and for regression networks which
are used as classifiers.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double mlperelclserror(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpermserror` function

/*************************************************************************
RMS error on the test set

INPUT PARAMETERS:
    Ensemble-   ensemble
    XY      -   test set
    NPoints -   test set size

RESULT:
    root mean square error.
    Its meaning for regression task is obvious. As for classification task
RMS error means error when estimating posterior probabilities.

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
double mlpermserror(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpeserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void mlpeserialize(const mlpensemble &obj, std::string &s_out);
void mlpeserialize(const mlpensemble &obj, std::ostream &s_out);

`mlpeunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void mlpeunserialize(const std::string &s_in, mlpensemble &obj);
void mlpeunserialize(const std::istream &s_in, mlpensemble &obj);

`mlptrain` subpackage

Classes

mlpcvreport
mlpreport
mlptrainer

Functions

mlpcontinuetraining
mlpcreatetrainer
mlpcreatetrainercls
mlpebagginglbfgs
mlpebagginglm
mlpetraines
mlpkfoldcv
mlpkfoldcvlbfgs
mlpkfoldcvlm
mlpsetalgobatch
mlpsetcond
mlpsetdataset
mlpsetdecay
mlpsetsparsedataset
mlpstarttraining
mlptrainensemblees
mlptraines
mlptrainlbfgs
mlptrainlm
mlptrainnetwork

Examples

nn_cls2		Binary classification problem
nn_cls3		Multiclass classification problem
nn_crossvalidation		Cross-validation
nn_ensembles_es		Early stopping ensembles
nn_parallel		Parallel training
nn_regr		Regression problem with one output (2=>1)
nn_regr_n		Regression problem with multiple outputs (2=>2)
nn_trainerobject		Advanced example on trainer object

`mlpcvreport` class

/*************************************************************************
Cross-validation estimates of generalization error
*************************************************************************/
class mlpcvreport
{
public:
    mlpcvreport();
    mlpcvreport(const mlpcvreport &rhs);
    mlpcvreport& operator=(const mlpcvreport &rhs);
    virtual ~mlpcvreport();
    double relclserror;
    double avgce;
    double rmserror;
    double avgerror;
    double avgrelerror;
};

`mlpreport` class

/*************************************************************************
Training report:
    * RelCLSError   -   fraction of misclassified cases.
    * AvgCE         -   acerage cross-entropy
    * RMSError      -   root-mean-square error
    * AvgError      -   average error
    * AvgRelError   -   average relative error
    * NGrad         -   number of gradient calculations
    * NHess         -   number of Hessian calculations
    * NCholesky     -   number of Cholesky decompositions

NOTE 1: RelCLSError/AvgCE are zero on regression problems.

NOTE 2: on classification problems  RMSError/AvgError/AvgRelError  contain
        errors in prediction of posterior probabilities
*************************************************************************/
class mlpreport
{
public:
    mlpreport();
    mlpreport(const mlpreport &rhs);
    mlpreport& operator=(const mlpreport &rhs);
    virtual ~mlpreport();
    double relclserror;
    double avgce;
    double rmserror;
    double avgerror;
    double avgrelerror;
    ae_int_t ngrad;
    ae_int_t nhess;
    ae_int_t ncholesky;
};

`mlptrainer` class

/*************************************************************************
Trainer object for neural network.

You should not try to access fields of this object directly -  use  ALGLIB
functions to work with this object.
*************************************************************************/
class mlptrainer
{
public:
    mlptrainer();
    mlptrainer(const mlptrainer &rhs);
    mlptrainer& operator=(const mlptrainer &rhs);
    virtual ~mlptrainer();
};

`mlpcontinuetraining` function

/*************************************************************************
IMPORTANT: this is an "expert" version of the MLPTrain() function.  We  do
           not recommend you to use it unless you are pretty sure that you
           need ability to monitor training progress.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

This function performs step-by-step training of the neural  network.  Here
"step-by-step" means that training starts  with  MLPStartTraining()  call,
and then user subsequently calls MLPContinueTraining() to perform one more
iteration of the training.

This  function  performs  one  more  iteration of the training and returns
either True (training continues) or False (training stopped). In case True
was returned, Network weights are updated according to the  current  state
of the optimization progress. In case False was  returned,  no  additional
updates is performed (previous update of  the  network weights moved us to
the final point, and no additional updates is needed).

EXAMPLE:
    >
    > [initialize network and trainer object]
    >
    > MLPStartTraining(Trainer, Network, True)
    > while MLPContinueTraining(Trainer, Network) do
    >     [visualize training progress]
    >

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural  network  structure,  which  is  used to  store
                    current state of the training process.

OUTPUT PARAMETERS:
    Network     -   weights of the neural network  are  rewritten  by  the
                    current approximation.

NOTE: this method uses sum-of-squares error function for training.

NOTE: it is expected that trainer object settings are NOT  changed  during
      step-by-step training, i.e. no  one  changes  stopping  criteria  or
      training set during training. It is possible and there is no defense
      against  such  actions,  but  algorithm  behavior  in  such cases is
      undefined and can be unpredictable.

NOTE: It  is  expected that Network is the same one which  was  passed  to
      MLPStartTraining() function.  However,  THIS  function  checks  only
      following:
      * that number of network inputs is consistent with trainer object
        settings
      * that number of network outputs/classes is consistent with  trainer
        object settings
      * that number of network weights is the same as number of weights in
        the network passed to MLPStartTraining() function
      Exception is thrown when these conditions are violated.

      It is also expected that you do not change state of the  network  on
      your own - the only party who has right to change network during its
      training is a trainer object. Any attempt to interfere with  trainer
      may lead to unpredictable results.


  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
bool mlpcontinuetraining(mlptrainer &s, multilayerperceptron &network, const xparams _xparams = alglib::xdefault);

`mlpcreatetrainer` function

/*************************************************************************
Creation of the network trainer object for regression networks

INPUT PARAMETERS:
    NIn         -   number of inputs, NIn>=1
    NOut        -   number of outputs, NOut>=1

OUTPUT PARAMETERS:
    S           -   neural network trainer object.
                    This structure can be used to train any regression
                    network with NIn inputs and NOut outputs.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpcreatetrainer(const ae_int_t nin, const ae_int_t nout, mlptrainer &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

`mlpcreatetrainercls` function

/*************************************************************************
Creation of the network trainer object for classification networks

INPUT PARAMETERS:
    NIn         -   number of inputs, NIn>=1
    NClasses    -   number of classes, NClasses>=2

OUTPUT PARAMETERS:
    S           -   neural network trainer object.
                    This structure can be used to train any classification
                    network with NIn inputs and NOut outputs.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpcreatetrainercls(const ae_int_t nin, const ae_int_t nclasses, mlptrainer &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mlpebagginglbfgs` function

/*************************************************************************
Training neural networks ensemble using  bootstrap  aggregating (bagging).
L-BFGS algorithm is used as base training method.

INPUT PARAMETERS:
    Ensemble    -   model with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay coefficient, >=0.001
    Restarts    -   restarts, >0.
    WStep       -   stopping criterion, same as in MLPTrainLBFGS
    MaxIts      -   stopping criterion, same as in MLPTrainLBFGS

OUTPUT PARAMETERS:
    Ensemble    -   trained model
    Info        -   return code:
                    * -8, if both WStep=0 and MaxIts=0
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report.
    OOBErrors   -   out-of-bag generalization error estimate

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpebagginglbfgs(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, const double wstep, const ae_int_t maxits, ae_int_t &info, mlpreport &rep, mlpcvreport &ooberrors, const xparams _xparams = alglib::xdefault);

`mlpebagginglm` function

/*************************************************************************
Training neural networks ensemble using  bootstrap  aggregating (bagging).
Modified Levenberg-Marquardt algorithm is used as base training method.

INPUT PARAMETERS:
    Ensemble    -   model with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay coefficient, >=0.001
    Restarts    -   restarts, >0.

OUTPUT PARAMETERS:
    Ensemble    -   trained model
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report.
    OOBErrors   -   out-of-bag generalization error estimate

  -- ALGLIB --
     Copyright 17.02.2009 by Bochkanov Sergey
*************************************************************************/
void mlpebagginglm(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, ae_int_t &info, mlpreport &rep, mlpcvreport &ooberrors, const xparams _xparams = alglib::xdefault);

`mlpetraines` function

/*************************************************************************
Training neural networks ensemble using early stopping.

INPUT PARAMETERS:
    Ensemble    -   model with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay coefficient, >=0.001
    Restarts    -   restarts, >0.

OUTPUT PARAMETERS:
    Ensemble    -   trained model
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NClasses-1].
                    * -1, if incorrect parameters was passed
                          (NPoints<0, Restarts<1).
                    *  6, if task has been solved.
    Rep         -   training report.
    OOBErrors   -   out-of-bag generalization error estimate

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void mlpetraines(mlpensemble &ensemble, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, ae_int_t &info, mlpreport &rep, const xparams _xparams = alglib::xdefault);

`mlpkfoldcv` function

/*************************************************************************
This function estimates generalization error using cross-validation on the
current dataset with current training settings.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural network. It must have same number of inputs and
                    output/classes as was specified during creation of the
                    trainer object. Network is not changed  during  cross-
                    validation and is not trained - it  is  used  only  as
                    representative of its architecture. I.e., we  estimate
                    generalization properties of  ARCHITECTURE,  not  some
                    specific network.
    NRestarts   -   number of restarts, >=0:
                    * NRestarts>0  means  that  for  each cross-validation
                      round   specified  number   of  random  restarts  is
                      performed,  with  best  network  being  chosen after
                      training.
                    * NRestarts=0 is same as NRestarts=1
    FoldsCount  -   number of folds in k-fold cross-validation:
                    * 2<=FoldsCount<=size of dataset
                    * recommended value: 10.
                    * values larger than dataset size will be silently
                      truncated down to dataset size

OUTPUT PARAMETERS:
    Rep         -   structure which contains cross-validation estimates:
                    * Rep.RelCLSError - fraction of misclassified cases.
                    * Rep.AvgCE - acerage cross-entropy
                    * Rep.RMSError - root-mean-square error
                    * Rep.AvgError - average error
                    * Rep.AvgRelError - average relative error

NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(),
      or subset with only one point  was  given,  zeros  are  returned  as
      estimates.

NOTE: this method performs FoldsCount cross-validation  rounds,  each  one
      with NRestarts random starts.  Thus,  FoldsCount*NRestarts  networks
      are trained in total.

NOTE: Rep.RelCLSError/Rep.AvgCE are zero on regression problems.

NOTE: on classification problems Rep.RMSError/Rep.AvgError/Rep.AvgRelError
      contain errors in prediction of posterior probabilities.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpkfoldcv(mlptrainer &s, const multilayerperceptron &network, const ae_int_t nrestarts, const ae_int_t foldscount, mlpreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mlpkfoldcvlbfgs` function

/*************************************************************************
Cross-validation estimate of generalization error.

Base algorithm - L-BFGS.

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry.   Network is
                    not changed during cross-validation -  it is used only
                    as a representative of its architecture.
    XY          -   training set.
    SSize       -   training set size
    Decay       -   weight  decay, same as in MLPTrainLBFGS
    Restarts    -   number of restarts, >0.
                    restarts are counted for each partition separately, so
                    total number of restarts will be Restarts*FoldsCount.
    WStep       -   stopping criterion, same as in MLPTrainLBFGS
    MaxIts      -   stopping criterion, same as in MLPTrainLBFGS
    FoldsCount  -   number of folds in k-fold cross-validation,
                    2<=FoldsCount<=SSize.
                    recommended value: 10.

OUTPUT PARAMETERS:
    Info        -   return code, same as in MLPTrainLBFGS
    Rep         -   report, same as in MLPTrainLM/MLPTrainLBFGS
    CVRep       -   generalization error estimates

  -- ALGLIB --
     Copyright 09.12.2007 by Bochkanov Sergey
*************************************************************************/
void mlpkfoldcvlbfgs(const multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, const double wstep, const ae_int_t maxits, const ae_int_t foldscount, ae_int_t &info, mlpreport &rep, mlpcvreport &cvrep, const xparams _xparams = alglib::xdefault);

`mlpkfoldcvlm` function

/*************************************************************************
Cross-validation estimate of generalization error.

Base algorithm - Levenberg-Marquardt.

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry.   Network is
                    not changed during cross-validation -  it is used only
                    as a representative of its architecture.
    XY          -   training set.
    SSize       -   training set size
    Decay       -   weight  decay, same as in MLPTrainLBFGS
    Restarts    -   number of restarts, >0.
                    restarts are counted for each partition separately, so
                    total number of restarts will be Restarts*FoldsCount.
    FoldsCount  -   number of folds in k-fold cross-validation,
                    2<=FoldsCount<=SSize.
                    recommended value: 10.

OUTPUT PARAMETERS:
    Info        -   return code, same as in MLPTrainLBFGS
    Rep         -   report, same as in MLPTrainLM/MLPTrainLBFGS
    CVRep       -   generalization error estimates

  -- ALGLIB --
     Copyright 09.12.2007 by Bochkanov Sergey
*************************************************************************/
void mlpkfoldcvlm(const multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, const ae_int_t foldscount, ae_int_t &info, mlpreport &rep, mlpcvreport &cvrep, const xparams _xparams = alglib::xdefault);

`mlpsetalgobatch` function

/*************************************************************************
This function sets training algorithm: batch training using L-BFGS will be
used.

This algorithm:
* the most robust for small-scale problems, but may be too slow for  large
  scale ones.
* perfoms full pass through the dataset before performing step
* uses conditions specified by MLPSetCond() for stopping
* is default one used by trainer object

INPUT PARAMETERS:
    S           -   trainer object

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpsetalgobatch(mlptrainer &s, const xparams _xparams = alglib::xdefault);

`mlpsetcond` function

/*************************************************************************
This function sets stopping criteria for the optimizer.

INPUT PARAMETERS:
    S           -   trainer object
    WStep       -   stopping criterion. Algorithm stops if  step  size  is
                    less than WStep. Recommended value - 0.01.  Zero  step
                    size means stopping after MaxIts iterations.
                    WStep>=0.
    MaxIts      -   stopping   criterion.  Algorithm  stops  after  MaxIts
                    epochs (full passes over entire dataset).  Zero MaxIts
                    means stopping when step is sufficiently small.
                    MaxIts>=0.

NOTE: by default, WStep=0.005 and MaxIts=0 are used. These values are also
      used when MLPSetCond() is called with WStep=0 and MaxIts=0.

NOTE: these stopping criteria are used for all kinds of neural training  -
      from "conventional" networks to early stopping ensembles. When  used
      for "conventional" networks, they are  used  as  the  only  stopping
      criteria. When combined with early stopping, they used as ADDITIONAL
      stopping criteria which can terminate early stopping algorithm.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpsetcond(mlptrainer &s, const double wstep, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`mlpsetdataset` function

/*************************************************************************
This function sets "current dataset" of the trainer object to  one  passed
by user.

INPUT PARAMETERS:
    S           -   trainer object
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed.
    NPoints     -   points count, >=0.

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
datasetformat is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpsetdataset(mlptrainer &s, const real_2d_array &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6] [7] [8]

`mlpsetdecay` function

/*************************************************************************
This function sets weight decay coefficient which is used for training.

INPUT PARAMETERS:
    S           -   trainer object
    Decay       -   weight  decay  coefficient,  >=0.  Weight  decay  term
                    'Decay*||Weights||^2' is added to error  function.  If
                    you don't know what Decay to choose, use 1.0E-3.
                    Weight decay can be set to zero,  in this case network
                    is trained without weight decay.

NOTE: by default network uses some small nonzero value for weight decay.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpsetdecay(mlptrainer &s, const double decay, const xparams _xparams = alglib::xdefault);

`mlpsetsparsedataset` function

/*************************************************************************
This function sets "current dataset" of the trainer object to  one  passed
by user (sparse matrix is used to store dataset).

INPUT PARAMETERS:
    S           -   trainer object
    XY          -   training  set,  see  below  for  information  on   the
                    training set format. This function checks  correctness
                    of  the  dataset  (no  NANs/INFs,  class  numbers  are
                    correct) and throws exception when  incorrect  dataset
                    is passed. Any  sparse  storage  format  can be  used:
                    Hash-table, CRS...
    NPoints     -   points count, >=0

DATASET FORMAT:

This  function  uses  two  different  dataset formats - one for regression
networks, another one for classification networks.

For regression networks with NIn inputs and NOut outputs following dataset
format is used:
* dataset is given by NPoints*(NIn+NOut) matrix
* each row corresponds to one example
* first NIn columns are inputs, next NOut columns are outputs

For classification networks with NIn inputs and NClasses clases  following
datasetformat is used:
* dataset is given by NPoints*(NIn+1) matrix
* each row corresponds to one example
* first NIn columns are inputs, last column stores class number (from 0 to
  NClasses-1).

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpsetsparsedataset(mlptrainer &s, const sparsematrix &xy, const ae_int_t npoints, const xparams _xparams = alglib::xdefault);

`mlpstarttraining` function

/*************************************************************************
IMPORTANT: this is an "expert" version of the MLPTrain() function.  We  do
           not recommend you to use it unless you are pretty sure that you
           need ability to monitor training progress.

This function performs step-by-step training of the neural  network.  Here
"step-by-step" means that training  starts  with  MLPStartTraining() call,
and then user subsequently calls MLPContinueTraining() to perform one more
iteration of the training.

After call to this function trainer object remembers network and  is ready
to  train  it.  However,  no  training  is  performed  until first call to
MLPContinueTraining() function. Subsequent calls  to MLPContinueTraining()
will advance training progress one iteration further.

EXAMPLE:
    >
    > ...initialize network and trainer object....
    >
    > MLPStartTraining(Trainer, Network, True)
    > while MLPContinueTraining(Trainer, Network) do
    >     ...visualize training progress...
    >

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural network. It must have same number of inputs and
                    output/classes as was specified during creation of the
                    trainer object.
    RandomStart -   randomize network before training or not:
                    * True  means  that  network  is  randomized  and  its
                      initial state (one which was passed to  the  trainer
                      object) is lost.
                    * False  means  that  training  is  started  from  the
                      current state of the network

OUTPUT PARAMETERS:
    Network     -   neural network which is ready to training (weights are
                    initialized, preprocessor is initialized using current
                    training set)

NOTE: this method uses sum-of-squares error function for training.

NOTE: it is expected that trainer object settings are NOT  changed  during
      step-by-step training, i.e. no  one  changes  stopping  criteria  or
      training set during training. It is possible and there is no defense
      against  such  actions,  but  algorithm  behavior  in  such cases is
      undefined and can be unpredictable.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlpstarttraining(mlptrainer &s, multilayerperceptron &network, const bool randomstart, const xparams _xparams = alglib::xdefault);

`mlptrainensemblees` function

/*************************************************************************
This function trains neural network ensemble passed to this function using
current dataset and early stopping training algorithm. Each early stopping
round performs NRestarts  random  restarts  (thus,  EnsembleSize*NRestarts
training rounds is performed in total).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   trainer object;
    Ensemble    -   neural network ensemble. It must have same  number  of
                    inputs and outputs/classes  as  was  specified  during
                    creation of the trainer object.
    NRestarts   -   number of restarts, >=0:
                    * NRestarts>0 means that specified  number  of  random
                      restarts are performed during each ES round;
                    * NRestarts=0 is silently replaced by 1.

OUTPUT PARAMETERS:
    Ensemble    -   trained ensemble;
    Rep         -   it contains all type of errors.

NOTE: this training method uses BOTH early stopping and weight decay!  So,
      you should select weight decay before starting training just as  you
      select it before training "conventional" networks.

NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(),
      or  single-point  dataset  was  passed,  ensemble  is filled by zero
      values.

NOTE: this method uses sum-of-squares error function for training.

  -- ALGLIB --
     Copyright 22.08.2012 by Bochkanov Sergey
*************************************************************************/
void mlptrainensemblees(mlptrainer &s, mlpensemble &ensemble, const ae_int_t nrestarts, mlpreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`mlptraines` function

/*************************************************************************
Neural network training using early stopping (base algorithm - L-BFGS with
regularization).

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry
    TrnXY       -   training set
    TrnSize     -   training set size, TrnSize>0
    ValXY       -   validation set
    ValSize     -   validation set size, ValSize>0
    Decay       -   weight decay constant, >=0.001
                    Decay term 'Decay*||Weights||^2' is added to error
                    function.
                    If you don't know what Decay to choose, use 0.001.
    Restarts    -   number of restarts, either:
                    * strictly positive number - algorithm make specified
                      number of restarts from random position.
                    * -1, in which case algorithm makes exactly one run
                      from the initial state of the network (no randomization).
                    If you don't know what Restarts to choose, choose one
                    one the following:
                    * -1 (deterministic start)
                    * +1 (one random restart)
                    * +5 (moderate amount of random restarts)

OUTPUT PARAMETERS:
    Network     -   trained neural network.
    Info        -   return code:
                    * -2, if there is a point with class number
                          outside of [0..NOut-1].
                    * -1, if wrong parameters specified
                          (NPoints<0, Restarts<1, ...).
                    *  2, task has been solved, stopping  criterion  met -
                          sufficiently small step size.  Not expected  (we
                          use  EARLY  stopping)  but  possible  and not an
                          error.
                    *  6, task has been solved, stopping  criterion  met -
                          increasing of validation set error.
    Rep         -   training report

NOTE:

Algorithm stops if validation set error increases for  a  long  enough  or
step size is small enought  (there  are  task  where  validation  set  may
decrease for eternity). In any case solution returned corresponds  to  the
minimum of validation set error.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void mlptraines(multilayerperceptron &network, const real_2d_array &trnxy, const ae_int_t trnsize, const real_2d_array &valxy, const ae_int_t valsize, const double decay, const ae_int_t restarts, ae_int_t &info, mlpreport &rep, const xparams _xparams = alglib::xdefault);

`mlptrainlbfgs` function

/*************************************************************************
Neural  network  training  using  L-BFGS  algorithm  with  regularization.
Subroutine  trains  neural  network  with  restarts from random positions.
Algorithm  is  well  suited  for  problems  of  any dimensionality (memory
requirements and step complexity are linear by weights number).

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay constant, >=0.001
                    Decay term 'Decay*||Weights||^2' is added to error
                    function.
                    If you don't know what Decay to choose, use 0.001.
    Restarts    -   number of restarts from random position, >0.
                    If you don't know what Restarts to choose, use 2.
    WStep       -   stopping criterion. Algorithm stops if  step  size  is
                    less than WStep. Recommended value - 0.01.  Zero  step
                    size means stopping after MaxIts iterations.
    MaxIts      -   stopping   criterion.  Algorithm  stops  after  MaxIts
                    iterations (NOT gradient  calculations).  Zero  MaxIts
                    means stopping when step is sufficiently small.

OUTPUT PARAMETERS:
    Network     -   trained neural network.
    Info        -   return code:
                    * -8, if both WStep=0 and MaxIts=0
                    * -2, if there is a point with class number
                          outside of [0..NOut-1].
                    * -1, if wrong parameters specified
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report

  -- ALGLIB --
     Copyright 09.12.2007 by Bochkanov Sergey
*************************************************************************/
void mlptrainlbfgs(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, const double wstep, const ae_int_t maxits, ae_int_t &info, mlpreport &rep, const xparams _xparams = alglib::xdefault);

`mlptrainlm` function

/*************************************************************************
Neural network training  using  modified  Levenberg-Marquardt  with  exact
Hessian calculation and regularization. Subroutine trains  neural  network
with restarts from random positions. Algorithm is well  suited  for  small
and medium scale problems (hundreds of weights).

INPUT PARAMETERS:
    Network     -   neural network with initialized geometry
    XY          -   training set
    NPoints     -   training set size
    Decay       -   weight decay constant, >=0.001
                    Decay term 'Decay*||Weights||^2' is added to error
                    function.
                    If you don't know what Decay to choose, use 0.001.
    Restarts    -   number of restarts from random position, >0.
                    If you don't know what Restarts to choose, use 2.

OUTPUT PARAMETERS:
    Network     -   trained neural network.
    Info        -   return code:
                    * -9, if internal matrix inverse subroutine failed
                    * -2, if there is a point with class number
                          outside of [0..NOut-1].
                    * -1, if wrong parameters specified
                          (NPoints<0, Restarts<1).
                    *  2, if task has been solved.
    Rep         -   training report

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void mlptrainlm(multilayerperceptron &network, const real_2d_array &xy, const ae_int_t npoints, const double decay, const ae_int_t restarts, ae_int_t &info, mlpreport &rep, const xparams _xparams = alglib::xdefault);

`mlptrainnetwork` function

/*************************************************************************
This function trains neural network passed to this function, using current
dataset (one which was passed to MLPSetDataset() or MLPSetSparseDataset())
and current training settings. Training  from  NRestarts  random  starting
positions is performed, best network is chosen.

Training is performed using current training algorithm.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S           -   trainer object
    Network     -   neural network. It must have same number of inputs and
                    output/classes as was specified during creation of the
                    trainer object.
    NRestarts   -   number of restarts, >=0:
                    * NRestarts>0 means that specified  number  of  random
                      restarts are performed, best network is chosen after
                      training
                    * NRestarts=0 means that current state of the  network
                      is used for training.

OUTPUT PARAMETERS:
    Network     -   trained network

NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(),
      network  is  filled  by zero  values.  Same  behavior  for functions
      MLPStartTraining and MLPContinueTraining.

NOTE: this method uses sum-of-squares error function for training.

  -- ALGLIB --
     Copyright 23.07.2012 by Bochkanov Sergey
*************************************************************************/
void mlptrainnetwork(mlptrainer &s, multilayerperceptron &network, const ae_int_t nrestarts, mlpreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4] [5] [6]

nn_cls2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Suppose that we want to classify numbers as positive (class 0) and negative
        // (class 1). We have training set which includes several strictly positive
        // or negative numbers - and zero.
        //
        // The problem is that we are not sure how to classify zero, so from time to
        // time we mark it as positive or negative (with equal probability). Other
        // numbers are marked in pure deterministic setting. How will neural network
        // cope with such classification task?
        //
        // NOTE: we use network with excessive amount of neurons, which guarantees
        //       almost exact reproduction of the training set. Generalization ability
        //       of such network is rather low, but we are not concerned with such
        //       questions in this basic demo.
        //
        mlptrainer trn;
        multilayerperceptron network;
        mlpreport rep;
        real_1d_array x = "[0]";
        real_1d_array y = "[0,0]";

        //
        // Training set. One row corresponds to one record [A => class(A)].
        //
        // Classes are denoted by numbers from 0 to 1, where 0 corresponds to positive
        // numbers and 1 to negative numbers.
        //
        // [ +1  0]
        // [ +2  0]
        // [ -1  1]
        // [ -2  1]
        // [  0  0]   !! sometimes we classify 0 as positive, sometimes as negative
        // [  0  1]   !!
        //
        real_2d_array xy = "[[+1,0],[+2,0],[-1,1],[-2,1],[0,0],[0,1]]";

        //
        //
        // When we solve classification problems, everything is slightly different from
        // the regression ones:
        //
        // 1. Network is created. Because we solve classification problem, we use
        //    mlpcreatec1() function instead of mlpcreate1(). This function creates
        //    classifier network with SOFTMAX-normalized outputs. This network returns
        //    vector of class membership probabilities which are normalized to be
        //    non-negative and sum to 1.0
        //
        // 2. We use mlpcreatetrainercls() function instead of mlpcreatetrainer() to
        //    create trainer object. Trainer object process dataset and neural network
        //    slightly differently to account for specifics of the classification
        //    problems.
        //
        // 3. Dataset is attached to trainer object. Note that dataset format is slightly
        //    different from one used for regression.
        //
        mlpcreatetrainercls(1, 2, trn);
        mlpcreatec1(1, 5, 2, network);
        mlpsetdataset(trn, xy, 6);

        //
        // Network is trained with 5 restarts from random positions
        //
        mlptrainnetwork(trn, network, 5, rep);

        //
        // Test our neural network on strictly positive and strictly negative numbers.
        //
        // IMPORTANT! Classifier network returns class membership probabilities instead
        // of class indexes. Network returns two values (probabilities) instead of one
        // (class index).
        //
        // Thus, for +1 we expect to get [P0,P1] = [1,0], where P0 is probability that
        // number is positive (belongs to class 0), and P1 is probability that number
        // is negative (belongs to class 1).
        //
        // For -1 we expect to get [P0,P1] = [0,1]
        //
        // Following properties are guaranteed by network architecture:
        // * P0>=0, P1>=0   non-negativity
        // * P0+P1=1        normalization
        //
        x = "[1]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [1.000,0.000]
        x = "[-1]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,1.000]

        //
        // But what our network will return for 0, which is between classes 0 and 1?
        //
        // In our dataset it has two different marks assigned (class 0 AND class 1).
        // So network will return something average between class 0 and class 1:
        //     0 => [0.5, 0.5]
        //
        x = "[0]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.500,0.500]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_cls3 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Suppose that we want to classify numbers as positive (class 0) and negative
        // (class 1). We also have one more class for zero (class 2).
        //
        // NOTE: we use network with excessive amount of neurons, which guarantees
        //       almost exact reproduction of the training set. Generalization ability
        //       of such network is rather low, but we are not concerned with such
        //       questions in this basic demo.
        //
        mlptrainer trn;
        multilayerperceptron network;
        mlpreport rep;
        real_1d_array x = "[0]";
        real_1d_array y = "[0,0,0]";

        //
        // Training set. One row corresponds to one record [A => class(A)].
        //
        // Classes are denoted by numbers from 0 to 2, where 0 corresponds to positive
        // numbers, 1 to negative numbers, 2 to zero
        //
        // [ +1  0]
        // [ +2  0]
        // [ -1  1]
        // [ -2  1]
        // [  0  2]
        //
        real_2d_array xy = "[[+1,0],[+2,0],[-1,1],[-2,1],[0,2]]";

        //
        //
        // When we solve classification problems, everything is slightly different from
        // the regression ones:
        //
        // 1. Network is created. Because we solve classification problem, we use
        //    mlpcreatec1() function instead of mlpcreate1(). This function creates
        //    classifier network with SOFTMAX-normalized outputs. This network returns
        //    vector of class membership probabilities which are normalized to be
        //    non-negative and sum to 1.0
        //
        // 2. We use mlpcreatetrainercls() function instead of mlpcreatetrainer() to
        //    create trainer object. Trainer object process dataset and neural network
        //    slightly differently to account for specifics of the classification
        //    problems.
        //
        // 3. Dataset is attached to trainer object. Note that dataset format is slightly
        //    different from one used for regression.
        //
        mlpcreatetrainercls(1, 3, trn);
        mlpcreatec1(1, 5, 3, network);
        mlpsetdataset(trn, xy, 5);

        //
        // Network is trained with 5 restarts from random positions
        //
        mlptrainnetwork(trn, network, 5, rep);

        //
        // Test our neural network on strictly positive and strictly negative numbers.
        //
        // IMPORTANT! Classifier network returns class membership probabilities instead
        // of class indexes. Network returns three values (probabilities) instead of one
        // (class index).
        //
        // Thus, for +1 we expect to get [P0,P1,P2] = [1,0,0],
        // for -1 we expect to get [P0,P1,P2] = [0,1,0],
        // and for 0 we will get [P0,P1,P2] = [0,0,1].
        //
        // Following properties are guaranteed by network architecture:
        // * P0>=0, P1>=0, P2>=0    non-negativity
        // * P0+P1+P2=1             normalization
        //
        x = "[1]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [1.000,0.000,0.000]
        x = "[-1]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,1.000,0.000]
        x = "[0]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,0.000,1.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_crossvalidation example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example shows how to perform cross-validation with ALGLIB
        //
        mlptrainer trn;
        multilayerperceptron network;
        mlpreport rep;

        //
        // Training set: f(x)=1/(x^2+1)
        // One row corresponds to one record [x,f(x)]
        //
        real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";

        //
        // Trainer object is created.
        // Dataset is attached to trainer object.
        //
        // NOTE: it is not good idea to perform cross-validation on sample
        //       as small as ours (13 examples). It is done for demonstration
        //       purposes only. Generalization error estimates won't be
        //       precise enough for practical purposes.
        //
        mlpcreatetrainer(1, 1, trn);
        mlpsetdataset(trn, xy, 13);

        //
        // The key property of the cross-validation is that it estimates
        // generalization properties of neural ARCHITECTURE. It does NOT
        // estimates generalization error of some specific network which
        // is passed to the k-fold CV routine.
        //
        // In our example we create 1x4x1 neural network and pass it to
        // CV routine without training it. Original state of the network
        // is not used for cross-validation - each round is restarted from
        // random initial state. Only geometry of network matters.
        //
        // We perform 5 restarts from different random positions for each
        // of the 10 cross-validation rounds.
        //
        mlpcreate1(1, 4, 1, network);
        mlpkfoldcv(trn, network, 5, 10, rep);

        //
        // Cross-validation routine stores estimates of the generalization
        // error to MLP report structure. You may examine its fields and
        // see estimates of different errors (RMS, CE, Avg).
        //
        // Because cross-validation is non-deterministic, in our manual we
        // can not say what values will be stored to rep after call to
        // mlpkfoldcv(). Every CV round will return slightly different
        // estimates.
        //
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_ensembles_es example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example shows how to train early stopping ensebles.
        //
        mlptrainer trn;
        mlpensemble ensemble;
        mlpreport rep;

        //
        // Training set: f(x)=1/(x^2+1)
        // One row corresponds to one record [x,f(x)]
        //
        real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";

        //
        // Trainer object is created.
        // Dataset is attached to trainer object.
        //
        // NOTE: it is not good idea to use early stopping ensemble on sample
        //       as small as ours (13 examples). It is done for demonstration
        //       purposes only. Ensemble training algorithm won't find good
        //       solution on such small sample.
        //
        mlpcreatetrainer(1, 1, trn);
        mlpsetdataset(trn, xy, 13);

        //
        // Ensemble is created and trained. Each of 50 network is trained
        // with 5 restarts.
        //
        mlpecreate1(1, 4, 1, 50, ensemble);
        mlptrainensemblees(trn, ensemble, 5, rep);
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_parallel example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example shows how to use parallel functionality of ALGLIB.
        // We generate simple 1-dimensional regression problem and show how
        // to use parallel training, parallel cross-validation, parallel
        // training of neural ensembles.
        //
        // We assume that you already know how to use ALGLIB in serial mode
        // and concentrate on its parallel capabilities.
        //
        // NOTE: it is not good idea to use parallel features on sample as small
        //       as ours (13 examples). It is done only for demonstration purposes.
        //
        mlptrainer trn;
        multilayerperceptron network;
        mlpensemble ensemble;
        mlpreport rep;
        real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";
        mlpcreatetrainer(1, 1, trn);
        mlpsetdataset(trn, xy, 13);
        mlpcreate1(1, 4, 1, network);
        mlpecreate1(1, 4, 1, 50, ensemble);

        //
        // Below we demonstrate how to perform:
        // * parallel training of individual networks
        // * parallel cross-validation
        // * parallel training of neural ensembles
        //
        // In order to use multithreading, you have to:
        // 1) Install SMP edition of ALGLIB.
        // 2) This step is specific for C++ users: you should activate OS-specific
        //    capabilities of ALGLIB by defining AE_OS=AE_POSIX (for *nix systems)
        //    or AE_OS=AE_WINDOWS (for Windows systems).
        //    C# users do not have to perform this step because C# programs are
        //    portable across different systems without OS-specific tuning.
        // 3) Tell ALGLIB that you want it to use multithreading by means of
        //    setnworkers() call:
        //          * alglib::setnworkers(0)  = use all cores
        //          * alglib::setnworkers(-1) = leave one core unused
        //          * alglib::setnworkers(-2) = leave two cores unused
        //          * alglib::setnworkers(+2) = use 2 cores (even if you have more)
        //    During runtime ALGLIB will automatically determine whether it is
        //    feasible to start worker threads and split your task between cores.
        //
        alglib::setnworkers(+2);

        //
        // First, we perform parallel training of individual network with 5
        // restarts from random positions. These 5 rounds of  training  are
        // executed in parallel manner,  with  best  network  chosen  after
        // training.
        //
        // ALGLIB can use additional way to speed up computations -  divide
        // dataset   into   smaller   subsets   and   process these subsets
        // simultaneously. It allows us  to  efficiently  parallelize  even
        // single training round. This operation is performed automatically
        // for large datasets, but our toy dataset is too small.
        //
        mlptrainnetwork(trn, network, 5, rep);

        //
        // Then, we perform parallel 10-fold cross-validation, with 5 random
        // restarts per each CV round. I.e., 5*10=50  networks  are trained
        // in total. All these operations can be parallelized.
        //
        // NOTE: again, ALGLIB can parallelize  calculation   of   gradient
        //       over entire dataset - but our dataset is too small.
        //
        mlpkfoldcv(trn, network, 5, 10, rep);

        //
        // Finally, we train early stopping ensemble of 50 neural networks,
        // each  of them is trained with 5 random restarts. I.e.,  5*50=250
        // networks aretrained in total.
        //
        mlptrainensemblees(trn, ensemble, 5, rep);
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_regr example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // The very simple example on neural network: network is trained to reproduce
        // small 2x2 multiplication table.
        //
        // NOTE: we use network with excessive amount of neurons, which guarantees
        //       almost exact reproduction of the training set. Generalization ability
        //       of such network is rather low, but we are not concerned with such
        //       questions in this basic demo.
        //
        mlptrainer trn;
        multilayerperceptron network;
        mlpreport rep;

        //
        // Training set:
        // * one row corresponds to one record A*B=C in the multiplication table
        // * first two columns store A and B, last column stores C
        //
        // [1 * 1 = 1]
        // [1 * 2 = 2]
        // [2 * 1 = 2]
        // [2 * 2 = 4]
        //
        real_2d_array xy = "[[1,1,1],[1,2,2],[2,1,2],[2,2,4]]";

        //
        // Network is created.
        // Trainer object is created.
        // Dataset is attached to trainer object.
        //
        mlpcreatetrainer(2, 1, trn);
        mlpcreate1(2, 5, 1, network);
        mlpsetdataset(trn, xy, 4);

        //
        // Network is trained with 5 restarts from random positions
        //
        mlptrainnetwork(trn, network, 5, rep);

        //
        // 2*2=?
        //
        real_1d_array x = "[2,2]";
        real_1d_array y = "[0]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [4.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_regr_n example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Network with 2 inputs and 2 outputs is trained to reproduce vector function:
        //     (x0,x1) => (x0+x1, x0*x1)
        //
        // Informally speaking, we want neural network to simultaneously calculate
        // both sum of two numbers and their product.
        //
        // NOTE: we use network with excessive amount of neurons, which guarantees
        //       almost exact reproduction of the training set. Generalization ability
        //       of such network is rather low, but we are not concerned with such
        //       questions in this basic demo.
        //
        mlptrainer trn;
        multilayerperceptron network;
        mlpreport rep;

        //
        // Training set. One row corresponds to one record [A,B,A+B,A*B].
        //
        // [ 1   1  1+1  1*1 ]
        // [ 1   2  1+2  1*2 ]
        // [ 2   1  2+1  2*1 ]
        // [ 2   2  2+2  2*2 ]
        //
        real_2d_array xy = "[[1,1,2,1],[1,2,3,2],[2,1,3,2],[2,2,4,4]]";

        //
        // Network is created.
        // Trainer object is created.
        // Dataset is attached to trainer object.
        //
        mlpcreatetrainer(2, 2, trn);
        mlpcreate1(2, 5, 2, network);
        mlpsetdataset(trn, xy, 4);

        //
        // Network is trained with 5 restarts from random positions
        //
        mlptrainnetwork(trn, network, 5, rep);

        //
        // 2+1=?
        // 2*1=?
        //
        real_1d_array x = "[2,1]";
        real_1d_array y = "[0,0]";
        mlpprocess(network, x, y);
        printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [3.000,2.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nn_trainerobject example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Trainer object is used to train network. It stores dataset, training settings,
        // and other information which is NOT part of neural network. You should use
        // trainer object as follows:
        // (1) you create trainer object and specify task type (classification/regression)
        //     and number of inputs/outputs
        // (2) you add dataset to the trainer object
        // (3) you may change training settings (stopping criteria or weight decay)
        // (4) finally, you may train one or more networks
        //
        // You may interleave stages 2...4 and repeat them many times. Trainer object
        // remembers its internal state and can be used several times after its creation
        // and initialization.
        //
        mlptrainer trn;

        //
        // Stage 1: object creation.
        //
        // We have to specify number of inputs and outputs. Trainer object can be used
        // only for problems with same number of inputs/outputs as was specified during
        // its creation.
        //
        // In case you want to train SOFTMAX-normalized network which solves classification
        // problems,  you  must  use  another  function  to  create  trainer  object:
        // mlpcreatetrainercls().
        //
        // Below we create trainer object which can be used to train regression networks
        // with 2 inputs and 1 output.
        //
        mlpcreatetrainer(2, 1, trn);

        //
        // Stage 2: specification of the training set
        //
        // By default trainer object stores empty dataset. So to solve your non-empty problem
        // you have to set dataset by passing to trainer dense or sparse matrix.
        //
        // One row of the matrix corresponds to one record A*B=C in the multiplication table.
        // First two columns store A and B, last column stores C
        //
        //     [1 * 1 = 1]   [ 1 1 1 ]
        //     [1 * 2 = 2]   [ 1 2 2 ]
        //     [2 * 1 = 2] = [ 2 1 2 ]
        //     [2 * 2 = 4]   [ 2 2 4 ]
        //
        real_2d_array xy = "[[1,1,1],[1,2,2],[2,1,2],[2,2,4]]";
        mlpsetdataset(trn, xy, 4);

        //
        // Stage 3: modification of the training parameters.
        //
        // You may modify parameters like weights decay or stopping criteria:
        // * we set moderate weight decay
        // * we choose iterations limit as stopping condition (another condition - step size -
        //   is zero, which means than this condition is not active)
        //
        double wstep = 0.000;
        ae_int_t maxits = 100;
        mlpsetdecay(trn, 0.01);
        mlpsetcond(trn, wstep, maxits);

        //
        // Stage 4: training.
        //
        // We will train several networks with different architecture using same trainer object.
        // We may change training parameters or even dataset, so different networks are trained
        // differently. But in this simple example we will train all networks with same settings.
        //
        // We create and train three networks:
        // * network 1 has 2x1 architecture     (2 inputs, no hidden neurons, 1 output)
        // * network 2 has 2x5x1 architecture   (2 inputs, 5 hidden neurons, 1 output)
        // * network 3 has 2x5x5x1 architecture (2 inputs, two hidden layers, 1 output)
        //
        // NOTE: these networks solve regression problems. For classification problems you
        //       should use mlpcreatec0/c1/c2 to create neural networks which have SOFTMAX-
        //       normalized outputs.
        //
        multilayerperceptron net1;
        multilayerperceptron net2;
        multilayerperceptron net3;
        mlpreport rep;

        mlpcreate0(2, 1, net1);
        mlpcreate1(2, 5, 1, net2);
        mlpcreate2(2, 5, 5, 1, net3);

        mlptrainnetwork(trn, net1, 5, rep);
        mlptrainnetwork(trn, net2, 5, rep);
        mlptrainnetwork(trn, net3, 5, rep);
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`nearestneighbor` subpackage

Classes

kdtree
kdtreerequestbuffer

Functions

kdtreebuild
kdtreebuildtagged
kdtreecreaterequestbuffer
kdtreequeryaknn
kdtreequerybox
kdtreequeryknn
kdtreequeryresultsdistances
kdtreequeryresultsdistancesi
kdtreequeryresultstags
kdtreequeryresultstagsi
kdtreequeryresultsx
kdtreequeryresultsxi
kdtreequeryresultsxy
kdtreequeryresultsxyi
kdtreequeryrnn
kdtreequeryrnnu
kdtreeserialize
kdtreetsqueryaknn
kdtreetsquerybox
kdtreetsqueryknn
kdtreetsqueryresultsdistances
kdtreetsqueryresultstags
kdtreetsqueryresultsx
kdtreetsqueryresultsxy
kdtreetsqueryrnn
kdtreetsqueryrnnu
kdtreeunserialize

Examples

nneighbor_d_1		Nearest neighbor search, KNN queries
nneighbor_d_2		Serialization of KD-trees

`kdtree` class

/*************************************************************************
KD-tree object.
*************************************************************************/
class kdtree
{
public:
    kdtree();
    kdtree(const kdtree &rhs);
    kdtree& operator=(const kdtree &rhs);
    virtual ~kdtree();
};

`kdtreerequestbuffer` class

/*************************************************************************
Buffer object which is used to perform nearest neighbor  requests  in  the
multithreaded mode (multiple threads working with same KD-tree object).

This object should be created with KDTreeCreateRequestBuffer().
*************************************************************************/
class kdtreerequestbuffer
{
public:
    kdtreerequestbuffer();
    kdtreerequestbuffer(const kdtreerequestbuffer &rhs);
    kdtreerequestbuffer& operator=(const kdtreerequestbuffer &rhs);
    virtual ~kdtreerequestbuffer();
};

`kdtreebuild` function

/*************************************************************************
KD-tree creation

This subroutine creates KD-tree from set of X-values and optional Y-values

INPUT PARAMETERS
    XY      -   dataset, array[0..N-1,0..NX+NY-1].
                one row corresponds to one point.
                first NX columns contain X-values, next NY (NY may be zero)
                columns may contain associated Y-values
    N       -   number of points, N>=0.
    NX      -   space dimension, NX>=1.
    NY      -   number of optional Y-values, NY>=0.
    NormType-   norm type:
                * 0 denotes infinity-norm
                * 1 denotes 1-norm
                * 2 denotes 2-norm (Euclidean norm)

OUTPUT PARAMETERS
    KDT     -   KD-tree


NOTES

1. KD-tree  creation  have O(N*logN) complexity and O(N*(2*NX+NY))  memory
   requirements.
2. Although KD-trees may be used with any combination of N  and  NX,  they
   are more efficient than brute-force search only when N >> 4^NX. So they
   are most useful in low-dimensional tasks (NX=2, NX=3). NX=1  is another
   inefficient case, because  simple  binary  search  (without  additional
   structures) is much more efficient in such tasks than KD-trees.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreebuild(const real_2d_array &xy, const ae_int_t n, const ae_int_t nx, const ae_int_t ny, const ae_int_t normtype, kdtree &kdt, const xparams _xparams = alglib::xdefault);
void kdtreebuild(const real_2d_array &xy, const ae_int_t nx, const ae_int_t ny, const ae_int_t normtype, kdtree &kdt, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`kdtreebuildtagged` function

/*************************************************************************
KD-tree creation

This  subroutine  creates  KD-tree  from set of X-values, integer tags and
optional Y-values

INPUT PARAMETERS
    XY      -   dataset, array[0..N-1,0..NX+NY-1].
                one row corresponds to one point.
                first NX columns contain X-values, next NY (NY may be zero)
                columns may contain associated Y-values
    Tags    -   tags, array[0..N-1], contains integer tags associated
                with points.
    N       -   number of points, N>=0
    NX      -   space dimension, NX>=1.
    NY      -   number of optional Y-values, NY>=0.
    NormType-   norm type:
                * 0 denotes infinity-norm
                * 1 denotes 1-norm
                * 2 denotes 2-norm (Euclidean norm)

OUTPUT PARAMETERS
    KDT     -   KD-tree

NOTES

1. KD-tree  creation  have O(N*logN) complexity and O(N*(2*NX+NY))  memory
   requirements.
2. Although KD-trees may be used with any combination of N  and  NX,  they
   are more efficient than brute-force search only when N >> 4^NX. So they
   are most useful in low-dimensional tasks (NX=2, NX=3). NX=1  is another
   inefficient case, because  simple  binary  search  (without  additional
   structures) is much more efficient in such tasks than KD-trees.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreebuildtagged(const real_2d_array &xy, const integer_1d_array &tags, const ae_int_t n, const ae_int_t nx, const ae_int_t ny, const ae_int_t normtype, kdtree &kdt, const xparams _xparams = alglib::xdefault);
void kdtreebuildtagged(const real_2d_array &xy, const integer_1d_array &tags, const ae_int_t nx, const ae_int_t ny, const ae_int_t normtype, kdtree &kdt, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreecreaterequestbuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel KD-tree requests.

KD-tree subpackage provides two sets of request functions - ones which use
internal buffer of KD-tree object  (these  functions  are  single-threaded
because they use same buffer, which can not shared between  threads),  and
ones which use external buffer.

This function is used to initialize external buffer.

INPUT PARAMETERS
    KDT         -   KD-tree which is associated with newly created buffer

OUTPUT PARAMETERS
    Buf         -   external buffer.


IMPORTANT: KD-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use buffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
void kdtreecreaterequestbuffer(const kdtree &kdt, kdtreerequestbuffer &buf, const xparams _xparams = alglib::xdefault);

`kdtreequeryaknn` function

/*************************************************************************
K-NN query: approximate K nearest neighbors

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryAKNN() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True
    Eps         -   approximation factor, Eps>=0. eps-approximate  nearest
                    neighbor  is  a  neighbor  whose distance from X is at
                    most (1+eps) times distance of true nearest neighbor.

RESULT
    number of actual neighbors found (either K or N, if K>N).

NOTES
    significant performance gain may be achieved only when Eps  is  is  on
    the order of magnitude of 1 or larger.

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
these results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreequeryaknn(kdtree &kdt, const real_1d_array &x, const ae_int_t k, const bool selfmatch, const double eps, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreequeryaknn(kdtree &kdt, const real_1d_array &x, const ae_int_t k, const double eps, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequerybox` function

/*************************************************************************
Box query: all points within user-specified box.

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryBox() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    BoxMin      -   lower bounds, array[0..NX-1].
    BoxMax      -   upper bounds, array[0..NX-1].


RESULT
    number of actual neighbors found (in [0,N]).

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
these results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() returns zeros for this request

NOTE: this particular query returns unordered results, because there is no
      meaningful way of  ordering  points.  Furthermore,  no 'distance' is
      associated with points - it is either INSIDE  or OUTSIDE (so request
      for distances will return zeros).

  -- ALGLIB --
     Copyright 14.05.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreequerybox(kdtree &kdt, const real_1d_array &boxmin, const real_1d_array &boxmax, const xparams _xparams = alglib::xdefault);

`kdtreequeryknn` function

/*************************************************************************
K-NN query: K nearest neighbors

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: KDTreeTsQueryKNN() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of actual neighbors found (either K or N, if K>N).

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
these results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreequeryknn(kdtree &kdt, const real_1d_array &x, const ae_int_t k, const bool selfmatch, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreequeryknn(kdtree &kdt, const real_1d_array &x, const ae_int_t k, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsdistances` function

/*************************************************************************
Distances from last query

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultsdistances().

INPUT PARAMETERS
    KDT     -   KD-tree
    R       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    R       -   filled with distances (in corresponding norm)

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultsdistances(const kdtree &kdt, real_1d_array &r, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsdistancesi` function

/*************************************************************************
Distances from last query; 'interactive' variant for languages like Python
which  support  constructs   like  "R = KDTreeQueryResultsDistancesI(KDT)"
and interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultsdistancesi(const kdtree &kdt, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`kdtreequeryresultstags` function

/*************************************************************************
Tags from last query

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultstags().

INPUT PARAMETERS
    KDT     -   KD-tree
    Tags    -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    Tags    -   filled with tags associated with points,
                or, when no tags were supplied, with zeros

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultstags(const kdtree &kdt, integer_1d_array &tags, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequeryresultstagsi` function

/*************************************************************************
Tags  from  last  query;  'interactive' variant for languages like  Python
which  support  constructs  like "Tags = KDTreeQueryResultsTagsI(KDT)" and
interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultstagsi(const kdtree &kdt, integer_1d_array &tags, const xparams _xparams = alglib::xdefault);

`kdtreequeryresultsx` function

/*************************************************************************
X-values from last query.

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultsx().

INPUT PARAMETERS
    KDT     -   KD-tree
    X       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    X       -   rows are filled with X-values

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultsx(const kdtree &kdt, real_2d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsxi` function

/*************************************************************************
X-values from last query; 'interactive' variant for languages like  Python
which   support    constructs   like  "X = KDTreeQueryResultsXI(KDT)"  and
interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultsxi(const kdtree &kdt, real_2d_array &x, const xparams _xparams = alglib::xdefault);

`kdtreequeryresultsxy` function

/*************************************************************************
X- and Y-values from last query

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - kdtreetsqueryresultsxy().

INPUT PARAMETERS
    KDT     -   KD-tree
    XY      -   possibly pre-allocated buffer. If XY is too small to store
                result, it is resized. If size(XY) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    XY      -   rows are filled with points: first NX columns with
                X-values, next NY columns - with Y-values.

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultsxy(const kdtree &kdt, real_2d_array &xy, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequeryresultsxyi` function

/*************************************************************************
XY-values from last query; 'interactive' variant for languages like Python
which   support    constructs   like "XY = KDTreeQueryResultsXYI(KDT)" and
interactive mode of interpreter.

This function allocates new array on each call,  so  it  is  significantly
slower than its 'non-interactive' counterpart, but it is  more  convenient
when you call it from command line.

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreequeryresultsxyi(const kdtree &kdt, real_2d_array &xy, const xparams _xparams = alglib::xdefault);

`kdtreequeryrnn` function

/*************************************************************************
R-NN query: all points within R-sphere centered at X, ordered by  distance
between point and X (by ascending).

NOTE: it is also possible to perform undordered queries performed by means
      of kdtreequeryrnnu() and kdtreetsqueryrnnu() functions. Such queries
      are faster because we do not have to use heap structure for sorting.

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: kdtreetsqueryrnn() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    R           -   radius of sphere (in corresponding norm), R>0
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of neighbors found, >=0

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
actual results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreequeryrnn(kdtree &kdt, const real_1d_array &x, const double r, const bool selfmatch, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreequeryrnn(kdtree &kdt, const real_1d_array &x, const double r, const xparams _xparams = alglib::xdefault);

Examples: [1]

`kdtreequeryrnnu` function

/*************************************************************************
R-NN query: all points within R-sphere  centered  at  X,  no  ordering  by
distance as undicated by "U" suffix (faster that ordered query, for  large
queries - significantly faster).

IMPORTANT: this function can not be used in multithreaded code because  it
           uses internal temporary buffer of kd-tree object, which can not
           be shared between multiple threads.  If  you  want  to  perform
           parallel requests, use function  which  uses  external  request
           buffer: kdtreetsqueryrnn() ("Ts" stands for "thread-safe").

INPUT PARAMETERS
    KDT         -   KD-tree
    X           -   point, array[0..NX-1].
    R           -   radius of sphere (in corresponding norm), R>0
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of neighbors found, >=0

This  subroutine  performs  query  and  stores  its result in the internal
structures of the KD-tree. You can use  following  subroutines  to  obtain
actual results:
* KDTreeQueryResultsX() to get X-values
* KDTreeQueryResultsXY() to get X- and Y-values
* KDTreeQueryResultsTags() to get tag values
* KDTreeQueryResultsDistances() to get distances

As indicated by "U" suffix, this function returns unordered results.

  -- ALGLIB --
     Copyright 01.11.2018 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreequeryrnnu(kdtree &kdt, const real_1d_array &x, const double r, const bool selfmatch, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreequeryrnnu(kdtree &kdt, const real_1d_array &x, const double r, const xparams _xparams = alglib::xdefault);

`kdtreeserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void kdtreeserialize(const kdtree &obj, std::string &s_out);
void kdtreeserialize(const kdtree &obj, std::ostream &s_out);

`kdtreetsqueryaknn` function

/*************************************************************************
K-NN query: approximate K nearest neighbors, using thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True
    Eps         -   approximation factor, Eps>=0. eps-approximate  nearest
                    neighbor  is  a  neighbor  whose distance from X is at
                    most (1+eps) times distance of true nearest neighbor.

RESULT
    number of actual neighbors found (either K or N, if K>N).

NOTES
    significant performance gain may be achieved only when Eps  is  is  on
    the order of magnitude of 1 or larger.

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreetsqueryaknn(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const ae_int_t k, const bool selfmatch, const double eps, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreetsqueryaknn(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const ae_int_t k, const double eps, const xparams _xparams = alglib::xdefault);

`kdtreetsquerybox` function

/*************************************************************************
Box query: all points within user-specified box, using thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    BoxMin      -   lower bounds, array[0..NX-1].
    BoxMax      -   upper bounds, array[0..NX-1].

RESULT
    number of actual neighbors found (in [0,N]).

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "ts" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() returns zeros for this query

NOTE: this particular query returns unordered results, because there is no
      meaningful way of  ordering  points.  Furthermore,  no 'distance' is
      associated with points - it is either INSIDE  or OUTSIDE (so request
      for distances will return zeros).

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 14.05.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreetsquerybox(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &boxmin, const real_1d_array &boxmax, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryknn` function

/*************************************************************************
K-NN query: K nearest neighbors, using external thread-local buffer.

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   kd-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    K           -   number of neighbors to return, K>=1
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of actual neighbors found (either K or N, if K>N).

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreetsqueryknn(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const ae_int_t k, const bool selfmatch, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreetsqueryknn(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const ae_int_t k, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryresultsdistances` function

/*************************************************************************
Distances from last query associated with kdtreerequestbuffer object.

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - KDTreeTsqueryresultsdistances().

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    R       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    R       -   filled with distances (in corresponding norm)

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreetsqueryresultsdistances(const kdtree &kdt, const kdtreerequestbuffer &buf, real_1d_array &r, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryresultstags` function

/*************************************************************************
Tags from last query associated with kdtreerequestbuffer object.

This function retuns results stored in  the  internal  buffer  of  kd-tree
object. If you performed buffered requests (ones which  use  instances  of
kdtreerequestbuffer class), you  should  call  buffered  version  of  this
function - KDTreeTsqueryresultstags().

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    Tags    -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    Tags    -   filled with tags associated with points,
                or, when no tags were supplied, with zeros

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreetsqueryresultstags(const kdtree &kdt, const kdtreerequestbuffer &buf, integer_1d_array &tags, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryresultsx` function

/*************************************************************************
X-values from last query associated with kdtreerequestbuffer object.

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    X       -   possibly pre-allocated buffer. If X is too small to store
                result, it is resized. If size(X) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    X       -   rows are filled with X-values

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsXY()            X- and Y-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreetsqueryresultsx(const kdtree &kdt, const kdtreerequestbuffer &buf, real_2d_array &x, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryresultsxy` function

/*************************************************************************
X- and Y-values from last query associated with kdtreerequestbuffer object.

INPUT PARAMETERS
    KDT     -   KD-tree
    Buf     -   request  buffer  object  created   for   this   particular
                instance of kd-tree structure.
    XY      -   possibly pre-allocated buffer. If XY is too small to store
                result, it is resized. If size(XY) is enough to store
                result, it is left unchanged.

OUTPUT PARAMETERS
    XY      -   rows are filled with points: first NX columns with
                X-values, next NY columns - with Y-values.

NOTES
1. points are ordered by distance from the query point (first = closest)
2. if  XY is larger than required to store result, only leading part  will
   be overwritten; trailing part will be left unchanged. So  if  on  input
   XY = [[A,B],[C,D]], and result is [1,2],  then  on  exit  we  will  get
   XY = [[1,2],[C,D]]. This is done purposely to increase performance;  if
   you want function  to  resize  array  according  to  result  size,  use
   function with same name and suffix 'I'.

SEE ALSO
* KDTreeQueryResultsX()             X-values
* KDTreeQueryResultsTags()          tag values
* KDTreeQueryResultsDistances()     distances

  -- ALGLIB --
     Copyright 28.02.2010 by Bochkanov Sergey
*************************************************************************/
void kdtreetsqueryresultsxy(const kdtree &kdt, const kdtreerequestbuffer &buf, real_2d_array &xy, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryrnn` function

/*************************************************************************
R-NN query: all points within  R-sphere  centered  at  X,  using  external
thread-local buffer, sorted by distance between point and X (by ascending)

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

NOTE: it is also possible to perform undordered queries performed by means
      of kdtreequeryrnnu() and kdtreetsqueryrnnu() functions. Such queries
      are faster because we do not have to use heap structure for sorting.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    R           -   radius of sphere (in corresponding norm), R>0
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of neighbors found, >=0

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreetsqueryrnn(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const double r, const bool selfmatch, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreetsqueryrnn(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const double r, const xparams _xparams = alglib::xdefault);

`kdtreetsqueryrnnu` function

/*************************************************************************
R-NN query: all points within  R-sphere  centered  at  X,  using  external
thread-local buffer, no ordering by distance as undicated  by  "U"  suffix
(faster that ordered query, for large queries - significantly faster).

You can call this function from multiple threads for same kd-tree instance,
assuming that different instances of buffer object are passed to different
threads.

INPUT PARAMETERS
    KDT         -   KD-tree
    Buf         -   request buffer  object  created  for  this  particular
                    instance of kd-tree structure with kdtreecreaterequestbuffer()
                    function.
    X           -   point, array[0..NX-1].
    R           -   radius of sphere (in corresponding norm), R>0
    SelfMatch   -   whether self-matches are allowed:
                    * if True, nearest neighbor may be the point itself
                      (if it exists in original dataset)
                    * if False, then only points with non-zero distance
                      are returned
                    * if not given, considered True

RESULT
    number of neighbors found, >=0

This  subroutine  performs  query  and  stores  its result in the internal
structures  of  the  buffer object. You can use following  subroutines  to
obtain these results (pay attention to "buf" in their names):
* KDTreeTsQueryResultsX() to get X-values
* KDTreeTsQueryResultsXY() to get X- and Y-values
* KDTreeTsQueryResultsTags() to get tag values
* KDTreeTsQueryResultsDistances() to get distances

As indicated by "U" suffix, this function returns unordered results.

IMPORTANT: kd-tree buffer should be used only with  KD-tree  object  which
           was used to initialize buffer. Any attempt to use biffer   with
           different object is dangerous - you  may  get  integrity  check
           failure (exception) because sizes of internal arrays do not fit
           to dimensions of KD-tree structure.

  -- ALGLIB --
     Copyright 18.03.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t kdtreetsqueryrnnu(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const double r, const bool selfmatch, const xparams _xparams = alglib::xdefault);
ae_int_t kdtreetsqueryrnnu(const kdtree &kdt, kdtreerequestbuffer &buf, const real_1d_array &x, const double r, const xparams _xparams = alglib::xdefault);

`kdtreeunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void kdtreeunserialize(const std::string &s_in, kdtree &obj);
void kdtreeunserialize(const std::istream &s_in, kdtree &obj);

nneighbor_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "alglibmisc.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array a = "[[0,0],[0,1],[1,0],[1,1]]";
        ae_int_t nx = 2;
        ae_int_t ny = 0;
        ae_int_t normtype = 2;
        kdtree kdt;
        real_1d_array x;
        real_1d_array x1;
        real_2d_array r = "[[]]";
        ae_int_t k;

        kdtreebuild(a, nx, ny, normtype, kdt);

        x = "[-1,0]";
        k = kdtreequeryknn(kdt, x, 1);
        printf("%d\n", int(k)); // EXPECTED: 1
        kdtreequeryresultsx(kdt, r);
        printf("%s\n", r.tostring(1).c_str()); // EXPECTED: [[0,0]]

        x1 = "[+0.9,0.1]";
        k = kdtreequeryknn(kdt, x1, 1);
        printf("%d\n", int(k)); // EXPECTED: 1
        kdtreequeryresultsx(kdt, r);
        printf("%s\n", r.tostring(1).c_str()); // EXPECTED: [[1,0]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

nneighbor_d_2 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "alglibmisc.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        real_2d_array a = "[[0,0],[0,1],[1,0],[1,1]]";
        ae_int_t nx = 2;
        ae_int_t ny = 0;
        ae_int_t normtype = 2;
        kdtree kdt0;
        kdtree kdt1;
        std::string s;
        real_1d_array x;
        real_2d_array r0 = "[[]]";
        real_2d_array r1 = "[[]]";

        //
        // Build tree and serialize it
        //
        kdtreebuild(a, nx, ny, normtype, kdt0);
        alglib::kdtreeserialize(kdt0, s);
        alglib::kdtreeunserialize(s, kdt1);

        //
        // Compare results from KNN queries
        //
        x = "[-1,0]";
        kdtreequeryknn(kdt0, x, 1);
        kdtreequeryresultsx(kdt0, r0);
        kdtreequeryknn(kdt1, x, 1);
        kdtreequeryresultsx(kdt1, r1);
        printf("%s\n", r0.tostring(1).c_str()); // EXPECTED: [[0,0]]
        printf("%s\n", r1.tostring(1).c_str()); // EXPECTED: [[0,0]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`nleq` subpackage

Classes

nleqreport
nleqstate

Functions

nleqcreatelm
nleqiteration
nleqrestartfrom
nleqresults
nleqresultsbuf
nleqsetcond
nleqsetstpmax
nleqsetxrep
nleqsolve

Examples

`nleqreport` class

/*************************************************************************

*************************************************************************/
class nleqreport
{
public:
    nleqreport();
    nleqreport(const nleqreport &rhs);
    nleqreport& operator=(const nleqreport &rhs);
    virtual ~nleqreport();
    ae_int_t iterationscount;
    ae_int_t nfunc;
    ae_int_t njac;
    ae_int_t terminationtype;
};

`nleqstate` class

/*************************************************************************

*************************************************************************/
class nleqstate
{
public:
    nleqstate();
    nleqstate(const nleqstate &rhs);
    nleqstate& operator=(const nleqstate &rhs);
    virtual ~nleqstate();
};

`nleqcreatelm` function

/*************************************************************************
                LEVENBERG-MARQUARDT-LIKE NONLINEAR SOLVER

DESCRIPTION:
This algorithm solves system of nonlinear equations
    F[0](x[0], ..., x[n-1])   = 0
    F[1](x[0], ..., x[n-1])   = 0
    ...
    F[M-1](x[0], ..., x[n-1]) = 0
with M/N do not necessarily coincide.  Algorithm  converges  quadratically
under following conditions:
    * the solution set XS is nonempty
    * for some xs in XS there exist such neighbourhood N(xs) that:
      * vector function F(x) and its Jacobian J(x) are continuously
        differentiable on N
      * ||F(x)|| provides local error bound on N, i.e. there  exists  such
        c1, that ||F(x)||>c1*distance(x,XS)
Note that these conditions are much more weaker than usual non-singularity
conditions. For example, algorithm will converge for any  affine  function
F (whether its Jacobian singular or not).


REQUIREMENTS:
Algorithm will request following information during its operation:
* function vector F[] and Jacobian matrix at given point X
* value of merit function f(x)=F[0]^2(x)+...+F[M-1]^2(x) at given point X


USAGE:
1. User initializes algorithm state with NLEQCreateLM() call
2. User tunes solver parameters with  NLEQSetCond(),  NLEQSetStpMax()  and
   other functions
3. User  calls  NLEQSolve()  function  which  takes  algorithm  state  and
   pointers (delegates, etc.) to callback functions which calculate  merit
   function value and Jacobian.
4. User calls NLEQResults() to get solution
5. Optionally, user may call NLEQRestartFrom() to  solve  another  problem
   with same parameters (N/M) but another starting  point  and/or  another
   function vector. NLEQRestartFrom() allows to reuse already  initialized
   structure.


INPUT PARAMETERS:
    N       -   space dimension, N>1:
                * if provided, only leading N elements of X are used
                * if not provided, determined automatically from size of X
    M       -   system size
    X       -   starting point


OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


NOTES:
1. you may tune stopping conditions with NLEQSetCond() function
2. if target function contains exp() or other fast growing functions,  and
   optimization algorithm makes too large steps which leads  to  overflow,
   use NLEQSetStpMax() function to bound algorithm's steps.
3. this  algorithm  is  a  slightly  modified implementation of the method
   described  in  'Levenberg-Marquardt  method  for constrained  nonlinear
   equations with strong local convergence properties' by Christian Kanzow
   Nobuo Yamashita and Masao Fukushima and further  developed  in  'On the
   convergence of a New Levenberg-Marquardt Method'  by  Jin-yan  Fan  and
   Ya-Xiang Yuan.


  -- ALGLIB --
     Copyright 20.08.2009 by Bochkanov Sergey
*************************************************************************/
void nleqcreatelm(const ae_int_t n, const ae_int_t m, const real_1d_array &x, nleqstate &state, const xparams _xparams = alglib::xdefault);
void nleqcreatelm(const ae_int_t m, const real_1d_array &x, nleqstate &state, const xparams _xparams = alglib::xdefault);

`nleqiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool nleqiteration(nleqstate &state, const xparams _xparams = alglib::xdefault);

`nleqrestartfrom` function

/*************************************************************************
This  subroutine  restarts  CG  algorithm from new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   structure used for reverse communication previously
                allocated with MinCGCreate call.
    X       -   new starting point.
    BndL    -   new lower bounds
    BndU    -   new upper bounds

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void nleqrestartfrom(nleqstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`nleqresults` function

/*************************************************************************
NLEQ solver results

INPUT PARAMETERS:
    State   -   algorithm state.

OUTPUT PARAMETERS:
    X       -   array[0..N-1], solution
    Rep     -   optimization report:
                * Rep.TerminationType completetion code:
                    * -4    ERROR:  algorithm   has   converged   to   the
                            stationary point Xf which is local minimum  of
                            f=F[0]^2+...+F[m-1]^2, but is not solution  of
                            nonlinear system.
                    *  1    sqrt(f)<=EpsF.
                    *  5    MaxIts steps was taken
                    *  7    stopping conditions are too stringent,
                            further improvement is impossible
                * Rep.IterationsCount contains iterations count
                * NFEV countains number of function calculations
                * ActiveConstraints contains number of active constraints

  -- ALGLIB --
     Copyright 20.08.2009 by Bochkanov Sergey
*************************************************************************/
void nleqresults(const nleqstate &state, real_1d_array &x, nleqreport &rep, const xparams _xparams = alglib::xdefault);

`nleqresultsbuf` function

/*************************************************************************
NLEQ solver results

Buffered implementation of NLEQResults(), which uses pre-allocated  buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 20.08.2009 by Bochkanov Sergey
*************************************************************************/
void nleqresultsbuf(const nleqstate &state, real_1d_array &x, nleqreport &rep, const xparams _xparams = alglib::xdefault);

`nleqsetcond` function

/*************************************************************************
This function sets stopping conditions for the nonlinear solver

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsF    -   >=0
                The subroutine finishes  its work if on k+1-th iteration
                the condition ||F||<=EpsF is satisfied
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations is unlimited.

Passing EpsF=0 and MaxIts=0 simultaneously will lead to  automatic
stopping criterion selection (small EpsF).

NOTES:

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void nleqsetcond(nleqstate &state, const double epsf, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`nleqsetstpmax` function

/*************************************************************************
This function sets maximum step length

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    StpMax  -   maximum step length, >=0. Set StpMax to 0.0,  if you don't
                want to limit step length.

Use this subroutine when target function  contains  exp()  or  other  fast
growing functions, and algorithm makes  too  large  steps  which  lead  to
overflow. This function allows us to reject steps that are too large  (and
therefore expose us to the possible overflow) without actually calculating
function value at the x+stp*d.

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void nleqsetstpmax(nleqstate &state, const double stpmax, const xparams _xparams = alglib::xdefault);

`nleqsetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to NLEQSolve().

  -- ALGLIB --
     Copyright 20.08.2010 by Bochkanov Sergey
*************************************************************************/
void nleqsetxrep(nleqstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

`nleqsolve` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear solver

These functions accept following parameters:
    state   -   algorithm state
    func    -   callback which calculates function (or merit function)
                value func at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


  -- ALGLIB --
     Copyright 20.03.2009 by Bochkanov Sergey
*************************************************************************/
void nleqsolve(nleqstate &state,
    void (*func)(const real_1d_array &x, double &func, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

`nls` subpackage

Classes

nlsreport
nlsstate

Functions

nlscreatedfo
nlsiteration
nlsoptimize
nlsrequesttermination
nlsrestartfrom
nlsresults
nlsresultsbuf
nlssetalgo2ps
nlssetalgodfolsa
nlssetbc
nlssetcond
nlssetscale
nlssetxrep

Examples

nls_derivative_free

Nonlinear least squares optimization using derivative-free algorithms

`nlsreport` class

/*************************************************************************
Optimization report, filled by NLSResults() function

FIELDS:
* TerminationType, completion code, which is a sum of a BASIC code and  an
ADDITIONAL code.

  The following basic codes denote failure:
    * -8    optimizer detected NAN/INF  either  in  the  function  itself,
            or its Jacobian; recovery was impossible, abnormal termination
            reported.
    * -3    box constraints are inconsistent

  The following basic codes denote success:
    *  2    relative step is no more than EpsX.
    *  5    MaxIts steps was taken
    *  7    stopping conditions are too stringent,
            further improvement is impossible
    *  8    terminated   by  user  who  called  NLSRequestTermination().
            X contains point which was "current accepted" when termination
            request was submitted.

  Additional codes can be set on success, but not on failure:
    * +800  if   during   algorithm   execution   the   solver encountered
            NAN/INF values in the target or  constraints  but  managed  to
            recover by reducing trust region radius,  the  solver  returns
            one of SUCCESS codes but adds +800 to the code.

* IterationsCount, contains iterations count
* NFunc, number of function calculations
*************************************************************************/
class nlsreport
{
public:
    nlsreport();
    nlsreport(const nlsreport &rhs);
    nlsreport& operator=(const nlsreport &rhs);
    virtual ~nlsreport();
    ae_int_t iterationscount;
    ae_int_t terminationtype;
    ae_int_t nfunc;
};

`nlsstate` class

/*************************************************************************
Nonlinear least squares solver
*************************************************************************/
class nlsstate
{
public:
    nlsstate();
    nlsstate(const nlsstate &rhs);
    nlsstate& operator=(const nlsstate &rhs);
    virtual ~nlsstate();
};

`nlscreatedfo` function

/*************************************************************************
                DERIVATIVE-FREE NONLINEAR LEAST SQUARES

DESCRIPTION:

This function creates a NLS  solver  configured  to  solve  a  constrained
nonlinear least squares problem

    min F(x) = f[0]^2 + f[1]^2 + ... + f[m-1]^2

where f[i] are available, but not their derivatives.

The  functions  f[i]  are  assumed  to  be   smooth,  but  may  have  some
amount of numerical noise (either  random  noise  or  deterministic  noise
arising from numerical simulations or other complex numerical processes).

INPUT PARAMETERS:
    N       -   dimension, N>1
                * if given, only leading N elements of X are used
                * if not given, automatically determined from size of X
    M       -   number of functions f[i], M>=1
    X       -   initial point, array[N]

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlscreatedfo(const ae_int_t n, const ae_int_t m, const real_1d_array &x, nlsstate &state, const xparams _xparams = alglib::xdefault);
void nlscreatedfo(const ae_int_t m, const real_1d_array &x, nlsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlsiteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool nlsiteration(nlsstate &state, const xparams _xparams = alglib::xdefault);

`nlsoptimize` function

/*************************************************************************
This family of functions is used to launch iterations of nonlinear optimizer

These functions accept following parameters:
    state   -   algorithm state
    fvec    -   callback which calculates function vector fi[]
                at given point x
    jac     -   callback which calculates function vector fi[]
                and Jacobian jac at given point x
    rep     -   optional callback which is called after each iteration
                can be NULL
    ptr     -   optional pointer which is passed to func/grad/hess/jac/rep
                can be NULL


CALLBACK PARALLELISM

The  NLS  optimizer  supports   parallel   model   evaluation   ('callback
parallelism').  This  feature,  which  is  present  in  commercial  ALGLIB
editions, greatly accelerates  optimization  when  using  a  solver  which
issues batch requests, i.e. multiple requests  for  target  values,  which
can be computed independently by different threads.

Callback parallelism is  usually   beneficial  when   processing  a  batch
request requires more than several  milliseconds.  It  also  requires  the
solver which issues requests in convenient batches, e.g. 2PS solver.

See ALGLIB Reference Manual, 'Working with commercial version' section for
more information.

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlsoptimize(nlsstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);
void nlsoptimize(nlsstate &state,
    void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr),
    void  (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr),
    void  (*rep)(const real_1d_array &x, double func, void *ptr) = NULL,
    void *ptr = NULL,
    const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlsrequesttermination` function

/*************************************************************************
This subroutine submits request for termination of running  optimizer.  It
should be called from user-supplied callback when user decides that it  is
time to "smoothly" terminate optimization process.  As  result,  optimizer
stops at point which was "current accepted" when termination  request  was
submitted and returns error code 8 (successful termination).

INPUT PARAMETERS:
    State   -   optimizer structure

NOTE: after  request  for  termination  optimizer  may   perform   several
      additional calls to user-supplied callbacks. It does  NOT  guarantee
      to stop immediately - it just guarantees that these additional calls
      will be discarded later.

NOTE: calling this function on optimizer which is NOT running will have no
      effect.

NOTE: multiple calls to this function are possible. First call is counted,
      subsequent calls are silently ignored.

  -- ALGLIB --
     Copyright 08.10.2014 by Bochkanov Sergey
*************************************************************************/
void nlsrequesttermination(nlsstate &state, const xparams _xparams = alglib::xdefault);

`nlsrestartfrom` function

/*************************************************************************
This  subroutine  restarts solver from the new point. All optimization
parameters are left unchanged.

This  function  allows  to  solve multiple  optimization  problems  (which
must have same number of dimensions) without object reallocation penalty.

INPUT PARAMETERS:
    State   -   optimizer
    X       -   new starting point.

  -- ALGLIB --
     Copyright 30.07.2010 by Bochkanov Sergey
*************************************************************************/
void nlsrestartfrom(nlsstate &state, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`nlsresults` function

/*************************************************************************
Nonlinear least squares solver results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    X       -   array[N], solution
    Rep     -   optimization  report;  includes  termination   codes   and
                additional information. Termination codes are returned  in
                rep.terminationtype field, its possible values are  listed
                below, see comments for this structure for more info.

                The termination code is a sum of a basic code (success  or
                failure)  and  one/several  additional  codes.  Additional
                codes are returned only for successful termination.

                The following basic codes can be returned:
                * -8    optimizer detected NAN/INF values in the target or
                        nonlinear constraints and failed to recover
                * -3    box constraints are inconsistent
                *  2    relative step is no more than EpsX.
                *  5    MaxIts steps was taken
                *  7    stopping conditions are too stringent,
                        further improvement is impossible
                *  8    terminated by user who called nlsrequesttermination().
                        X contains point which was "current accepted" when
                        termination request was submitted.

                The following additional codes can be returned  (added  to
                a basic code):
                * +800      if   during  algorithm  execution  the  solver
                            encountered NAN/INF values in  the  target  or
                            constraints but managed to recover by reducing
                            trust region radius, the  solver  returns  one
                            of SUCCESS codes but adds +800 to the code.

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlsresults(const nlsstate &state, real_1d_array &x, nlsreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlsresultsbuf` function

/*************************************************************************
Buffered implementation of NLSResults(), which uses pre-allocated buffer
to store X[]. If buffer size is  too  small,  it  resizes  buffer.  It  is
intended to be used in the inner cycles of performance critical algorithms
where array reallocation penalty is too large to be ignored.

  -- ALGLIB --
     Copyright 10.03.2009 by Bochkanov Sergey
*************************************************************************/
void nlsresultsbuf(const nlsstate &state, real_1d_array &x, nlsreport &rep, const xparams _xparams = alglib::xdefault);

`nlssetalgo2ps` function

/*************************************************************************
This function sets the derivative-free NLS optimization algorithm  to  the
2PS (2-Point Stencil) algorithm.

This solver is recommended for the following cases:
* an expensive target function is minimized  by the commercial ALGLIB with
  callback  parallelism  activated  (see  ALGLIB Reference Manual for more
  information about parallel callbacks)
* an inexpensive target function is minimized by any ALGLIB edition  (free
  or commercial)

This function works only with solvers created with nlscreatedfo(), i.e. in
the derivative-free mode.

See the end of this comment for more information about the algorithm.

INPUT PARAMETERS:
    State           -   solver; must be created with nlscreatedfo() call -
                        passing  an  object   initialized   with   another
                        constructor function will result in an exception.
    NNoisyRestarts  -   number of restarts performed to combat a noise  in
                        the target. (see below, section 'RESTARTS', for  a
                        detailed discussion):
                        * 0     means that no restarts is performed, the
                                solver stops as soon as stopping  criteria
                                are met. Recommended for noise-free tasks.
                        * >0    means  that when the stopping criteria are
                                met, the solver will  perform  a  restart:
                                increase the trust   radius  and  resample
                                points. It often helps to  solve  problems
                                with random or deterministic noise.

ALGORITHM DESCRIPTION AND DISCUSSION

The 2PS algorithm is a derivative-free model-based nonlinear least squares
solver which builds local models by evaluating the target at  N additional
points around the current one, with geometry similar to the 2-point finite
difference stencil.

Similarly to the Levenberg-Marquardt algorithm, the solver shows quadratic
convergence despite the fact that it builds linear models.

When compared with the DFO-LSA solver, the 2PS algorithm has the following
distinctive properties:
* the 2PS algorithm performs more target function evaluations per iteration
  (at least N+1 instead of 1-2 usually performed by the DFO-LSA)
* 2PS requires several times less iterations than the DFO-LSA because each
  iteration extracts and utilizes more information about the  target. This
  difference tends to exaggerate when N increases
* contrary to that, DFO-LSA is much better at reuse of previously computed
  points. Thus, DFO-LSA needs several times less target  evaluations  than
  2PS, usually about 3-4 times less (this ratio seems to be more  or  less
  constant independently of N).

The summary is that:
* for  expensive  targets  2PS  provides better parallelism potential than
  DFO-LSA because the former issues many  simultaneous  target  evaluation
  requests which can be easily parallelized. It  is  possible  for  2PS to
  outperform DFO-LSA by parallelism  alone,  despite  the  fact  that  the
  latter needs 3-4 times less target function evaluations.
* for inexpensive targets 2PS may win  because  it  needs many  times less
  iterations, and thus the overhead associated with the working set updates
  is also many times less.

RESTARTS

Restarts is a strategy used to deal with random and deterministic noise in
the target/constraints.

Noise in the objective function can be random, arising from measurement or
simulation   uncertainty,  or  deterministic,   resulting   from   complex
underlying phenomena like numerical errors or branches in the target.  Its
influence is especially high at last stages of the optimization, when  all
computations are performed with small values of a trust radius.

Restarts allow the optimization algorithm to be robust against both  types
of noise by temporarily increasing a trust radius in order  to  capture  a
global structure of the target and avoid being trapped  by  noise-produced
local features.

A restart is usually performed when the stopping criteria  are  triggered.
Instead of stopping, the solver increases  trust  radius  to  its  initial
value and tries to rebuild a model.

If you decide to optimize with restarts,  it  is  recommended  to  perform
a small amount of restarts, up to 5. Generally, restarts do not allow  one
to completely solve the problem of noise, but still   it  is  possible  to
achieve some additional progress.

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlssetalgo2ps(nlsstate &state, const ae_int_t nnoisyrestarts, const xparams _xparams = alglib::xdefault);
void nlssetalgo2ps(nlsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlssetalgodfolsa` function

/*************************************************************************
This function sets the derivative-free NLS optimization algorithm  to  the
DFO-LSA algorithm, an ALGLIB implementation (with several modifications)of
the original DFO-LS algorithm  by  Cartis, C., Fiala, J., Marteau, B.  and
Roberts, L. ('Improving the  Flexibility  and  Robustness  of  Model-Based
Derivative-Free Optimization Solvers', 2019). The A in DFO-LSA stands  for
ALGLIB, in order to distinguish our slightly modified implementation  from
the original algorithm.

This solver is recommended for the following  case:  an  expensive  target
function is minimized without parallelism being used (either  free  ALGLIB
is used or commercial one is used but the target callback is non-reentrant
i.e. it can not be simultaneously called from multiple threads)

This function works only with solvers created with nlscreatedfo(), i.e. in
the derivative-free mode.

See the end of this comment for more information about the algorithm.

INPUT PARAMETERS:
    State           -   solver; must be created with nlscreatedfo() call -
                        passing  an  object   initialized   with   another
                        constructor function will result in an exception.
    NNoisyRestarts  -   number of restarts performed to combat a noise  in
                        the target. (see below, section 'RESTARTS', for  a
                        detailed discussion):
                        * 0     means that no restarts is performed, the
                                solver stops as soon as stopping  criteria
                                are met. Recommended for noise-free tasks.
                        * >0    means  that when the stopping criteria are
                                met, the solver will  perform  a  restart:
                                increase the trust   radius  and  resample
                                points. It often helps to  solve  problems
                                with random or deterministic noise.

ALGORITHM DESCRIPTION AND DISCUSSION

The DFO-LSA algorithm is a derivative-free model-based  NLS  solver  which
builds local models by remembering N+1 previously computed  target  values
and updating them as optimization progresses.

Similarly to the Levenberg-Marquardt algorithm, the solver shows quadratic
convergence  despite  the  fact  that  it  builds   linear   models.   Our
implementation generally follows the same lines as the  original  DFO-LSA,
with several modifications to trust radius  update  strategies,  stability
fixes (unlike original DFO-LS, our implementation can handle  and  recover
from the target breaking down due to infeasible arguments) and other minor
implementation details.

When compared with the 2PS solver, the DFO-LSA algorithm has the following
distinctive properties:
* the 2PS algorithm performs more target function evaluations per iteration
  (at least N+1 instead of 1-2 usually performed by DFO-LSA)
* 2PS requires several times less iterations  than  DFO-LSA  because  each
  iterations extracts and utilizes more information about the target. This
  difference tends to exaggerate when N increases
* contrary to that, DFO-LSA is much better at reuse of previously computed
  points. Thus, DFO-LSA needs several times less target  evaluations  than
  2PS, usually about 3-4 times less (this ratio seems to be more  or  less
  constant independently of N).

The summary is that:
* for  expensive  targets DFO-LSA is much more efficient than 2PS  because
  it reuses previously computed target values as much as possible.
* however, DFO-LSA has little parallelism potential because  (unlike  2PS)
  it  does  not  evaluate the target  in several points simultaneously and
  independently
* additionally, because DFO-LSA performs many times more  iterations  than
  2PS, iteration overhead (working set updates and matrix  inversions)  is
  an issue here. For inexpensive targets it is possible for DFO-LSA to  be
  outperformed by 2PS merely because of the linear algebra cost.

RESTARTS

Restarts is a strategy used to deal with random and deterministic noise in
the target/constraints.

Noise in the objective function can be random, arising from measurement or
simulation   uncertainty,  or  deterministic,   resulting   from   complex
underlying phenomena like numerical errors or branches in the target.  Its
influence is especially high at last stages of the optimization, when  all
computations are performed with small values of a trust radius.

Restarts allow the optimization algorithm to be robust against both  types
of noise by temporarily increasing a trust radius in order  to  capture  a
global structure of the target and avoid being trapped  by  noise-produced
local features.

A restart is usually performed when the stopping criteria  are  triggered.
Instead of stopping, the solver increases  trust  radius  to  its  initial
value and tries to rebuild a model.

If you decide to optimize with restarts,  it  is  recommended  to  perform
a small amount of restarts, up to 5. Generally, restarts do not allow  one
to completely solve the problem of noise, but still   it  is  possible  to
achieve some additional progress.

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlssetalgodfolsa(nlsstate &state, const ae_int_t nnoisyrestarts, const xparams _xparams = alglib::xdefault);
void nlssetalgodfolsa(nlsstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlssetbc` function

/*************************************************************************
This function sets box constraints

Box constraints are inactive by default (after initial creation). They are
preserved until explicitly turned off with another SetBC() call.

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    BndL    -   lower bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very small number or -INF (the latter is recommended).
    BndU    -   upper bounds, array[N].
                If some (all) variables are unbounded, you may specify
                very large number or +INF (the latter is recommended).

NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th
variable will be "frozen" at X[i]=BndL[i]=BndU[i].

NOTE 2: unless  explicitly  mentioned  in  the   specific   NLS  algorithm
        description, the following holds:
        * box constraints are always satisfied exactly
        * the target is NOT evaluated outside of the box-constrained area

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlssetbc(nlsstate &state, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlssetcond` function

/*************************************************************************
This function sets stopping conditions

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    EpsX    -   stop when the scaled trust region radius is  smaller  than
                EpsX.
    MaxIts  -   maximum number of iterations. If MaxIts=0, the  number  of
                iterations   is    unlimited.

Passing  EpsX=0  and  MaxIts=0  (simultaneously)  will  lead  to automatic
stopping criterion selection (small EpsX).

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlssetcond(nlsstate &state, const double epsx, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

Examples: [1]

`nlssetscale` function

/*************************************************************************
This function sets variable scales

ALGLIB optimizers use scaling matrices to test stopping  conditions  (step
size and gradient are scaled before comparison with tolerances).  Scale of
the I-th variable is a translation invariant measure of:
a) "how large" the variable is
b) how large the step should be to make significant changes in the function

Generally, scale is NOT considered to be a  form  of  preconditioner.  But
derivative-free optimizers often use scaling matrix both  in  the stopping
condition tests and as a preconditioner.

Proper scaling is very important for the algorithm performance. It is less
important for the quality of results, but still has some influence (it  is
easier  to  converge  when  variables  are  properly  scaled, so premature
stopping is possible when very badly scalled variables are  combined  with
relaxed stopping conditions).

INPUT PARAMETERS:
    State   -   structure stores algorithm state
    S       -   array[N], non-zero scaling coefficients
                S[i] may be negative, sign doesn't matter.

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlssetscale(nlsstate &state, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`nlssetxrep` function

/*************************************************************************
This function turns on/off reporting.

INPUT PARAMETERS:
    State   -   structure which stores algorithm state
    NeedXRep-   whether iteration reports are needed or not

If NeedXRep is True, algorithm will call rep() callback function if  it is
provided to NLSOptimize().

  -- ALGLIB --
     Copyright 15.10.2023 by Bochkanov Sergey
*************************************************************************/
void nlssetxrep(nlsstate &state, const bool needxrep, const xparams _xparams = alglib::xdefault);

nls_derivative_free example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"

using namespace alglib;
void  function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
    //
    // this callback calculates
    // f0(x0,x1) = 100*(x0+3)^4,
    // f1(x0,x1) = (x1-3)^4
    //
    fi[0] = 10*pow(x[0]+3,2);
    fi[1] = pow(x[1]-3,2);
}
int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where 
        //
        //     f0(x0,x1) = 10*(x0+3)^2
        //     f1(x0,x1) = (x1-3)^2
        //
        // subject to box constraints
        //
        //     -1 <= x0 <= +1
        //     -1 <= x1 <= +1
        //
        // using DFO mode of the NLS optimizer.
        //
        // IMPORTANT: the  NLS  optimizer   supports   parallel  model  evaluation
        //            ('callback parallelism'). This feature, which  is present in
        //            commercial ALGLIB editions, greatly accelerates optimization
        //            when  using  a  solver  which  issues  batch  requests, i.e.
        //            multiple requests  for  target values, which can be computed
        //            independently by different threads.
        //
        //            Callback parallelism is usually  beneficial when  processing
        //            a  batch  request  requires  more than several milliseconds.
        //            This particular  example,  of  course,  is  not  suited  for
        //            callback parallelism.
        //
        //            It  also  requires  the  solver  which  issues  requests  in
        //            convenient batches, e.g. 2PS solver.
        //
        //            See ALGLIB Reference Manual, 'Working with commercial version'
        //            section,  and  comments  on  nlsoptimize() function for more
        //            information.
        //
        real_1d_array x = "[0,0]";
        real_1d_array s = "[1,1]";
        real_1d_array bndl = "[-1,-1]";
        real_1d_array bndu = "[+1,+1]";
        double epsx = 0.0000001;
        ae_int_t maxits = 0;
        nlsstate state;
        nlsreport rep;

        //
        // Create optimizer, tell it to:
        // * use derivative-free mode
        // * use unit scale for all variables (s is a unit vector)
        // * stop after short enough step (less than epsx)
        //
        nlscreatedfo(2, x, state);
        nlssetcond(state, epsx, maxits);
        nlssetscale(state, s);
        nlssetbc(state, bndl, bndu);

        //
        // Choose a derivative-free nonlinear least squares algorithm. ALGLIB
        // supports the following solvers:
        //
        // * DFO-LSA  - a modified version of DFO-LS (Cartis, Fiala, Marteau,
        //   Roberts), with "A" standing for ALGLIB in order to distinguish it
        //   from the original version. This algorithm achieves the smallest
        //   function evaluations count, but has relatively high iteration
        //   overhead and no callback parallelism potential (it issues target
        //   evaluation requests one by one, so they can not be parallelized).
        //   Recommended for expensive targets with no parallelism support.
        //
        // * 2PS (two-point stencil) - an easily parallelized algorithm
        //   developed by ALGLIB Project. It needs about 3x-4x more target
        //   evaluations than DFO-LSA (the ratio has no strong dependence on
        //   the problem size), however it issues target evaluation requests
        //   in large batches, so they can be computed in parallel. Additionally
        //   it has low iteration overhead, so it can be better suited for
        //   problems with cheap targets that DFO-LSA.
        //
        // Both solvers demonstrate quadratic convergence similarly to the
        // Levenberg-Marquardt method.
        //
        // The summary is:
        // * expensive target, no parallelism => DFO-LSA 
        // * expensive target, parallel callbacks => 2PS
        // * inexpensive target => most likely 2PS, maybe DFO-LSA
        //
        // The code below sets the algorithm to be DFO-LSA, then switches
        // it to 2PS.
        //
        nlssetalgodfolsa(state);
        nlssetalgo2ps(state);

        //
        // Solve the problem.
        //
        // The code below does not use parallelism. If you want to activate
        // callback parallelism, use commercial edition of ALGLIB and pass
        // alglib::parallelcallbacks as an additional parameter to nlsoptimize().
        //
        // Callback parallelism is intended for expensive problems where one
        // batch (~N target evaluations) takes tens and hundreds of milliseconds
        // to compute.
        //
        alglib::nlsoptimize(state, function1_fvec);

        //
        // Test optimization results
        //
        nlsresults(state, x, rep);
        printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,+1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`normaldistr` subpackage

Functions

bivariatenormalcdf
bivariatenormalpdf
errorfunction
errorfunctionc
inverf
invnormalcdf
invnormaldistribution
normalcdf
normaldistribution
normalpdf

Examples

`bivariatenormalcdf` function

/*************************************************************************
Bivariate normal CDF

Returns the area under the bivariate Gaussian  PDF  with  correlation
parameter equal to Rho, integrated from minus infinity to (x,y):


                                          x      y
                                          -      -
                            1            | |    | |
    bvn(x,y,rho) = -------------------   |      |   f(u,v,rho)*du*dv
                    2pi*sqrt(1-rho^2)  | |    | |
                                        -      -
                                       -INF   -INF


where

                      (    u^2 - 2*rho*u*v + v^2  )
    f(u,v,rho)   = exp( - ----------------------- )
                      (        2*(1-rho^2)        )


with -1<rho<+1 and arbitrary x, y.

This subroutine uses high-precision approximation scheme proposed  by
Alan Genz in "Numerical  Computation  of  Rectangular  Bivariate  and
Trivariate Normal and  t  probabilities",  which  computes  CDF  with
absolute error roughly equal to 1e-14.

This function won't fail as long as Rho is in (-1,+1) range.

  -- ALGLIB --
     Copyright 15.11.2019 by Bochkanov Sergey
*************************************************************************/
double bivariatenormalcdf(const double x, const double y, const double rho, const xparams _xparams = alglib::xdefault);

`bivariatenormalpdf` function

/*************************************************************************
Bivariate normal PDF

Returns probability density function of the bivariate  Gaussian  with
correlation parameter equal to Rho:

                         1              (    x^2 - 2*rho*x*y + y^2  )
    f(x,y,rho) = ----------------- * exp( - ----------------------- )
                 2pi*sqrt(1-rho^2)      (        2*(1-rho^2)        )


with -1<rho<+1 and arbitrary x, y.

This function won't fail as long as Rho is in (-1,+1) range.

  -- ALGLIB --
     Copyright 15.11.2019 by Bochkanov Sergey
*************************************************************************/
double bivariatenormalpdf(const double x, const double y, const double rho, const xparams _xparams = alglib::xdefault);

`errorfunction` function

/*************************************************************************
Error function

The integral is

                          x
                           -
                2         | |          2
  erf(x)  =  --------     |    exp( - t  ) dt.
             sqrt(pi)   | |
                         -
                          0

For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise
erf(x) = 1 - erfc(x).


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,1         30000       3.7e-16     1.0e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunction(const double x, const xparams _xparams = alglib::xdefault);

`errorfunctionc` function

/*************************************************************************
Complementary error function

 1 - erf(x) =

                          inf.
                            -
                 2         | |          2
  erfc(x)  =  --------     |    exp( - t  ) dt
              sqrt(pi)   | |
                          -
                           x


For small x, erfc(x) = 1 - erf(x); otherwise rational
approximations are computed.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE      0,26.6417   30000       5.7e-14     1.5e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double errorfunctionc(const double x, const xparams _xparams = alglib::xdefault);

`inverf` function

/*************************************************************************
Inverse of the error function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double inverf(const double e, const xparams _xparams = alglib::xdefault);

`invnormalcdf` function

/*************************************************************************
Inverse of Normal CDF

Returns the argument, x, for which the area under the
Gaussian probability density function (integrated from
minus infinity to x) is equal to y.


For small arguments 0 < y < exp(-2), the program computes
z = sqrt( -2.0 * log(y) );  then the approximation is
x = z - log(z)/z  - (1/z) P(1/z) / Q(1/z).
There are two rational functions P/Q, one for 0 < y < exp(-32)
and the other for y up to exp(-2).  For larger arguments,
w = y - 0.5, and  x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).

ACCURACY:

                     Relative error:
arithmetic   domain        # trials      peak         rms
   IEEE     0.125, 1        20000       7.2e-16     1.3e-16
   IEEE     3e-308, 0.135   50000       4.6e-16     9.8e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double invnormalcdf(const double y0, const xparams _xparams = alglib::xdefault);

`invnormaldistribution` function

/*************************************************************************
Same as invnormalcdf(), deprecated name
*************************************************************************/
double invnormaldistribution(const double y0, const xparams _xparams = alglib::xdefault);

`normalcdf` function

/*************************************************************************
Normal distribution CDF

Returns the area under the Gaussian probability density
function, integrated from minus infinity to x:

                           x
                            -
                  1        | |          2
   ndtr(x)  = ---------    |    exp( - t /2 ) dt
              sqrt(2pi)  | |
                          -
                         -inf.

            =  ( 1 + erf(z) ) / 2
            =  erfc(z) / 2

where z = x/sqrt(2). Computation is via the functions
erf and erfc.


ACCURACY:

                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -13,0        30000       3.4e-14     6.7e-15

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double normalcdf(const double x, const xparams _xparams = alglib::xdefault);

`normaldistribution` function

/*************************************************************************
Same as normalcdf(), obsolete name.
*************************************************************************/
double normaldistribution(const double x, const xparams _xparams = alglib::xdefault);

`normalpdf` function

/*************************************************************************
Normal distribution PDF

Returns Gaussian probability density function:

               1
   f(x)  = --------- * exp(-x^2/2)
           sqrt(2pi)

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double normalpdf(const double x, const xparams _xparams = alglib::xdefault);

`normestimator` subpackage

Classes

normestimatorstate

Functions

normestimatorcreate
normestimatorestimatesparse
normestimatorresults
normestimatorsetseed

Examples

`normestimatorstate` class

/*************************************************************************
This object stores state of the iterative norm estimation algorithm.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class normestimatorstate
{
public:
    normestimatorstate();
    normestimatorstate(const normestimatorstate &rhs);
    normestimatorstate& operator=(const normestimatorstate &rhs);
    virtual ~normestimatorstate();
};

`normestimatorcreate` function

/*************************************************************************
This procedure initializes matrix norm estimator.

USAGE:
1. User initializes algorithm state with NormEstimatorCreate() call
2. User calls NormEstimatorEstimateSparse() (or NormEstimatorIteration())
3. User calls NormEstimatorResults() to get solution.

INPUT PARAMETERS:
    M       -   number of rows in the matrix being estimated, M>0
    N       -   number of columns in the matrix being estimated, N>0
    NStart  -   number of random starting vectors
                recommended value - at least 5.
    NIts    -   number of iterations to do with best starting vector
                recommended value - at least 5.

OUTPUT PARAMETERS:
    State   -   structure which stores algorithm state


NOTE: this algorithm is effectively deterministic, i.e. it always  returns
same result when repeatedly called for the same matrix. In fact, algorithm
uses randomized starting vectors, but internal  random  numbers  generator
always generates same sequence of the random values (it is a  feature, not
bug).

Algorithm can be made non-deterministic with NormEstimatorSetSeed(0) call.

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorcreate(const ae_int_t m, const ae_int_t n, const ae_int_t nstart, const ae_int_t nits, normestimatorstate &state, const xparams _xparams = alglib::xdefault);

`normestimatorestimatesparse` function

/*************************************************************************
This function estimates norm of the sparse M*N matrix A.

INPUT PARAMETERS:
    State       -   norm estimator state, must be initialized with a  call
                    to NormEstimatorCreate()
    A           -   sparse M*N matrix, must be converted to CRS format
                    prior to calling this function.

After this function  is  over  you can call NormEstimatorResults() to get
estimate of the norm(A).

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorestimatesparse(normestimatorstate &state, const sparsematrix &a, const xparams _xparams = alglib::xdefault);

`normestimatorresults` function

/*************************************************************************
Matrix norm estimation results

INPUT PARAMETERS:
    State   -   algorithm state

OUTPUT PARAMETERS:
    Nrm     -   estimate of the matrix norm, Nrm>=0

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorresults(const normestimatorstate &state, double &nrm, const xparams _xparams = alglib::xdefault);

`normestimatorsetseed` function

/*************************************************************************
This function changes seed value used by algorithm. In some cases we  need
deterministic processing, i.e. subsequent calls must return equal results,
in other cases we need non-deterministic algorithm which returns different
results for the same matrix on every pass.

Setting zero seed will lead to non-deterministic algorithm, while non-zero
value will make our algorithm deterministic.

INPUT PARAMETERS:
    State       -   norm estimator state, must be initialized with a  call
                    to NormEstimatorCreate()
    SeedVal     -   seed value, >=0. Zero value = non-deterministic algo.

  -- ALGLIB --
     Copyright 06.12.2011 by Bochkanov Sergey
*************************************************************************/
void normestimatorsetseed(normestimatorstate &state, const ae_int_t seedval, const xparams _xparams = alglib::xdefault);

`odesolver` subpackage

Classes

odesolverreport
odesolverstate

Functions

odesolveriteration
odesolverresults
odesolverrkck
odesolversolve

Examples

odesolver_d1

Solving y'=-y with ODE solver

`odesolverreport` class

/*************************************************************************

*************************************************************************/
class odesolverreport
{
public:
    odesolverreport();
    odesolverreport(const odesolverreport &rhs);
    odesolverreport& operator=(const odesolverreport &rhs);
    virtual ~odesolverreport();
    ae_int_t nfev;
    ae_int_t terminationtype;
};

`odesolverstate` class

/*************************************************************************

*************************************************************************/
class odesolverstate
{
public:
    odesolverstate();
    odesolverstate(const odesolverstate &rhs);
    odesolverstate& operator=(const odesolverstate &rhs);
    virtual ~odesolverstate();
};

`odesolveriteration` function

/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
bool odesolveriteration(odesolverstate &state, const xparams _xparams = alglib::xdefault);

`odesolverresults` function

/*************************************************************************
ODE solver results

Called after OdeSolverIteration returned False.

INPUT PARAMETERS:
    State   -   algorithm state (used by OdeSolverIteration).

OUTPUT PARAMETERS:
    M       -   number of tabulated values, M>=1
    XTbl    -   array[0..M-1], values of X
    YTbl    -   array[0..M-1,0..N-1], values of Y in X[i]
    Rep     -   solver report:
                * Rep.TerminationType completetion code:
                    * -2    X is not ordered  by  ascending/descending  or
                            there are non-distinct X[],  i.e.  X[i]=X[i+1]
                    * -1    incorrect parameters were specified
                    *  1    task has been solved
                * Rep.NFEV contains number of function calculations

  -- ALGLIB --
     Copyright 01.09.2009 by Bochkanov Sergey
*************************************************************************/
void odesolverresults(const odesolverstate &state, ae_int_t &m, real_1d_array &xtbl, real_2d_array &ytbl, odesolverreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`odesolverrkck` function

/*************************************************************************
Cash-Karp adaptive ODE solver.

This subroutine solves ODE  Y'=f(Y,x)  with  initial  conditions  Y(xs)=Ys
(here Y may be single variable or vector of N variables).

INPUT PARAMETERS:
    Y       -   initial conditions, array[0..N-1].
                contains values of Y[] at X[0]
    N       -   system size
    X       -   points at which Y should be tabulated, array[0..M-1]
                integrations starts at X[0], ends at X[M-1],  intermediate
                values at X[i] are returned too.
                SHOULD BE ORDERED BY ASCENDING OR BY DESCENDING!
    M       -   number of intermediate points + first point + last point:
                * M>2 means that you need both Y(X[M-1]) and M-2 values at
                  intermediate points
                * M=2 means that you want just to integrate from  X[0]  to
                  X[1] and don't interested in intermediate values.
                * M=1 means that you don't want to integrate :)
                  it is degenerate case, but it will be handled correctly.
                * M<1 means error
    Eps     -   tolerance (absolute/relative error on each  step  will  be
                less than Eps). When passing:
                * Eps>0, it means desired ABSOLUTE error
                * Eps<0, it means desired RELATIVE error.  Relative errors
                  are calculated with respect to maximum values of  Y seen
                  so far. Be careful to use this criterion  when  starting
                  from Y[] that are close to zero.
    H       -   initial  step  lenth,  it  will  be adjusted automatically
                after the first  step.  If  H=0,  step  will  be  selected
                automatically  (usualy  it  will  be  equal  to  0.001  of
                min(x[i]-x[j])).

OUTPUT PARAMETERS
    State   -   structure which stores algorithm state between  subsequent
                calls of OdeSolverIteration. Used for reverse communication.
                This structure should be passed  to the OdeSolverIteration
                subroutine.

SEE ALSO
    AutoGKSmoothW, AutoGKSingular, AutoGKIteration, AutoGKResults.


  -- ALGLIB --
     Copyright 01.09.2009 by Bochkanov Sergey
*************************************************************************/
void odesolverrkck(const real_1d_array &y, const ae_int_t n, const real_1d_array &x, const ae_int_t m, const double eps, const double h, odesolverstate &state, const xparams _xparams = alglib::xdefault);
void odesolverrkck(const real_1d_array &y, const real_1d_array &x, const double eps, const double h, odesolverstate &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`odesolversolve` function

/*************************************************************************
This function is used to start iterations of the ODE solver

It accepts following parameters:
    diff    -   callback which calculates dy/dx for given y and x
    ptr     -   optional pointer which is passed to diff; can be NULL


  -- ALGLIB --
     Copyright 01.09.2009 by Bochkanov Sergey
*************************************************************************/
void odesolversolve(odesolverstate &state,
    void (*diff)(const real_1d_array &y, double x, real_1d_array &dy, void *ptr),
    void *ptr = NULL, const xparams _xparams = alglib::xdefault);

Examples: [1]

odesolver_d1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "diffequations.h"

using namespace alglib;
void ode_function_1_diff(const real_1d_array &y, double x, real_1d_array &dy, void *ptr) 
{
    // this callback calculates f(y[],x)=-y[0]
    dy[0] = -y[0];
}
int main(int argc, char **argv)
{
    try
    {
        real_1d_array y = "[1]";
        real_1d_array x = "[0, 1, 2, 3]";
        double eps = 0.00001;
        double h = 0;
        odesolverstate s;
        ae_int_t m;
        real_1d_array xtbl;
        real_2d_array ytbl;
        odesolverreport rep;
        odesolverrkck(y, x, eps, h, s);
        alglib::odesolversolve(s, ode_function_1_diff);
        odesolverresults(s, m, xtbl, ytbl, rep);
        printf("%d\n", int(m)); // EXPECTED: 4
        printf("%s\n", xtbl.tostring(2).c_str()); // EXPECTED: [0, 1, 2, 3]
        printf("%s\n", ytbl.tostring(2).c_str()); // EXPECTED: [[1], [0.367], [0.135], [0.050]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`optguardapi` subpackage

Classes

optguardnonc0report
optguardnonc1test0report
optguardnonc1test1report
optguardreport

Examples

`optguardnonc0report` class

/*************************************************************************
This  structure  is  used  for  detailed   reporting  about  suspected  C0
continuity violation.

=== WHAT IS TESTED =======================================================

C0 test  studies  function  values (not gradient!)  obtained  during  line
searches and monitors estimate of the Lipschitz  constant.  Sudden  spikes
usually indicate that discontinuity was detected.


=== WHAT IS REPORTED =====================================================

Actually, report retrieval function returns TWO report structures:

* one for most suspicious point found so far (one with highest  change  in
  the function value), so called "strongest" report
* another one for most detailed line search (more function  evaluations  =
  easier to understand what's going on) which triggered  test #0 criteria,
  so called "longest" report

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* fidx - is an index of the function (0 for  target  function, 1 or higher
  for nonlinear constraints) which is suspected of being "non-C1"
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.
* inneriter, outeriter - inner and outer iteration indexes (can be -1 if no
  iteration information was specified)

You can plot function values stored in stp[]  and  f[]  arrays  and  study
behavior of your function by your own eyes, just  to  be  sure  that  test
correctly reported C1 violation.

  -- ALGLIB --
     Copyright 19.11.2018 by Bochkanov Sergey
*************************************************************************/
class optguardnonc0report
{
public:
    optguardnonc0report();
    optguardnonc0report(const optguardnonc0report &rhs);
    optguardnonc0report& operator=(const optguardnonc0report &rhs);
    virtual ~optguardnonc0report();
    bool positive;
    ae_int_t fidx;
    real_1d_array x0;
    real_1d_array d;
    ae_int_t n;
    real_1d_array stp;
    real_1d_array f;
    ae_int_t cnt;
    ae_int_t stpidxa;
    ae_int_t stpidxb;
    ae_int_t inneriter;
    ae_int_t outeriter;
};

`optguardnonc1test0report` class

/*************************************************************************
This  structure  is  used  for  detailed   reporting  about  suspected  C1
continuity violation as flagged by C1 test #0 (OptGuard  has several tests
for C1 continuity, this report is used by #0).

=== WHAT IS TESTED =======================================================

C1 test #0 studies function values (not gradient!)  obtained  during  line
searches and monitors behavior of directional  derivative  estimate.  This
test is less powerful than test #1, but it does  not  depend  on  gradient
values  and  thus  it  is  more  robust  against  artifacts  introduced by
numerical differentiation.


=== WHAT IS REPORTED =====================================================

Actually, report retrieval function returns TWO report structures:

* one for most suspicious point found so far (one with highest  change  in
  the directional derivative), so called "strongest" report
* another one for most detailed line search (more function  evaluations  =
  easier to understand what's going on) which triggered  test #0 criteria,
  so called "longest" report

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* fidx - is an index of the function (0 for  target  function, 1 or higher
  for nonlinear constraints) which is suspected of being "non-C1"
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], f[] - arrays of length CNT which store step lengths and  function
  values at these points; f[i] is evaluated in x0+stp[i]*d.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.
* inneriter, outeriter - inner and outer iteration indexes (can be -1 if no
  iteration information was specified)

You can plot function values stored in stp[]  and  f[]  arrays  and  study
behavior of your function by your own eyes, just  to  be  sure  that  test
correctly reported C1 violation.

  -- ALGLIB --
     Copyright 19.11.2018 by Bochkanov Sergey
*************************************************************************/
class optguardnonc1test0report
{
public:
    optguardnonc1test0report();
    optguardnonc1test0report(const optguardnonc1test0report &rhs);
    optguardnonc1test0report& operator=(const optguardnonc1test0report &rhs);
    virtual ~optguardnonc1test0report();
    bool positive;
    ae_int_t fidx;
    real_1d_array x0;
    real_1d_array d;
    ae_int_t n;
    real_1d_array stp;
    real_1d_array f;
    ae_int_t cnt;
    ae_int_t stpidxa;
    ae_int_t stpidxb;
    ae_int_t inneriter;
    ae_int_t outeriter;
};

`optguardnonc1test1report` class

/*************************************************************************
This  structure  is  used  for  detailed   reporting  about  suspected  C1
continuity violation as flagged by C1 test #1 (OptGuard  has several tests
for C1 continuity, this report is used by #1).

=== WHAT IS TESTED =======================================================

C1 test #1 studies individual  components  of  the  gradient  as  recorded
during line searches. Upon discovering discontinuity in the gradient  this
test records specific component which was suspected (or  one  with  highest
indication of discontinuity if multiple components are suspected).

When precise analytic gradient is provided this test is more powerful than
test #0  which  works  with  function  values  and  ignores  user-provided
gradient.  However,  test  #0  becomes  more   powerful   when   numerical
differentiation is employed (in such cases test #1 detects  higher  levels
of numerical noise and becomes too conservative).

This test also tells specific components of the gradient which violate  C1
continuity, which makes it more informative than #0, which just tells that
continuity is violated.


=== WHAT IS REPORTED =====================================================

Actually, report retrieval function returns TWO report structures:

* one for most suspicious point found so far (one with highest  change  in
  the directional derivative), so called "strongest" report
* another one for most detailed line search (more function  evaluations  =
  easier to understand what's going on) which triggered  test #1 criteria,
  so called "longest" report

In both cases following fields are returned:

* positive - is TRUE  when test flagged suspicious point;  FALSE  if  test
  did not notice anything (in the latter cases fields below are empty).
* fidx - is an index of the function (0 for  target  function, 1 or higher
  for nonlinear constraints) which is suspected of being "non-C1"
* vidx - is an index of the variable in [0,N) with nonsmooth derivative
* x0[], d[] - arrays of length N which store initial point  and  direction
  for line search (d[] can be normalized, but does not have to)
* stp[], g[] - arrays of length CNT which store step lengths and  gradient
  values at these points; g[i] is evaluated in  x0+stp[i]*d  and  contains
  vidx-th component of the gradient.
* stpidxa, stpidxb - we  suspect  that  function  violates  C1  continuity
  between steps #stpidxa and #stpidxb (usually we have  stpidxb=stpidxa+3,
  with  most  likely  position  of  the  violation  between  stpidxa+1 and
  stpidxa+2.
* inneriter, outeriter - inner and outer iteration indexes (can be -1 if  no
  iteration information was specified)

You can plot function values stored in stp[]  and  g[]  arrays  and  study
behavior of your function by your own eyes, just  to  be  sure  that  test
correctly reported C1 violation.

  -- ALGLIB --
     Copyright 19.11.2018 by Bochkanov Sergey
*************************************************************************/
class optguardnonc1test1report
{
public:
    optguardnonc1test1report();
    optguardnonc1test1report(const optguardnonc1test1report &rhs);
    optguardnonc1test1report& operator=(const optguardnonc1test1report &rhs);
    virtual ~optguardnonc1test1report();
    bool positive;
    ae_int_t fidx;
    ae_int_t vidx;
    real_1d_array x0;
    real_1d_array d;
    ae_int_t n;
    real_1d_array stp;
    real_1d_array g;
    ae_int_t cnt;
    ae_int_t stpidxa;
    ae_int_t stpidxb;
    ae_int_t inneriter;
    ae_int_t outeriter;
};

`optguardreport` class

/*************************************************************************
This structure is used to store  OptGuard  report,  i.e.  report  on   the
properties of the nonlinear function being optimized with ALGLIB.

After you tell your optimizer to activate OptGuard  this technology starts
to silently monitor function values and gradients/Jacobians  being  passed
all around during your optimization session. Depending on specific set  of
checks enabled OptGuard may perform additional function evaluations  (say,
about 3*N evaluations if you want to check analytic gradient for errors).

Upon discovering that something strange happens  (function  values  and/or
gradient components change too sharply and/or unexpectedly) OptGuard  sets
one of the "suspicion  flags" (without interrupting optimization session).
After optimization is done, you can examine OptGuard report.

Following report fields can be set:
* nonc0suspected
* nonc1suspected
* badgradsuspected


=== WHAT CAN BE DETECTED WITH OptGuard INTEGRITY CHECKER =================

Following  types  of  errors  in your target function (constraints) can be
caught:
a) discontinuous functions ("non-C0" part of the report)
b) functions with discontinuous derivative ("non-C1" part of the report)
c) errors in the analytic gradient provided by user

These types of errors result in optimizer  stopping  well  before reaching
solution (most often - right after encountering discontinuity).

Type A errors are usually  coding  errors  during  implementation  of  the
target function. Most "normal" problems involve continuous functions,  and
anyway you can't reliably optimize discontinuous function.

Type B errors are either coding errors or (in case code itself is correct)
evidence of the fact  that  your  problem  is  an  "incorrect"  one.  Most
optimizers (except for ones provided by MINNS subpackage) do  not  support
nonsmooth problems.

Type C errors are coding errors which often prevent optimizer from  making
even one step  or result in optimizing stopping  too  early,  as  soon  as
actual descent direction becomes too different from one suggested by user-
supplied gradient.


=== WHAT IS REPORTED =====================================================

Following set of report fields deals with discontinuous  target functions,
ones not belonging to C0 continuity class:

* nonc0suspected - is a flag which is set upon discovering some indication
  of the discontinuity. If this flag is false, the rest of "non-C0" fields
  should be ignored
* nonc0fidx - is an index of the function (0 for  target  function,  1  or
  higher for nonlinear constraints) which is suspected of being "non-C0"
* nonc0lipshitzc - a Lipchitz constant for a function which was  suspected
  of being non-continuous.
* nonc0test0positive -  set  to  indicate  specific  test  which  detected
  continuity violation (test #0)

Following set of report fields deals with discontinuous gradient/Jacobian,
i.e. with functions violating C1 continuity:

* nonc1suspected - is a flag which is set upon discovering some indication
  of the discontinuity. If this flag is false, the rest of "non-C1" fields
  should be ignored
* nonc1fidx - is an index of the function (0 for  target  function,  1  or
  higher for nonlinear constraints) which is suspected of being "non-C1"
* nonc1lipshitzc - a Lipchitz constant for a function gradient  which  was
  suspected of being non-smooth.
* nonc1test0positive -  set  to  indicate  specific  test  which  detected
  continuity violation (test #0)
* nonc1test1positive -  set  to  indicate  specific  test  which  detected
  continuity violation (test #1)

Following set of report fields deals with errors in the gradient:
* badgradsuspected - is a flad which is set upon discovering an  error  in
  the analytic gradient supplied by user
* badgradfidx - index  of   the  function  with bad gradient (0 for target
  function, 1 or higher for nonlinear constraints)
* badgradvidx - index of the variable
* badgradxbase - location where Jacobian is tested
* following  matrices  store  user-supplied  Jacobian  and  its  numerical
  differentiation version (which is assumed to be  free  from  the  coding
  errors), both of them computed near the initial point:
  * badgraduser, an array[K,N], analytic Jacobian supplied by user
  * badgradnum,  an array[K,N], numeric  Jacobian computed by ALGLIB
  Here K is a total number of  nonlinear  functions  (target  +  nonlinear
  constraints), N is a variable number.
  The  element  of  badgraduser[] with index [badgradfidx,badgradvidx]  is
  assumed to be wrong.

More detailed error log can  be  obtained  from  optimizer  by  explicitly
requesting reports for tests C0.0, C1.0, C1.1.

  -- ALGLIB --
     Copyright 19.11.2018 by Bochkanov Sergey
*************************************************************************/
class optguardreport
{
public:
    optguardreport();
    optguardreport(const optguardreport &rhs);
    optguardreport& operator=(const optguardreport &rhs);
    virtual ~optguardreport();
    bool nonc0suspected;
    bool nonc0test0positive;
    ae_int_t nonc0fidx;
    double nonc0lipschitzc;
    bool nonc1suspected;
    bool nonc1test0positive;
    bool nonc1test1positive;
    ae_int_t nonc1fidx;
    double nonc1lipschitzc;
    bool badgradsuspected;
    ae_int_t badgradfidx;
    ae_int_t badgradvidx;
    real_1d_array badgradxbase;
    real_2d_array badgraduser;
    real_2d_array badgradnum;
};

`opts` subpackage

Classes

lptestproblem
qpxproblem

Functions

lptestproblemcreate
lptestproblemgetm
lptestproblemgetn
lptestproblemgettargetf
lptestproblemhasknowntarget
lptestproblemserialize
lptestproblemsetbc
lptestproblemsetcost
lptestproblemsetlc2
lptestproblemsetscale
lptestproblemunserialize
qpxproblemaddqc2
qpxproblemcreate
qpxproblemgetbc
qpxproblemgetinitialpoint
qpxproblemgetlc2
qpxproblemgetlinearterm
qpxproblemgetmcc
qpxproblemgetmlc
qpxproblemgetmqc
qpxproblemgetn
qpxproblemgetorigin
qpxproblemgetqc2i
qpxproblemgetquadraticterm
qpxproblemgetscale
qpxproblemgettotalconstraints
qpxproblemhasinitialpoint
qpxproblemhasorigin
qpxproblemhasquadraticterm
qpxproblemhasscale
qpxproblemisquadraticobjective
qpxproblemsetbc
qpxproblemsetinitialpoint
qpxproblemsetlc2
qpxproblemsetlinearterm
qpxproblemsetorigin
qpxproblemsetquadraticterm
qpxproblemsetscale
xdbgminlpcreatefromtestproblem

Examples

`lptestproblem` class

/*************************************************************************
This is a test problem class  intended  for  internal  performance  tests.
Never use it directly in your projects.
*************************************************************************/
class lptestproblem
{
public:
    lptestproblem();
    lptestproblem(const lptestproblem &rhs);
    lptestproblem& operator=(const lptestproblem &rhs);
    virtual ~lptestproblem();
};

`qpxproblem` class

/*************************************************************************
A general QP problem (a linear/quadratic target subject to a mix  of  box,
linear, quadratic and conic constraints).

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
class qpxproblem
{
public:
    qpxproblem();
    qpxproblem(const qpxproblem &rhs);
    qpxproblem& operator=(const qpxproblem &rhs);
    virtual ~qpxproblem();
};

`lptestproblemcreate` function

/*************************************************************************
Initialize test LP problem.

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
void lptestproblemcreate(const ae_int_t n, const bool hasknowntarget, const double targetf, lptestproblem &p, const xparams _xparams = alglib::xdefault);

`lptestproblemgetm` function

/*************************************************************************
Query test problem info

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
ae_int_t lptestproblemgetm(lptestproblem &p, const xparams _xparams = alglib::xdefault);

`lptestproblemgetn` function

/*************************************************************************
Query test problem info

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
ae_int_t lptestproblemgetn(lptestproblem &p, const xparams _xparams = alglib::xdefault);

`lptestproblemgettargetf` function

/*************************************************************************
Query test problem info

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
double lptestproblemgettargetf(lptestproblem &p, const xparams _xparams = alglib::xdefault);

`lptestproblemhasknowntarget` function

/*************************************************************************
Query test problem info

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
bool lptestproblemhasknowntarget(lptestproblem &p, const xparams _xparams = alglib::xdefault);

`lptestproblemserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void lptestproblemserialize(const lptestproblem &obj, std::string &s_out);
void lptestproblemserialize(const lptestproblem &obj, std::ostream &s_out);

`lptestproblemsetbc` function

/*************************************************************************
Set box constraints for test LP problem

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
void lptestproblemsetbc(lptestproblem &p, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`lptestproblemsetcost` function

/*************************************************************************
Set cost for test LP problem

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
void lptestproblemsetcost(lptestproblem &p, const real_1d_array &c, const xparams _xparams = alglib::xdefault);

`lptestproblemsetlc2` function

/*************************************************************************
Set box constraints for test LP problem

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
void lptestproblemsetlc2(lptestproblem &p, const sparsematrix &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t m, const xparams _xparams = alglib::xdefault);

`lptestproblemsetscale` function

/*************************************************************************
Set scale for test LP problem

This function is intended for internal use by ALGLIB.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
void lptestproblemsetscale(lptestproblem &p, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`lptestproblemunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void lptestproblemunserialize(const std::string &s_in, lptestproblem &obj);
void lptestproblemunserialize(const std::istream &s_in, lptestproblem &obj);

`qpxproblemaddqc2` function

/*************************************************************************
Append two-sided quadratic constraint, same format as minqpaddqc2()

  -- ALGLIB --
     Copyright 19.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemaddqc2(qpxproblem &p, const sparsematrix &q, const bool isupper, const real_1d_array &b, const double cl, const double cu, const bool applyorigin, const xparams _xparams = alglib::xdefault);

`qpxproblemcreate` function

/*************************************************************************
Initialize QPX problem.

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemcreate(const ae_int_t n, qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemgetbc` function

/*************************************************************************
Get box constraints

  -- ALGLIB --
     Copyright 20.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetbc(qpxproblem &p, real_1d_array &bndl, real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`qpxproblemgetinitialpoint` function

/*************************************************************************
Get initial point

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetinitialpoint(qpxproblem &p, real_1d_array &x0, const xparams _xparams = alglib::xdefault);

`qpxproblemgetlc2` function

/*************************************************************************
Get linear constraints

  -- ALGLIB --
     Copyright 20.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetlc2(qpxproblem &p, sparsematrix &a, real_1d_array &al, real_1d_array &au, ae_int_t &m, const xparams _xparams = alglib::xdefault);

`qpxproblemgetlinearterm` function

/*************************************************************************
Get linear term

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetlinearterm(qpxproblem &p, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`qpxproblemgetmcc` function

/*************************************************************************
Get conic constraints count

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t qpxproblemgetmcc(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemgetmlc` function

/*************************************************************************
Get linear constraints count

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t qpxproblemgetmlc(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemgetmqc` function

/*************************************************************************
Get quadratic constraints count

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t qpxproblemgetmqc(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemgetn` function

/*************************************************************************
Get variables count

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t qpxproblemgetn(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemgetorigin` function

/*************************************************************************
Get origin

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetorigin(qpxproblem &p, real_1d_array &xorigin, const xparams _xparams = alglib::xdefault);

`qpxproblemgetqc2i` function

/*************************************************************************
Get IDX-th two-sided quadratic constraint, same format as minqpaddqc2(),
except for the fact that it always returns isUpper=False, even if the
original matrix was an upper triangular one.

NOTE: this function is not optimized for big matrices. Whilst still having
      O(max(N,Nonzeros)) running time, it may  be  somewhat  slow  due  to
      dynamic structures being used internally.


  -- ALGLIB --
     Copyright 19.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetqc2i(qpxproblem &p, const ae_int_t idx, sparsematrix &q, bool &isupper, real_1d_array &b, double &cl, double &cu, bool &applyorigin, const xparams _xparams = alglib::xdefault);

`qpxproblemgetquadraticterm` function

/*************************************************************************
Get quadratic term, returns zero matrix if no quadratic term is present

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetquadraticterm(qpxproblem &p, sparsematrix &q, bool &isupper, const xparams _xparams = alglib::xdefault);

`qpxproblemgetscale` function

/*************************************************************************
Get scale

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemgetscale(qpxproblem &p, real_1d_array &s, const xparams _xparams = alglib::xdefault);

`qpxproblemgettotalconstraints` function

/*************************************************************************
Get total constraints count

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
ae_int_t qpxproblemgettotalconstraints(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemhasinitialpoint` function

/*************************************************************************
Get initial point presence

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
bool qpxproblemhasinitialpoint(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemhasorigin` function

/*************************************************************************
Get origin presence

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
bool qpxproblemhasorigin(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemhasquadraticterm` function

/*************************************************************************
Returns False if no quadratic term was specified, or quadratic term is
numerically zero.

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
bool qpxproblemhasquadraticterm(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemhasscale` function

/*************************************************************************
Get scale presence

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
bool qpxproblemhasscale(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemisquadraticobjective` function

/*************************************************************************
Returns objective type: True for zero/linear/constant.

Present version does not return False.

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
bool qpxproblemisquadraticobjective(qpxproblem &p, const xparams _xparams = alglib::xdefault);

`qpxproblemsetbc` function

/*************************************************************************
Set box constraints

  -- ALGLIB --
     Copyright 20.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetbc(qpxproblem &p, const real_1d_array &bndl, const real_1d_array &bndu, const xparams _xparams = alglib::xdefault);

`qpxproblemsetinitialpoint` function

/*************************************************************************
Set initial point

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetinitialpoint(qpxproblem &p, const real_1d_array &x0, const xparams _xparams = alglib::xdefault);

`qpxproblemsetlc2` function

/*************************************************************************
Set linear constraints

  -- ALGLIB --
     Copyright 20.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetlc2(qpxproblem &p, const sparsematrix &a, const real_1d_array &al, const real_1d_array &au, const ae_int_t m, const xparams _xparams = alglib::xdefault);

`qpxproblemsetlinearterm` function

/*************************************************************************
Set linear term

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetlinearterm(qpxproblem &p, const real_1d_array &c, const xparams _xparams = alglib::xdefault);

`qpxproblemsetorigin` function

/*************************************************************************
Set origin

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetorigin(qpxproblem &p, const real_1d_array &xorigin, const xparams _xparams = alglib::xdefault);

`qpxproblemsetquadraticterm` function

/*************************************************************************
Set quadratic term; Q can be in any sparse matrix format.

Only one triangle (lower or upper) is referenced by this function.

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetquadraticterm(qpxproblem &p, const sparsematrix &q, const bool isupper, const xparams _xparams = alglib::xdefault);

`qpxproblemsetscale` function

/*************************************************************************
Set scale

  -- ALGLIB --
     Copyright 25.08.2024 by Bochkanov Sergey
*************************************************************************/
void qpxproblemsetscale(qpxproblem &p, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`xdbgminlpcreatefromtestproblem` function

/*************************************************************************
This is internal function intended to  be  used  only  by  ALGLIB  itself.
Although for technical reasons it is made publicly available (and has  its
own manual entry), you should never call it.

  -- ALGLIB --
     Copyright 11.01.2011 by Bochkanov Sergey
*************************************************************************/
void xdbgminlpcreatefromtestproblem(const lptestproblem &p, minlpstate &state, const xparams _xparams = alglib::xdefault);

`ortfac` subpackage

Functions

cmatrixlq
cmatrixlqunpackl
cmatrixlqunpackq
cmatrixqr
cmatrixqrunpackq
cmatrixqrunpackr
hmatrixtd
hmatrixtdunpackq
rmatrixbd
rmatrixbdmultiplybyp
rmatrixbdmultiplybyq
rmatrixbdunpackdiagonals
rmatrixbdunpackpt
rmatrixbdunpackq
rmatrixhessenberg
rmatrixhessenbergunpackh
rmatrixhessenbergunpackq
rmatrixlq
rmatrixlqunpackl
rmatrixlqunpackq
rmatrixqr
rmatrixqrunpackq
rmatrixqrunpackr
smatrixtd
smatrixtdunpackq

Examples

`cmatrixlq` function

/*************************************************************************
LQ decomposition of a rectangular complex matrix of size MxN

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1]
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices Q and L in compact form
    Tau -   array of scalar factors which are used to form matrix Q. Array
            whose indexes range within [0.. Min(M,N)-1]

Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size MxN.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
void cmatrixlq(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau, const xparams _xparams = alglib::xdefault);

`cmatrixlqunpackl` function

/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A

Input parameters:
    A       -   matrices Q and L in compact form.
                Output of CMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    L       -   matrix L, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixlqunpackl(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_2d_array &l, const xparams _xparams = alglib::xdefault);

`cmatrixlqunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from LQ decomposition of a complex matrix A.

Input parameters:
    A           -   matrices Q and R in compact form.
                    Output of CMatrixLQ subroutine .
    M           -   number of rows in matrix A. M>=0.
    N           -   number of columns in matrix A. N>=0.
    Tau         -   scalar factors which are used to form Q.
                    Output of CMatrixLQ subroutine .
    QRows       -   required number of rows in matrix Q. N>=QColumns>=0.

Output parameters:
    Q           -   first QRows rows of matrix Q.
                    Array whose index ranges within [0..QRows-1, 0..N-1].
                    If QRows=0, array isn't changed.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixlqunpackq(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, const complex_1d_array &tau, const ae_int_t qrows, complex_2d_array &q, const xparams _xparams = alglib::xdefault);

`cmatrixqr` function

/*************************************************************************
QR decomposition of a rectangular complex matrix of size MxN

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1]
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices Q and R in compact form
    Tau -   array of scalar factors which are used to form matrix Q. Array
            whose indexes range within [0.. Min(M,N)-1]

Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size MxN.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994
*************************************************************************/
void cmatrixqr(complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_1d_array &tau, const xparams _xparams = alglib::xdefault);

`cmatrixqrunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from QR decomposition of a complex matrix A.

Input parameters:
    A           -   matrices Q and R in compact form.
                    Output of CMatrixQR subroutine .
    M           -   number of rows in matrix A. M>=0.
    N           -   number of columns in matrix A. N>=0.
    Tau         -   scalar factors which are used to form Q.
                    Output of CMatrixQR subroutine .
    QColumns    -   required number of columns in matrix Q. M>=QColumns>=0.

Output parameters:
    Q           -   first QColumns columns of matrix Q.
                    Array whose index ranges within [0..M-1, 0..QColumns-1].
                    If QColumns=0, array isn't changed.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixqrunpackq(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, const complex_1d_array &tau, const ae_int_t qcolumns, complex_2d_array &q, const xparams _xparams = alglib::xdefault);

`cmatrixqrunpackr` function

/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A

Input parameters:
    A       -   matrices Q and R in compact form.
                Output of CMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    R       -   matrix R, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixqrunpackr(const complex_2d_array &a, const ae_int_t m, const ae_int_t n, complex_2d_array &r, const xparams _xparams = alglib::xdefault);

`hmatrixtd` function

/*************************************************************************
Reduction of a Hermitian matrix which is given  by  its  higher  or  lower
triangular part to a real  tridiagonal  matrix  using  unitary  similarity
transformation: Q'*A*Q = T.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix to be transformed
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format. If IsUpper = True, then matrix A is  given
                by its upper triangle, and the lower triangle is not  used
                and not modified by the algorithm, and vice versa
                if IsUpper = False.

Output parameters:
    A       -   matrices T and Q in  compact form (see lower)
    Tau     -   array of factors which are forming matrices H(i)
                array with elements [0..N-2].
    D       -   main diagonal of real symmetric matrix T.
                array with elements [0..N-1].
    E       -   secondary diagonal of real symmetric matrix T.
                array with elements [0..N-2].


  If IsUpper=True, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-2) . . . H(2) H(0).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a complex scalar, and v is a complex vector with
  v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
  A(0:i-1,i+1), and tau in TAU(i).

  If IsUpper=False, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(0) H(2) . . . H(n-2).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a complex scalar, and v is a complex vector with
  v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v1  v2  v3 )              (  d                  )
    (      d   e   v2  v3 )              (  e   d              )
    (          d   e   v3 )              (  v0  e   d          )
    (              d   e  )              (  v0  v1  e   d      )
    (                  d  )              (  v0  v1  v2  e   d  )

where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1992
*************************************************************************/
void hmatrixtd(complex_2d_array &a, const ae_int_t n, const bool isupper, complex_1d_array &tau, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);

`hmatrixtdunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces a Hermitian matrix to a real  tridiagonal
form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   the result of a HMatrixTD subroutine
    N       -   size of matrix A.
    IsUpper -   storage format (a parameter of HMatrixTD subroutine)
    Tau     -   the result of a HMatrixTD subroutine

Output parameters:
    Q       -   transformation matrix.
                array with elements [0..N-1, 0..N-1].

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void hmatrixtdunpackq(const complex_2d_array &a, const ae_int_t n, const bool isupper, const complex_1d_array &tau, complex_2d_array &q, const xparams _xparams = alglib::xdefault);

`rmatrixbd` function

/*************************************************************************
Reduction of a rectangular matrix to  bidiagonal form

The algorithm reduces the rectangular matrix A to  bidiagonal form by
orthogonal transformations P and Q: A = Q*B*(P^T).

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   source matrix. array[0..M-1, 0..N-1]
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.

Output parameters:
    A       -   matrices Q, B, P in compact form (see below).
    TauQ    -   scalar factors which are used to form matrix Q.
    TauP    -   scalar factors which are used to form matrix P.

The main diagonal and one of the  secondary  diagonals  of  matrix  A  are
replaced with bidiagonal  matrix  B.  Other  elements  contain  elementary
reflections which form MxM matrix Q and NxN matrix P, respectively.

If M>=N, B is the upper  bidiagonal  MxN  matrix  and  is  stored  in  the
corresponding  elements  of  matrix  A.  Matrix  Q  is  represented  as  a
product   of   elementary   reflections   Q = H(0)*H(1)*...*H(n-1),  where
H(i) = 1-tau*v*v'. Here tau is a scalar which is stored  in  TauQ[i],  and
vector v has the following  structure:  v(0:i-1)=0, v(i)=1, v(i+1:m-1)  is
stored   in   elements   A(i+1:m-1,i).   Matrix   P  is  as  follows:  P =
G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i],
u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1).

If M<N, B is the  lower  bidiagonal  MxN  matrix  and  is  stored  in  the
corresponding   elements  of  matrix  A.  Q = H(0)*H(1)*...*H(m-2),  where
H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1)
is    stored    in   elements   A(i+2:m-1,i).    P = G(0)*G(1)*...*G(m-1),
G(i) = 1-tau*u*u', tau is stored in  TauP,  u(0:i-1)=0, u(i)=1, u(i+1:n-1)
is stored in A(i,i+1:n-1).

EXAMPLE:

m=6, n=5 (m > n):               m=5, n=6 (m < n):

(  d   e   u1  u1  u1 )         (  d   u1  u1  u1  u1  u1 )
(  v1  d   e   u2  u2 )         (  e   d   u2  u2  u2  u2 )
(  v1  v2  d   e   u3 )         (  v1  e   d   u3  u3  u3 )
(  v1  v2  v3  d   e  )         (  v1  v2  e   d   u4  u4 )
(  v1  v2  v3  v4  d  )         (  v1  v2  v3  e   d   u5 )
(  v1  v2  v3  v4  v5 )

Here vi and ui are vectors which form H(i) and G(i), and d and e -
are the diagonal and off-diagonal elements of matrix B.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     September 30, 1994.
     Sergey Bochkanov, ALGLIB project, translation from FORTRAN to
     pseudocode, 2007-2010.
*************************************************************************/
void rmatrixbd(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tauq, real_1d_array &taup, const xparams _xparams = alglib::xdefault);

`rmatrixbdmultiplybyp` function

/*************************************************************************
Multiplication by matrix P which reduces matrix A to  bidiagonal form.

The algorithm allows pre- or post-multiply by P or P'.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of RMatrixBD subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUP        -   scalar factors which are used to form P.
                    Output of RMatrixBD subroutine.
    Z           -   multiplied matrix.
                    Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
    ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                    ZRows=N, otherwise ZRows can be arbitrary.
    ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                    ZColumns=N, otherwise ZColumns can be arbitrary.
    FromTheRight -  pre- or post-multiply.
    DoTranspose -   multiply by P or P'.

Output parameters:
    Z - product of Z and P.
                Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
                If ZRows=0 or ZColumns=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyp(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &taup, real_2d_array &z, const ae_int_t zrows, const ae_int_t zcolumns, const bool fromtheright, const bool dotranspose, const xparams _xparams = alglib::xdefault);

`rmatrixbdmultiplybyq` function

/*************************************************************************
Multiplication by matrix Q which reduces matrix A to  bidiagonal form.

The algorithm allows pre- or post-multiply by Q or Q'.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUQ        -   scalar factors which are used to form Q.
                    Output of ToBidiagonal subroutine.
    Z           -   multiplied matrix.
                    array[0..ZRows-1,0..ZColumns-1]
    ZRows       -   number of rows in matrix Z. If FromTheRight=False,
                    ZRows=M, otherwise ZRows can be arbitrary.
    ZColumns    -   number of columns in matrix Z. If FromTheRight=True,
                    ZColumns=M, otherwise ZColumns can be arbitrary.
    FromTheRight -  pre- or post-multiply.
    DoTranspose -   multiply by Q or Q'.

Output parameters:
    Z           -   product of Z and Q.
                    Array[0..ZRows-1,0..ZColumns-1]
                    If ZRows=0 or ZColumns=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyq(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &tauq, real_2d_array &z, const ae_int_t zrows, const ae_int_t zcolumns, const bool fromtheright, const bool dotranspose, const xparams _xparams = alglib::xdefault);

`rmatrixbdunpackdiagonals` function

/*************************************************************************
Unpacking of the main and secondary diagonals of bidiagonal decomposition
of matrix A.

Input parameters:
    B   -   output of RMatrixBD subroutine.
    M   -   number of rows in matrix B.
    N   -   number of columns in matrix B.

Output parameters:
    IsUpper -   True, if the matrix is upper bidiagonal.
                otherwise IsUpper is False.
    D       -   the main diagonal.
                Array whose index ranges within [0..Min(M,N)-1].
    E       -   the secondary diagonal (upper or lower, depending on
                the value of IsUpper).
                Array index ranges within [0..Min(M,N)-1], the last
                element is not used.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackdiagonals(const real_2d_array &b, const ae_int_t m, const ae_int_t n, bool &isupper, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);

`rmatrixbdunpackpt` function

/*************************************************************************
Unpacking matrix P which reduces matrix A to bidiagonal form.
The subroutine returns transposed matrix P.

Input parameters:
    QP      -   matrices Q and P in compact form.
                Output of ToBidiagonal subroutine.
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.
    TAUP    -   scalar factors which are used to form P.
                Output of ToBidiagonal subroutine.
    PTRows  -   required number of rows of matrix P^T. N >= PTRows >= 0.

Output parameters:
    PT      -   first PTRows columns of matrix P^T
                Array[0..PTRows-1, 0..N-1]
                If PTRows=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackpt(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &taup, const ae_int_t ptrows, real_2d_array &pt, const xparams _xparams = alglib::xdefault);

`rmatrixbdunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    QP          -   matrices Q and P in compact form.
                    Output of ToBidiagonal subroutine.
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    TAUQ        -   scalar factors which are used to form Q.
                    Output of ToBidiagonal subroutine.
    QColumns    -   required number of columns in matrix Q.
                    M>=QColumns>=0.

Output parameters:
    Q           -   first QColumns columns of matrix Q.
                    Array[0..M-1, 0..QColumns-1]
                    If QColumns=0, the array is not modified.

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackq(const real_2d_array &qp, const ae_int_t m, const ae_int_t n, const real_1d_array &tauq, const ae_int_t qcolumns, real_2d_array &q, const xparams _xparams = alglib::xdefault);

`rmatrixhessenberg` function

/*************************************************************************
Reduction of a square matrix to  upper Hessenberg form: Q'*A*Q = H,
where Q is an orthogonal matrix, H - Hessenberg matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix A with elements [0..N-1, 0..N-1]
    N       -   size of matrix A.

Output parameters:
    A       -   matrices Q and P in  compact form (see below).
    Tau     -   array of scalar factors which are used to form matrix Q.
                Array whose index ranges within [0..N-2]

Matrix H is located on the main diagonal, on the lower secondary  diagonal
and above the main diagonal of matrix A. The elements which are used to
form matrix Q are situated in array Tau and below the lower secondary
diagonal of matrix A as follows:

Matrix Q is represented as a product of elementary reflections

Q = H(0)*H(2)*...*H(n-2),

where each H(i) is given by

H(i) = 1 - tau * v * (v^T)

where tau is a scalar stored in Tau[I]; v - is a real vector,
so that v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) stored in A(i+2:n-1,i).

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1992
*************************************************************************/
void rmatrixhessenberg(real_2d_array &a, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);

`rmatrixhessenbergunpackh` function

/*************************************************************************
Unpacking matrix H (the result of matrix A reduction to upper Hessenberg form)

Input parameters:
    A   -   output of RMatrixHessenberg subroutine.
    N   -   size of matrix A.

Output parameters:
    H   -   matrix H. Array whose indexes range within [0..N-1, 0..N-1].

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixhessenbergunpackh(const real_2d_array &a, const ae_int_t n, real_2d_array &h, const xparams _xparams = alglib::xdefault);

`rmatrixhessenbergunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces matrix A to upper Hessenberg form

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A   -   output of RMatrixHessenberg subroutine.
    N   -   size of matrix A.
    Tau -   scalar factors which are used to form Q.
            Output of RMatrixHessenberg subroutine.

Output parameters:
    Q   -   matrix Q.
            Array whose indexes range within [0..N-1, 0..N-1].

  -- ALGLIB --
     2005-2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixhessenbergunpackq(const real_2d_array &a, const ae_int_t n, const real_1d_array &tau, real_2d_array &q, const xparams _xparams = alglib::xdefault);

`rmatrixlq` function

/*************************************************************************
LQ decomposition of a rectangular matrix of size MxN

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices L and Q in compact form (see below)
    Tau -   array of scalar factors which are used to form
            matrix Q. Array whose index ranges within [0..Min(M,N)-1].

Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size M x N.

The elements of matrix L are located on and below  the  main  diagonal  of
matrix A. The elements which are located in Tau array and above  the  main
diagonal of matrix A are used to form matrix Q as follows:

Matrix Q is represented as a product of elementary reflections

Q = H(k-1)*H(k-2)*...*H(1)*H(0),

where k = min(m,n), and each H(i) is of the form

H(i) = 1 - tau * v * (v^T)

where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1)=0,
v(i) = 1, v(i+1:n-1) stored in A(i,i+1:n-1).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlq(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);

`rmatrixlqunpackl` function

/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A

Input parameters:
    A       -   matrices Q and L in compact form.
                Output of RMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    L       -   matrix L, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlqunpackl(const real_2d_array &a, const ae_int_t m, const ae_int_t n, real_2d_array &l, const xparams _xparams = alglib::xdefault);

`rmatrixlqunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from the LQ decomposition of a matrix A

Input parameters:
    A       -   matrices L and Q in compact form.
                Output of RMatrixLQ subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.
    Tau     -   scalar factors which are used to form Q.
                Output of the RMatrixLQ subroutine.
    QRows   -   required number of rows in matrix Q. N>=QRows>=0.

Output parameters:
    Q       -   first QRows rows of matrix Q. Array whose indexes range
                within [0..QRows-1, 0..N-1]. If QRows=0, the array remains
                unchanged.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlqunpackq(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const real_1d_array &tau, const ae_int_t qrows, real_2d_array &q, const xparams _xparams = alglib::xdefault);

`rmatrixqr` function

/*************************************************************************
QR decomposition of a rectangular matrix of size MxN

Input parameters:
    A   -   matrix A whose indexes range within [0..M-1, 0..N-1].
    M   -   number of rows in matrix A.
    N   -   number of columns in matrix A.

Output parameters:
    A   -   matrices Q and R in compact form (see below).
    Tau -   array of scalar factors which are used to form
            matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)].

Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.

The elements of matrix R are located on and above the main diagonal of
matrix A. The elements which are located in Tau array and below the main
diagonal of matrix A are used to form matrix Q as follows:

Matrix Q is represented as a product of elementary reflections

Q = H(0)*H(2)*...*H(k-1),

where k = min(m,n), and each H(i) is in the form

H(i) = 1 - tau * v * (v^T)

where tau is a scalar stored in Tau[I]; v - real vector,
so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i).

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixqr(real_2d_array &a, const ae_int_t m, const ae_int_t n, real_1d_array &tau, const xparams _xparams = alglib::xdefault);

`rmatrixqrunpackq` function

/*************************************************************************
Partial unpacking of matrix Q from the QR decomposition of a matrix A

Input parameters:
    A       -   matrices Q and R in compact form.
                Output of RMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.
    Tau     -   scalar factors which are used to form Q.
                Output of the RMatrixQR subroutine.
    QColumns -  required number of columns of matrix Q. M>=QColumns>=0.

Output parameters:
    Q       -   first QColumns columns of matrix Q.
                Array whose indexes range within [0..M-1, 0..QColumns-1].
                If QColumns=0, the array remains unchanged.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixqrunpackq(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const real_1d_array &tau, const ae_int_t qcolumns, real_2d_array &q, const xparams _xparams = alglib::xdefault);

`rmatrixqrunpackr` function

/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A

Input parameters:
    A       -   matrices Q and R in compact form.
                Output of RMatrixQR subroutine.
    M       -   number of rows in given matrix A. M>=0.
    N       -   number of columns in given matrix A. N>=0.

Output parameters:
    R       -   matrix R, array[0..M-1, 0..N-1].

  -- ALGLIB routine --
     17.02.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixqrunpackr(const real_2d_array &a, const ae_int_t m, const ae_int_t n, real_2d_array &r, const xparams _xparams = alglib::xdefault);

`smatrixtd` function

/*************************************************************************
Reduction of a symmetric matrix which is given by its higher or lower
triangular part to a tridiagonal matrix using orthogonal similarity
transformation: Q'*A*Q=T.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   matrix to be transformed
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format. If IsUpper = True, then matrix A is given
                by its upper triangle, and the lower triangle is not used
                and not modified by the algorithm, and vice versa
                if IsUpper = False.

Output parameters:
    A       -   matrices T and Q in  compact form (see lower)
    Tau     -   array of factors which are forming matrices H(i)
                array with elements [0..N-2].
    D       -   main diagonal of symmetric matrix T.
                array with elements [0..N-1].
    E       -   secondary diagonal of symmetric matrix T.
                array with elements [0..N-2].


  If IsUpper=True, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n-2) . . . H(2) H(0).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
  A(0:i-1,i+1), and tau in TAU(i).

  If IsUpper=False, the matrix Q is represented as a product of elementary
  reflectors

     Q = H(0) H(2) . . . H(n-2).

  Each H(i) has the form

     H(i) = I - tau * v * v'

  where tau is a real scalar, and v is a real vector with
  v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
  and tau in TAU(i).

  The contents of A on exit are illustrated by the following examples
  with n = 5:

  if UPLO = 'U':                       if UPLO = 'L':

    (  d   e   v1  v2  v3 )              (  d                  )
    (      d   e   v2  v3 )              (  e   d              )
    (          d   e   v3 )              (  v0  e   d          )
    (              d   e  )              (  v0  v1  e   d      )
    (                  d  )              (  v0  v1  v2  e   d  )

  where d and e denote diagonal and off-diagonal elements of T, and vi
  denotes an element of the vector defining H(i).

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1992
*************************************************************************/
void smatrixtd(real_2d_array &a, const ae_int_t n, const bool isupper, real_1d_array &tau, real_1d_array &d, real_1d_array &e, const xparams _xparams = alglib::xdefault);

`smatrixtdunpackq` function

/*************************************************************************
Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
form.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

Input parameters:
    A       -   the result of a SMatrixTD subroutine
    N       -   size of matrix A.
    IsUpper -   storage format (a parameter of SMatrixTD subroutine)
    Tau     -   the result of a SMatrixTD subroutine

Output parameters:
    Q       -   transformation matrix.
                array with elements [0..N-1, 0..N-1].

  -- ALGLIB --
     Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void smatrixtdunpackq(const real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &tau, real_2d_array &q, const xparams _xparams = alglib::xdefault);

`parametric` subpackage

Classes

pspline2interpolant
pspline3interpolant

Functions

parametricrdpfixed
pspline2arclength
pspline2build
pspline2buildperiodic
pspline2calc
pspline2diff
pspline2diff2
pspline2parametervalues
pspline2tangent
pspline3arclength
pspline3build
pspline3buildperiodic
pspline3calc
pspline3diff
pspline3diff2
pspline3parametervalues
pspline3tangent

Examples

parametric_rdp

Parametric Ramer-Douglas-Peucker approximation

`pspline2interpolant` class

/*************************************************************************
Parametric spline inteprolant: 2-dimensional curve.

You should not try to access its members directly - use PSpline2XXXXXXXX()
functions instead.
*************************************************************************/
class pspline2interpolant
{
public:
    pspline2interpolant();
    pspline2interpolant(const pspline2interpolant &rhs);
    pspline2interpolant& operator=(const pspline2interpolant &rhs);
    virtual ~pspline2interpolant();
};

`pspline3interpolant` class

/*************************************************************************
Parametric spline inteprolant: 3-dimensional curve.

You should not try to access its members directly - use PSpline3XXXXXXXX()
functions instead.
*************************************************************************/
class pspline3interpolant
{
public:
    pspline3interpolant();
    pspline3interpolant(const pspline3interpolant &rhs);
    pspline3interpolant& operator=(const pspline3interpolant &rhs);
    virtual ~pspline3interpolant();
};

`parametricrdpfixed` function

/*************************************************************************
This  subroutine fits piecewise linear curve to points with Ramer-Douglas-
Peucker algorithm. This  function  performs PARAMETRIC fit, i.e. it can be
used to fit curves like circles.

On  input  it  accepts dataset which describes parametric multidimensional
curve X(t), with X being vector, and t taking values in [0,N), where N  is
a number of points in dataset. As result, it returns reduced  dataset  X2,
which can be used to build  parametric  curve  X2(t),  which  approximates
X(t) with desired precision (or has specified number of sections).


INPUT PARAMETERS:
    X       -   array of multidimensional points:
                * at least N elements, leading N elements are used if more
                  than N elements were specified
                * order of points is IMPORTANT because  it  is  parametric
                  fit
                * each row of array is one point which has D coordinates
    N       -   number of elements in X
    D       -   number of dimensions (elements per row of X)
    StopM   -   stopping condition - desired number of sections:
                * at most M sections are generated by this function
                * less than M sections can be generated if we have N<M
                  (or some X are non-distinct).
                * zero StopM means that algorithm does not stop after
                  achieving some pre-specified section count
    StopEps -   stopping condition - desired precision:
                * algorithm stops after error in each section is at most Eps
                * zero Eps means that algorithm does not stop after
                  achieving some pre-specified precision

OUTPUT PARAMETERS:
    X2      -   array of corner points for piecewise approximation,
                has length NSections+1 or zero (for NSections=0).
    Idx2    -   array of indexes (parameter values):
                * has length NSections+1 or zero (for NSections=0).
                * each element of Idx2 corresponds to same-numbered
                  element of X2
                * each element of Idx2 is index of  corresponding  element
                  of X2 at original array X, i.e. I-th  row  of  X2  is
                  Idx2[I]-th row of X.
                * elements of Idx2 can be treated as parameter values
                  which should be used when building new parametric curve
                * Idx2[0]=0, Idx2[NSections]=N-1
    NSections-  number of sections found by algorithm, NSections<=M,
                NSections can be zero for degenerate datasets
                (N<=1 or all X[] are non-distinct).

NOTE: algorithm stops after:
      a) dividing curve into StopM sections
      b) achieving required precision StopEps
      c) dividing curve into N-1 sections
      If both StopM and StopEps are non-zero, algorithm is stopped by  the
      FIRST criterion which is satisfied. In case both StopM  and  StopEps
      are zero, algorithm stops because of (c).

  -- ALGLIB --
     Copyright 02.10.2014 by Bochkanov Sergey
*************************************************************************/
void parametricrdpfixed(const real_2d_array &x, const ae_int_t n, const ae_int_t d, const ae_int_t stopm, const double stopeps, real_2d_array &x2, integer_1d_array &idx2, ae_int_t &nsections, const xparams _xparams = alglib::xdefault);

Examples: [1]

`pspline2arclength` function

/*************************************************************************
This function  calculates  arc length, i.e. length of  curve  between  t=a
and t=b.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    A,B -   parameter values corresponding to arc ends:
            * B>A will result in positive length returned
            * B<A will result in negative length returned

RESULT:
    length of arc starting at T=A and ending at T=B.


  -- ALGLIB PROJECT --
     Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double pspline2arclength(const pspline2interpolant &p, const double a, const double b, const xparams _xparams = alglib::xdefault);

`pspline2build` function

/*************************************************************************
This function  builds  non-periodic 2-dimensional parametric spline  which
starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]).

INPUT PARAMETERS:
    XY  -   points, array[0..N-1,0..1].
            XY[I,0:1] corresponds to the Ith point.
            Order of points is important!
    N   -   points count, N>=5 for Akima splines, N>=2 for other types  of
            splines.
    ST  -   spline type:
            * 0     Akima spline
            * 1     parabolically terminated Catmull-Rom spline (Tension=0)
            * 2     parabolically terminated cubic spline
    PT  -   parameterization type:
            * 0     uniform
            * 1     chord length
            * 2     centripetal

OUTPUT PARAMETERS:
    P   -   parametric spline interpolant


NOTES:
* this function  assumes  that  there all consequent points  are distinct.
  I.e. (x0,y0)<>(x1,y1),  (x1,y1)<>(x2,y2),  (x2,y2)<>(x3,y3)  and  so on.
  However, non-consequent points may coincide, i.e. we can  have  (x0,y0)=
  =(x2,y2).

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2build(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline2interpolant &p, const xparams _xparams = alglib::xdefault);

`pspline2buildperiodic` function

/*************************************************************************
This  function  builds  periodic  2-dimensional  parametric  spline  which
starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then
back to (X[0],Y[0]).

INPUT PARAMETERS:
    XY  -   points, array[0..N-1,0..1].
            XY[I,0:1] corresponds to the Ith point.
            XY[N-1,0:1] must be different from XY[0,0:1].
            Order of points is important!
    N   -   points count, N>=3 for other types of splines.
    ST  -   spline type:
            * 1     Catmull-Rom spline (Tension=0) with cyclic boundary conditions
            * 2     cubic spline with cyclic boundary conditions
    PT  -   parameterization type:
            * 0     uniform
            * 1     chord length
            * 2     centripetal

OUTPUT PARAMETERS:
    P   -   parametric spline interpolant


NOTES:
* this function  assumes  that there all consequent points  are  distinct.
  I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2),  (x2,y2)<>(x3,y3)  and  so  on.
  However, non-consequent points may coincide, i.e. we can  have  (x0,y0)=
  =(x2,y2).
* last point of sequence is NOT equal to the first  point.  You  shouldn't
  make curve "explicitly periodic" by making them equal.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2buildperiodic(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline2interpolant &p, const xparams _xparams = alglib::xdefault);

`pspline2calc` function

/*************************************************************************
This function  calculates  the value of the parametric spline for a  given
value of parameter T

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-position
    Y   -   Y-position


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2calc(const pspline2interpolant &p, const double t, double &x, double &y, const xparams _xparams = alglib::xdefault);

`pspline2diff` function

/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT).

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   X-derivative
    Y   -   Y-value
    DY  -   Y-derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2diff(const pspline2interpolant &p, const double t, double &x, double &dx, double &y, double &dy, const xparams _xparams = alglib::xdefault);

`pspline2diff2` function

/*************************************************************************
This function calculates first and second derivative with respect to T.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   derivative
    D2X -   second derivative
    Y   -   Y-value
    DY  -   derivative
    D2Y -   second derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2diff2(const pspline2interpolant &p, const double t, double &x, double &dx, double &d2x, double &y, double &dy, double &d2y, const xparams _xparams = alglib::xdefault);

`pspline2parametervalues` function

/*************************************************************************
This function returns vector of parameter values correspoding to points.

I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P)  we
have
    (X[0],Y[0]) = PSpline2Calc(P,U[0]),
    (X[1],Y[1]) = PSpline2Calc(P,U[1]),
    (X[2],Y[2]) = PSpline2Calc(P,U[2]),
    ...

INPUT PARAMETERS:
    P   -   parametric spline interpolant

OUTPUT PARAMETERS:
    N   -   array size
    T   -   array[0..N-1]


NOTES:
* for non-periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]=1
* for periodic splines     U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]<1

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2parametervalues(const pspline2interpolant &p, ae_int_t &n, real_1d_array &t, const xparams _xparams = alglib::xdefault);

`pspline2tangent` function

/*************************************************************************
This function  calculates  tangent vector for a given value of parameter T

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X    -   X-component of tangent vector (normalized)
    Y    -   Y-component of tangent vector (normalized)

NOTE:
    X^2+Y^2 is either 1 (for non-zero tangent vector) or 0.


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline2tangent(const pspline2interpolant &p, const double t, double &x, double &y, const xparams _xparams = alglib::xdefault);

`pspline3arclength` function

/*************************************************************************
This function  calculates  arc length, i.e. length of  curve  between  t=a
and t=b.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    A,B -   parameter values corresponding to arc ends:
            * B>A will result in positive length returned
            * B<A will result in negative length returned

RESULT:
    length of arc starting at T=A and ending at T=B.


  -- ALGLIB PROJECT --
     Copyright 30.05.2010 by Bochkanov Sergey
*************************************************************************/
double pspline3arclength(const pspline3interpolant &p, const double a, const double b, const xparams _xparams = alglib::xdefault);

`pspline3build` function

/*************************************************************************
This function  builds  non-periodic 3-dimensional parametric spline  which
starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]).

Same as PSpline2Build() function, but for 3D, so we  won't  duplicate  its
description here.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3build(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline3interpolant &p, const xparams _xparams = alglib::xdefault);

`pspline3buildperiodic` function

/*************************************************************************
This  function  builds  periodic  3-dimensional  parametric  spline  which
starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1])
and then back to (X[0],Y[0],Z[0]).

Same as PSpline2Build() function, but for 3D, so we  won't  duplicate  its
description here.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3buildperiodic(const real_2d_array &xy, const ae_int_t n, const ae_int_t st, const ae_int_t pt, pspline3interpolant &p, const xparams _xparams = alglib::xdefault);

`pspline3calc` function

/*************************************************************************
This function  calculates  the value of the parametric spline for a  given
value of parameter T.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-position
    Y   -   Y-position
    Z   -   Z-position


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3calc(const pspline3interpolant &p, const double t, double &x, double &y, double &z, const xparams _xparams = alglib::xdefault);

`pspline3diff` function

/*************************************************************************
This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT).

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   X-derivative
    Y   -   Y-value
    DY  -   Y-derivative
    Z   -   Z-value
    DZ  -   Z-derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3diff(const pspline3interpolant &p, const double t, double &x, double &dx, double &y, double &dy, double &z, double &dz, const xparams _xparams = alglib::xdefault);

`pspline3diff2` function

/*************************************************************************
This function calculates first and second derivative with respect to T.

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X   -   X-value
    DX  -   derivative
    D2X -   second derivative
    Y   -   Y-value
    DY  -   derivative
    D2Y -   second derivative
    Z   -   Z-value
    DZ  -   derivative
    D2Z -   second derivative


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3diff2(const pspline3interpolant &p, const double t, double &x, double &dx, double &d2x, double &y, double &dy, double &d2y, double &z, double &dz, double &d2z, const xparams _xparams = alglib::xdefault);

`pspline3parametervalues` function

/*************************************************************************
This function returns vector of parameter values correspoding to points.

Same as PSpline2ParameterValues(), but for 3D.

  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3parametervalues(const pspline3interpolant &p, ae_int_t &n, real_1d_array &t, const xparams _xparams = alglib::xdefault);

`pspline3tangent` function

/*************************************************************************
This function  calculates  tangent vector for a given value of parameter T

INPUT PARAMETERS:
    P   -   parametric spline interpolant
    T   -   point:
            * T in [0,1] corresponds to interval spanned by points
            * for non-periodic splines T<0 (or T>1) correspond to parts of
              the curve before the first (after the last) point
            * for periodic splines T<0 (or T>1) are projected  into  [0,1]
              by making T=T-floor(T).

OUTPUT PARAMETERS:
    X    -   X-component of tangent vector (normalized)
    Y    -   Y-component of tangent vector (normalized)
    Z    -   Z-component of tangent vector (normalized)

NOTE:
    X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0.


  -- ALGLIB PROJECT --
     Copyright 28.05.2010 by Bochkanov Sergey
*************************************************************************/
void pspline3tangent(const pspline3interpolant &p, const double t, double &x, double &y, double &z, const xparams _xparams = alglib::xdefault);

parametric_rdp example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use RDP algorithm to approximate parametric 2D curve given by
        // locations in t=0,1,2,3 (see below), which form piecewise linear
        // trajectory through D-dimensional space (2-dimensional in our example).
        // 
        //     |
        //     |
        //     -     *     *     X2................X3
        //     |                .
        //     |               .
        //     -     *     *  .  *     *     *     *
        //     |             .
        //     |            .
        //     -     *     X1    *     *     *     *
        //     |      .....
        //     |  ....
        //     X0----|-----|-----|-----|-----|-----|---
        //
        ae_int_t npoints = 4;
        ae_int_t ndimensions = 2;
        real_2d_array x = "[[0,0],[2,1],[3,3],[6,3]]";

        //
        // Approximation of parametric curve is performed by another parametric curve
        // with lesser amount of points. It allows to work with "compressed"
        // representation, which needs smaller amount of memory. Say, in our example
        // (we allow points with error smaller than 0.8) approximation will have
        // just two sequential sections connecting X0 with X2, and X2 with X3.
        // 
        //     |
        //     |
        //     -     *     *     X2................X3
        //     |               . 
        //     |             .  
        //     -     *     .     *     *     *     *
        //     |         .    
        //     |       .     
        //     -     .     X1    *     *     *     *
        //     |   .       
        //     | .    
        //     X0----|-----|-----|-----|-----|-----|---
        //
        //
        real_2d_array y;
        integer_1d_array idxy;
        ae_int_t nsections;
        ae_int_t limitcnt = 0;
        double limiteps = 0.8;
        parametricrdpfixed(x, npoints, ndimensions, limitcnt, limiteps, y, idxy, nsections);
        printf("%d\n", int(nsections)); // EXPECTED: 2
        printf("%s\n", idxy.tostring().c_str()); // EXPECTED: [0,2,3]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`pca` subpackage

Functions

pcabuildbasis
pcatruncatedsubspace
pcatruncatedsubspacesparse

Examples

`pcabuildbasis` function

/*************************************************************************
Principal components analysis

This function builds orthogonal basis  where  first  axis  corresponds  to
direction with maximum variance, second axis  maximizes  variance  in  the
subspace orthogonal to first axis and so on.

This function builds FULL basis, i.e. returns N vectors  corresponding  to
ALL directions, no matter how informative. If you need  just a  few  (say,
10 or 50) of the most important directions, you may find it faster to  use
one of the reduced versions:
* pcatruncatedsubspace() - for subspace iteration based method

It should be noted that, unlike LDA, PCA does not use class labels.

INPUT PARAMETERS:
    X           -   dataset, array[NPoints,NVars].
                    matrix contains ONLY INDEPENDENT VARIABLES.
    NPoints     -   dataset size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1

OUTPUT PARAMETERS:
    S2          -   array[NVars]. variance values corresponding
                    to basis vectors.
    V           -   array[NVars,NVars]
                    matrix, whose columns store basis vectors.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 25.08.2008 by Bochkanov Sergey
*************************************************************************/
void pcabuildbasis(const real_2d_array &x, const ae_int_t npoints, const ae_int_t nvars, real_1d_array &s2, real_2d_array &v, const xparams _xparams = alglib::xdefault);
void pcabuildbasis(const real_2d_array &x, real_1d_array &s2, real_2d_array &v, const xparams _xparams = alglib::xdefault);

`pcatruncatedsubspace` function

/*************************************************************************
Principal components analysis

This function performs truncated PCA, i.e. returns just a few most important
directions.

Internally it uses iterative eigensolver which is very efficient when only
a minor fraction of full basis is required. Thus, if you need full  basis,
it is better to use pcabuildbasis() function.

It should be noted that, unlike LDA, PCA does not use class labels.

INPUT PARAMETERS:
    X           -   dataset, array[0..NPoints-1,0..NVars-1].
                    matrix contains ONLY INDEPENDENT VARIABLES.
    NPoints     -   dataset size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NNeeded     -   number of requested components, in [1,NVars] range;
                    this function is efficient only for NNeeded<<NVars.
    Eps         -   desired  precision  of  vectors  returned;  underlying
                    solver will stop iterations as soon as absolute  error
                    in corresponding singular values  reduces  to  roughly
                    eps*MAX(lambda[]), with lambda[] being array of  eigen
                    values.
                    Zero value means that  algorithm  performs  number  of
                    iterations  specified  by  maxits  parameter,  without
                    paying attention to precision.
    MaxIts      -   number of iterations performed by  subspace  iteration
                    method. Zero value means that no  limit  on  iteration
                    count is placed (eps-based stopping condition is used).


OUTPUT PARAMETERS:
    S2          -   array[NNeeded]. Variance values corresponding
                    to basis vectors.
    V           -   array[NVars,NNeeded]
                    matrix, whose columns store basis vectors.

NOTE: passing eps=0 and maxits=0 results in small eps  being  selected  as
stopping condition. Exact value of automatically selected eps is  version-
-dependent.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 10.01.2017 by Bochkanov Sergey
*************************************************************************/
void pcatruncatedsubspace(const real_2d_array &x, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nneeded, const double eps, const ae_int_t maxits, real_1d_array &s2, real_2d_array &v, const xparams _xparams = alglib::xdefault);
void pcatruncatedsubspace(const real_2d_array &x, const ae_int_t nneeded, const double eps, const ae_int_t maxits, real_1d_array &s2, real_2d_array &v, const xparams _xparams = alglib::xdefault);

`pcatruncatedsubspacesparse` function

/*************************************************************************
Sparse truncated principal components analysis

This function performs sparse truncated PCA, i.e. returns just a few  most
important principal components for a sparse input X.

Internally it uses iterative eigensolver which is very efficient when only
a minor fraction of full basis is required.

It should be noted that, unlike LDA, PCA does not use class labels.

INPUT PARAMETERS:
    X           -   sparse dataset, sparse  npoints*nvars  matrix.  It  is
                    recommended to use CRS sparse storage format;  non-CRS
                    input will be internally converted to CRS.
                    Matrix contains ONLY INDEPENDENT VARIABLES,  and  must
                    be EXACTLY npoints*nvars.
    NPoints     -   dataset size, NPoints>=0
    NVars       -   number of independent variables, NVars>=1
    NNeeded     -   number of requested components, in [1,NVars] range;
                    this function is efficient only for NNeeded<<NVars.
    Eps         -   desired  precision  of  vectors  returned;  underlying
                    solver will stop iterations as soon as absolute  error
                    in corresponding singular values  reduces  to  roughly
                    eps*MAX(lambda[]), with lambda[] being array of  eigen
                    values.
                    Zero value means that  algorithm  performs  number  of
                    iterations  specified  by  maxits  parameter,  without
                    paying attention to precision.
    MaxIts      -   number of iterations performed by  subspace  iteration
                    method. Zero value means that no  limit  on  iteration
                    count is placed (eps-based stopping condition is used).


OUTPUT PARAMETERS:
    S2          -   array[NNeeded]. Variance values corresponding
                    to basis vectors.
    V           -   array[NVars,NNeeded]
                    matrix, whose columns store basis vectors.

NOTE: passing eps=0 and maxits=0 results in small eps  being  selected  as
      a stopping condition. Exact value of automatically selected  eps  is
      version-dependent.

NOTE: zero  MaxIts  is  silently  replaced  by some reasonable value which
      prevents eternal loops (possible when inputs are degenerate and  too
      stringent stopping criteria are specified). In  current  version  it
      is 50+2*NVars.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB --
     Copyright 10.01.2017 by Bochkanov Sergey
*************************************************************************/
void pcatruncatedsubspacesparse(const sparsematrix &x, const ae_int_t npoints, const ae_int_t nvars, const ae_int_t nneeded, const double eps, const ae_int_t maxits, real_1d_array &s2, real_2d_array &v, const xparams _xparams = alglib::xdefault);

`poissondistr` subpackage

Functions

invpoissondistribution
poissoncdistribution
poissondistribution

Examples

`invpoissondistribution` function

/*************************************************************************
Inverse Poisson distribution

Finds the Poisson variable x such that the integral
from 0 to x of the Poisson density is equal to the
given probability y.

This is accomplished using the inverse gamma integral
function and the relation

   m = igami( k+1, y ).

ACCURACY:

See inverse incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invpoissondistribution(const ae_int_t k, const double y, const xparams _xparams = alglib::xdefault);

`poissoncdistribution` function

/*************************************************************************
Complemented Poisson distribution

Returns the sum of the terms k+1 to infinity of the Poisson
distribution:

 inf.       j
  --   -m  m
  >   e    --
  --       j!
 j=k+1

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the formula

y = pdtrc( k, m ) = igam( k+1, m ).

The arguments must both be positive.

ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissoncdistribution(const ae_int_t k, const double m, const xparams _xparams = alglib::xdefault);

`poissondistribution` function

/*************************************************************************
Poisson distribution

Returns the sum of the first k+1 terms of the Poisson
distribution:

  k         j
  --   -m  m
  >   e    --
  --       j!
 j=0

The terms are not summed directly; instead the incomplete
gamma integral is employed, according to the relation

y = pdtr( k, m ) = igamc( k+1, m ).

The arguments must both be positive.
ACCURACY:

See incomplete gamma function

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double poissondistribution(const ae_int_t k, const double m, const xparams _xparams = alglib::xdefault);

`polint` subpackage

Functions

polynomialbar2cheb
polynomialbar2pow
polynomialbuild
polynomialbuildcheb1
polynomialbuildcheb2
polynomialbuildeqdist
polynomialcalccheb1
polynomialcalccheb2
polynomialcalceqdist
polynomialcheb2bar
polynomialpow2bar

Examples

polint_d_calcdiff		Interpolation and differentiation using barycentric representation
polint_d_conv		Conversion between power basis and barycentric representation
polint_d_spec		Polynomial interpolation on special grids (equidistant, Chebyshev I/II)

`polynomialbar2cheb` function

/*************************************************************************
Conversion from barycentric representation to Chebyshev basis.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    P   -   polynomial in barycentric form
    A,B -   base interval for Chebyshev polynomials (see below)
            A<>B

OUTPUT PARAMETERS
    T   -   coefficients of Chebyshev representation;
            P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 },
            where Ti - I-th Chebyshev polynomial.

NOTES:
    barycentric interpolant passed as P may be either polynomial  obtained
    from  polynomial  interpolation/ fitting or rational function which is
    NOT polynomial. We can't distinguish between these two cases, and this
    algorithm just tries to work assuming that P IS a polynomial.  If not,
    algorithm will return results, but they won't have any meaning.

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2cheb(const barycentricinterpolant &p, const double a, const double b, real_1d_array &t, const xparams _xparams = alglib::xdefault);

`polynomialbar2pow` function

/*************************************************************************
Conversion from barycentric representation to power basis.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    P   -   polynomial in barycentric form
    C   -   offset (see below); 0.0 is used as default value.
    S   -   scale (see below);  1.0 is used as default value. S<>0.

OUTPUT PARAMETERS
    A   -   coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
    N   -   number of coefficients (polynomial degree plus 1)

NOTES:
1.  this function accepts offset and scale, which can be  set  to  improve
    numerical properties of polynomial. For example, if P was obtained  as
    result of interpolation on [-1,+1],  you  can  set  C=0  and  S=1  and
    represent  P  as sum of 1, x, x^2, x^3 and so on. In most cases you it
    is exactly what you need.

    However, if your interpolation model was built on [999,1001], you will
    see significant growth of numerical errors when using {1, x, x^2, x^3}
    as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3
    will be better option. Such representation can be  obtained  by  using
    1000.0 as offset C and 1.0 as scale S.

2.  power basis is ill-conditioned and tricks described above can't  solve
    this problem completely. This function  will  return  coefficients  in
    any  case,  but  for  N>8  they  will  become unreliable. However, N's
    less than 5 are pretty safe.

3.  barycentric interpolant passed as P may be either polynomial  obtained
    from  polynomial  interpolation/ fitting or rational function which is
    NOT polynomial. We can't distinguish between these two cases, and this
    algorithm just tries to work assuming that P IS a polynomial.  If not,
    algorithm will return results, but they won't have any meaning.

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialbar2pow(const barycentricinterpolant &p, const double c, const double s, real_1d_array &a, const xparams _xparams = alglib::xdefault);
void polynomialbar2pow(const barycentricinterpolant &p, real_1d_array &a, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialbuild` function

/*************************************************************************
Lagrange intepolant: generation of the model on the general grid.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    X   -   abscissas, array[0..N-1]
    Y   -   function values, array[0..N-1]
    N   -   number of points, N>=1

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuild(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);
void polynomialbuild(const real_1d_array &x, const real_1d_array &y, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialbuildcheb1` function

/*************************************************************************
Lagrange intepolant on Chebyshev grid (first kind).
This function has O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    Y   -   function values at the nodes, array[0..N-1],
            Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)))
    N   -   number of points, N>=1
            for N=1 a constant model is constructed.

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb1(const double a, const double b, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);
void polynomialbuildcheb1(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialbuildcheb2` function

/*************************************************************************
Lagrange intepolant on Chebyshev grid (second kind).
This function has O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    Y   -   function values at the nodes, array[0..N-1],
            Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)))
    N   -   number of points, N>=1
            for N=1 a constant model is constructed.

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildcheb2(const double a, const double b, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);
void polynomialbuildcheb2(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialbuildeqdist` function

/*************************************************************************
Lagrange intepolant: generation of the model on equidistant grid.
This function has O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    Y   -   function values at the nodes, array[0..N-1]
    N   -   number of points, N>=1
            for N=1 a constant model is constructed.

OUTPUT PARAMETERS
    P   -   barycentric model which represents Lagrange interpolant
            (see ratint unit info and BarycentricCalc() description for
            more information).

  -- ALGLIB --
     Copyright 03.12.2009 by Bochkanov Sergey
*************************************************************************/
void polynomialbuildeqdist(const double a, const double b, const real_1d_array &y, const ae_int_t n, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);
void polynomialbuildeqdist(const double a, const double b, const real_1d_array &y, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialcalccheb1` function

/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (first kind)
with O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    F   -   function values, array[0..N-1]
    N   -   number of points on Chebyshev grid (first kind),
            X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))
            for N=1 a constant model is constructed.
    T   -   position where P(x) is calculated

RESULT
    value of the Lagrange interpolant at T

IMPORTANT
    this function provides fast interface which is not overflow-safe
    nor it is very precise.
    the best option is to use  PolIntBuildCheb1()/BarycentricCalc()
    subroutines unless you are pretty sure that your data will not result
    in overflow.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb1(const double a, const double b, const real_1d_array &f, const ae_int_t n, const double t, const xparams _xparams = alglib::xdefault);
double polynomialcalccheb1(const double a, const double b, const real_1d_array &f, const double t, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialcalccheb2` function

/*************************************************************************
Fast polynomial interpolation function on Chebyshev points (second kind)
with O(N) complexity.

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    F   -   function values, array[0..N-1]
    N   -   number of points on Chebyshev grid (second kind),
            X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))
            for N=1 a constant model is constructed.
    T   -   position where P(x) is calculated

RESULT
    value of the Lagrange interpolant at T

IMPORTANT
    this function provides fast interface which is not overflow-safe
    nor it is very precise.
    the best option is to use PolIntBuildCheb2()/BarycentricCalc()
    subroutines unless you are pretty sure that your data will not result
    in overflow.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalccheb2(const double a, const double b, const real_1d_array &f, const ae_int_t n, const double t, const xparams _xparams = alglib::xdefault);
double polynomialcalccheb2(const double a, const double b, const real_1d_array &f, const double t, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialcalceqdist` function

/*************************************************************************
Fast equidistant polynomial interpolation function with O(N) complexity

INPUT PARAMETERS:
    A   -   left boundary of [A,B]
    B   -   right boundary of [A,B]
    F   -   function values, array[0..N-1]
    N   -   number of points on equidistant grid, N>=1
            for N=1 a constant model is constructed.
    T   -   position where P(x) is calculated

RESULT
    value of the Lagrange interpolant at T

IMPORTANT
    this function provides fast interface which is not overflow-safe
    nor it is very precise.
    the best option is to use  PolynomialBuildEqDist()/BarycentricCalc()
    subroutines unless you are pretty sure that your data will not result
    in overflow.

  -- ALGLIB --
     Copyright 02.12.2009 by Bochkanov Sergey
*************************************************************************/
double polynomialcalceqdist(const double a, const double b, const real_1d_array &f, const ae_int_t n, const double t, const xparams _xparams = alglib::xdefault);
double polynomialcalceqdist(const double a, const double b, const real_1d_array &f, const double t, const xparams _xparams = alglib::xdefault);

Examples: [1]

`polynomialcheb2bar` function

/*************************************************************************
Conversion from Chebyshev basis to barycentric representation.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    T   -   coefficients of Chebyshev representation;
            P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N },
            where Ti - I-th Chebyshev polynomial.
    N   -   number of coefficients:
            * if given, only leading N elements of T are used
            * if not given, automatically determined from size of T
    A,B -   base interval for Chebyshev polynomials (see above)
            A<B

OUTPUT PARAMETERS
    P   -   polynomial in barycentric form

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialcheb2bar(const real_1d_array &t, const ae_int_t n, const double a, const double b, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);
void polynomialcheb2bar(const real_1d_array &t, const double a, const double b, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);

`polynomialpow2bar` function

/*************************************************************************
Conversion from power basis to barycentric representation.
This function has O(N^2) complexity.

INPUT PARAMETERS:
    A   -   coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 }
    N   -   number of coefficients (polynomial degree plus 1)
            * if given, only leading N elements of A are used
            * if not given, automatically determined from size of A
    C   -   offset (see below); 0.0 is used as default value.
    S   -   scale (see below);  1.0 is used as default value. S<>0.

OUTPUT PARAMETERS
    P   -   polynomial in barycentric form


NOTES:
1.  this function accepts offset and scale, which can be  set  to  improve
    numerical properties of polynomial. For example, if you interpolate on
    [-1,+1],  you  can  set C=0 and S=1 and convert from sum of 1, x, x^2,
    x^3 and so on. In most cases you it is exactly what you need.

    However, if your interpolation model was built on [999,1001], you will
    see significant growth of numerical errors when using {1, x, x^2, x^3}
    as  input  basis.  Converting  from  sum  of  1, (x-1000), (x-1000)^2,
    (x-1000)^3 will be better option (you have to specify 1000.0 as offset
    C and 1.0 as scale S).

2.  power basis is ill-conditioned and tricks described above can't  solve
    this problem completely. This function  will  return barycentric model
    in any case, but for N>8 accuracy well degrade. However, N's less than
    5 are pretty safe.

  -- ALGLIB --
     Copyright 30.09.2010 by Bochkanov Sergey
*************************************************************************/
void polynomialpow2bar(const real_1d_array &a, const ae_int_t n, const double c, const double s, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);
void polynomialpow2bar(const real_1d_array &a, barycentricinterpolant &p, const xparams _xparams = alglib::xdefault);

Examples: [1]

polint_d_calcdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate polynomial interpolation and differentiation
        // of y=x^2-x sampled at [0,1,2]. Barycentric representation of polynomial is used.
        //
        real_1d_array x = "[0,1,2]";
        real_1d_array y = "[0,0,2]";
        double t = -1;
        double v;
        double dv;
        double d2v;
        barycentricinterpolant p;

        // barycentric model is created
        polynomialbuild(x, y, p);

        // barycentric interpolation is demonstrated
        v = barycentriccalc(p, t);
        printf("%.4f\n", double(v)); // EXPECTED: 2.0

        // barycentric differentation is demonstrated
        barycentricdiff1(p, t, v, dv);
        printf("%.4f\n", double(v)); // EXPECTED: 2.0
        printf("%.4f\n", double(dv)); // EXPECTED: -3.0

        // second derivatives with barycentric representation
        barycentricdiff1(p, t, v, dv);
        printf("%.4f\n", double(v)); // EXPECTED: 2.0
        printf("%.4f\n", double(dv)); // EXPECTED: -3.0
        barycentricdiff2(p, t, v, dv, d2v);
        printf("%.4f\n", double(v)); // EXPECTED: 2.0
        printf("%.4f\n", double(dv)); // EXPECTED: -3.0
        printf("%.4f\n", double(d2v)); // EXPECTED: 2.0
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

polint_d_conv example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate conversion of y=x^2-x
        // between power basis and barycentric representation.
        //
        real_1d_array a = "[0,-1,+1]";
        double t = 2;
        real_1d_array a2;
        double v;
        barycentricinterpolant p;

        //
        // a=[0,-1,+1] is decomposition of y=x^2-x in the power basis:
        //
        //     y = 0 - 1*x + 1*x^2
        //
        // We convert it to the barycentric form.
        //
        polynomialpow2bar(a, p);

        // now we have barycentric interpolation; we can use it for interpolation
        v = barycentriccalc(p, t);
        printf("%.2f\n", double(v)); // EXPECTED: 2.0

        // we can also convert back from barycentric representation to power basis
        polynomialbar2pow(p, a2);
        printf("%s\n", a2.tostring(2).c_str()); // EXPECTED: [0,-1,+1]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

polint_d_spec example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Temporaries:
        // * values of y=x^2-x sampled at three special grids:
        //   * equdistant grid spanning [0,2],     x[i] = 2*i/(N-1), i=0..N-1
        //   * Chebyshev-I grid spanning [-1,+1],  x[i] = 1 + Cos(PI*(2*i+1)/(2*n)), i=0..N-1
        //   * Chebyshev-II grid spanning [-1,+1], x[i] = 1 + Cos(PI*i/(n-1)), i=0..N-1
        // * barycentric interpolants for these three grids
        // * vectors to store coefficients of quadratic representation
        //
        real_1d_array y_eqdist = "[0,0,2]";
        real_1d_array y_cheb1 = "[-0.116025,0.000000,1.616025]";
        real_1d_array y_cheb2 = "[0,0,2]";
        barycentricinterpolant p_eqdist;
        barycentricinterpolant p_cheb1;
        barycentricinterpolant p_cheb2;
        real_1d_array a_eqdist;
        real_1d_array a_cheb1;
        real_1d_array a_cheb2;

        //
        // First, we demonstrate construction of barycentric interpolants on
        // special grids. We unpack power representation to ensure that
        // interpolant was built correctly.
        //
        // In all three cases we should get same quadratic function.
        //
        polynomialbuildeqdist(0.0, 2.0, y_eqdist, p_eqdist);
        polynomialbar2pow(p_eqdist, a_eqdist);
        printf("%s\n", a_eqdist.tostring(4).c_str()); // EXPECTED: [0,-1,+1]

        polynomialbuildcheb1(-1, +1, y_cheb1, p_cheb1);
        polynomialbar2pow(p_cheb1, a_cheb1);
        printf("%s\n", a_cheb1.tostring(4).c_str()); // EXPECTED: [0,-1,+1]

        polynomialbuildcheb2(-1, +1, y_cheb2, p_cheb2);
        polynomialbar2pow(p_cheb2, a_cheb2);
        printf("%s\n", a_cheb2.tostring(4).c_str()); // EXPECTED: [0,-1,+1]

        //
        // Now we demonstrate polynomial interpolation without construction 
        // of the barycentricinterpolant structure.
        //
        // We calculate interpolant value at x=-2.
        // In all three cases we should get same f=6
        //
        double t = -2;
        double v;
        v = polynomialcalceqdist(0.0, 2.0, y_eqdist, t);
        printf("%.4f\n", double(v)); // EXPECTED: 6.0

        v = polynomialcalccheb1(-1, +1, y_cheb1, t);
        printf("%.4f\n", double(v)); // EXPECTED: 6.0

        v = polynomialcalccheb2(-1, +1, y_cheb2, t);
        printf("%.4f\n", double(v)); // EXPECTED: 6.0
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`polynomialsolver` subpackage

Classes

polynomialsolverreport

Functions

polynomialsolve

Examples

`polynomialsolverreport` class

/*************************************************************************

*************************************************************************/
class polynomialsolverreport
{
public:
    polynomialsolverreport();
    polynomialsolverreport(const polynomialsolverreport &rhs);
    polynomialsolverreport& operator=(const polynomialsolverreport &rhs);
    virtual ~polynomialsolverreport();
    double maxerr;
};

`polynomialsolve` function

/*************************************************************************
Polynomial root finding.

This function returns all roots of the polynomial
    P(x) = a0 + a1*x + a2*x^2 + ... + an*x^n
Both real and complex roots are returned (see below).

INPUT PARAMETERS:
    A       -   array[N+1], polynomial coefficients:
                * A[0] is constant term
                * A[N] is a coefficient of X^N
    N       -   polynomial degree

OUTPUT PARAMETERS:
    X       -   array of complex roots:
                * for isolated real root, X[I] is strictly real: IMAGE(X[I])=0
                * complex roots are always returned in pairs - roots occupy
                  positions I and I+1, with:
                  * X[I+1]=Conj(X[I])
                  * IMAGE(X[I]) > 0
                  * IMAGE(X[I+1]) = -IMAGE(X[I]) < 0
                * multiple real roots may have non-zero imaginary part due
                  to roundoff errors. There is no reliable way to distinguish
                  real root of multiplicity 2 from two  complex  roots  in
                  the presence of roundoff errors.
    Rep     -   report, additional information, following fields are set:
                * Rep.MaxErr - max( |P(xi)| )  for  i=0..N-1.  This  field
                  allows to quickly estimate "quality" of the roots  being
                  returned.

NOTE:   this function uses companion matrix method to find roots. In  case
        internal EVD  solver  fails  do  find  eigenvalues,  exception  is
        generated.

NOTE:   roots are not "polished" and  no  matrix  balancing  is  performed
        for them.

  -- ALGLIB --
     Copyright 24.02.2014 by Bochkanov Sergey
*************************************************************************/
void polynomialsolve(const real_1d_array &a, const ae_int_t n, complex_1d_array &x, polynomialsolverreport &rep, const xparams _xparams = alglib::xdefault);

`psif` subpackage

Functions

psi

Examples

`psi` function

/*************************************************************************
Psi (digamma) function

             d      -
  psi(x)  =  -- ln | (x)
             dx

is the logarithmic derivative of the gamma function.
For integer x,
                  n-1
                   -
psi(n) = -EUL  +   >  1/k.
                   -
                  k=1

This formula is used for 0 < n <= 10.  If x is negative, it
is transformed to a positive argument by the reflection
formula  psi(1-x) = psi(x) + pi cot(pi x).
For general positive x, the argument is made greater than 10
using the recurrence  psi(x+1) = psi(x) + 1/x.
Then the following asymptotic expansion is applied:

                          inf.   B
                           -      2k
psi(x) = log(x) - 1/2x -   >   -------
                           -        2k
                          k=1   2k x

where the B2k are Bernoulli numbers.

ACCURACY:
   Relative error (except absolute when |psi| < 1):
arithmetic   domain     # trials      peak         rms
   IEEE      0,30        30000       1.3e-15     1.4e-16
   IEEE      -30,0       40000       1.5e-15     2.2e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier
*************************************************************************/
double psi(const double x, const xparams _xparams = alglib::xdefault);

`ratint` subpackage

Classes

barycentricinterpolant

Functions

barycentricbuildfloaterhormann
barycentricbuildxyw
barycentriccalc
barycentricdiff1
barycentricdiff2
barycentriclintransx
barycentriclintransy
barycentricunpack

Examples

`barycentricinterpolant` class

/*************************************************************************
Barycentric interpolant.
*************************************************************************/
class barycentricinterpolant
{
public:
    barycentricinterpolant();
    barycentricinterpolant(const barycentricinterpolant &rhs);
    barycentricinterpolant& operator=(const barycentricinterpolant &rhs);
    virtual ~barycentricinterpolant();
};

`barycentricbuildfloaterhormann` function

/*************************************************************************
Rational interpolant without poles

The subroutine constructs the rational interpolating function without real
poles  (see  'Barycentric rational interpolation with no  poles  and  high
rates of approximation', Michael S. Floater. and  Kai  Hormann,  for  more
information on this subject).

Input parameters:
    X   -   interpolation nodes, array[0..N-1].
    Y   -   function values, array[0..N-1].
    N   -   number of nodes, N>0.
    D   -   order of the interpolation scheme, 0 <= D <= N-1.
            D<0 will cause an error.
            D>=N it will be replaced with D=N-1.
            if you don't know what D to choose, use small value about 3-5.

Output parameters:
    B   -   barycentric interpolant.

Note:
    this algorithm always succeeds and calculates the weights  with  close
    to machine precision.

  -- ALGLIB PROJECT --
     Copyright 17.06.2007 by Bochkanov Sergey
*************************************************************************/
void barycentricbuildfloaterhormann(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t d, barycentricinterpolant &b, const xparams _xparams = alglib::xdefault);

`barycentricbuildxyw` function

/*************************************************************************
Rational interpolant from X/Y/W arrays

F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))

INPUT PARAMETERS:
    X   -   interpolation nodes, array[0..N-1]
    F   -   function values, array[0..N-1]
    W   -   barycentric weights, array[0..N-1]
    N   -   nodes count, N>0

OUTPUT PARAMETERS:
    B   -   barycentric interpolant built from (X, Y, W)

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricbuildxyw(const real_1d_array &x, const real_1d_array &y, const real_1d_array &w, const ae_int_t n, barycentricinterpolant &b, const xparams _xparams = alglib::xdefault);

`barycentriccalc` function

/*************************************************************************
Rational interpolation using barycentric formula

F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i]))

Input parameters:
    B   -   barycentric interpolant built with one of model building
            subroutines.
    T   -   interpolation point

Result:
    barycentric interpolant F(t)

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
double barycentriccalc(const barycentricinterpolant &b, const double t, const xparams _xparams = alglib::xdefault);

`barycentricdiff1` function

/*************************************************************************
Differentiation of barycentric interpolant: first derivative.

Algorithm used in this subroutine is very robust and should not fail until
provided with values too close to MaxRealNumber  (usually  MaxRealNumber/N
or greater will overflow).

INPUT PARAMETERS:
    B   -   barycentric interpolant built with one of model building
            subroutines.
    T   -   interpolation point

OUTPUT PARAMETERS:
    F   -   barycentric interpolant at T
    DF  -   first derivative

NOTE


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricdiff1(const barycentricinterpolant &b, const double t, double &f, double &df, const xparams _xparams = alglib::xdefault);

`barycentricdiff2` function

/*************************************************************************
Differentiation of barycentric interpolant: first/second derivatives.

INPUT PARAMETERS:
    B   -   barycentric interpolant built with one of model building
            subroutines.
    T   -   interpolation point

OUTPUT PARAMETERS:
    F   -   barycentric interpolant at T
    DF  -   first derivative
    D2F -   second derivative

NOTE: this algorithm may fail due to overflow/underflor if  used  on  data
whose values are close to MaxRealNumber or MinRealNumber.  Use more robust
BarycentricDiff1() subroutine in such cases.


  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricdiff2(const barycentricinterpolant &b, const double t, double &f, double &df, double &d2f, const xparams _xparams = alglib::xdefault);

`barycentriclintransx` function

/*************************************************************************
This subroutine performs linear transformation of the argument.

INPUT PARAMETERS:
    B       -   rational interpolant in barycentric form
    CA, CB  -   transformation coefficients: x = CA*t + CB

OUTPUT PARAMETERS:
    B       -   transformed interpolant with X replaced by T

  -- ALGLIB PROJECT --
     Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriclintransx(barycentricinterpolant &b, const double ca, const double cb, const xparams _xparams = alglib::xdefault);

`barycentriclintransy` function

/*************************************************************************
This  subroutine   performs   linear  transformation  of  the  barycentric
interpolant.

INPUT PARAMETERS:
    B       -   rational interpolant in barycentric form
    CA, CB  -   transformation coefficients: B2(x) = CA*B(x) + CB

OUTPUT PARAMETERS:
    B       -   transformed interpolant

  -- ALGLIB PROJECT --
     Copyright 19.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentriclintransy(barycentricinterpolant &b, const double ca, const double cb, const xparams _xparams = alglib::xdefault);

`barycentricunpack` function

/*************************************************************************
Extracts X/Y/W arrays from rational interpolant

INPUT PARAMETERS:
    B   -   barycentric interpolant

OUTPUT PARAMETERS:
    N   -   nodes count, N>0
    X   -   interpolation nodes, array[0..N-1]
    F   -   function values, array[0..N-1]
    W   -   barycentric weights, array[0..N-1]

  -- ALGLIB --
     Copyright 17.08.2009 by Bochkanov Sergey
*************************************************************************/
void barycentricunpack(const barycentricinterpolant &b, ae_int_t &n, real_1d_array &x, real_1d_array &y, real_1d_array &w, const xparams _xparams = alglib::xdefault);

`rbf` subpackage

Classes

rbfcalcbuffer
rbfmodel
rbfreport

Functions

rbfbuildmodel
rbfcalc
rbfcalc1
rbfcalc2
rbfcalc3
rbfcalcbuf
rbfcreate
rbfcreatecalcbuffer
rbfdiff
rbfdiff1
rbfdiff2
rbfdiff3
rbfdiffbuf
rbffastcalc
rbfgetmodelversion
rbfgridcalc2
rbfgridcalc2v
rbfgridcalc2vsubset
rbfgridcalc3v
rbfgridcalc3vsubset
rbfhess
rbfhessbuf
rbfpeekprogress
rbfrequesttermination
rbfserialize
rbfsetalgobiharmonic
rbfsetalgohierarchical
rbfsetalgomultilayer
rbfsetalgomultiquadricauto
rbfsetalgomultiquadricmanual
rbfsetalgoqnn
rbfsetalgothinplatespline
rbfsetconstterm
rbfsetfastevaltol
rbfsetlinterm
rbfsetpoints
rbfsetpointsandscales
rbfsetv2bf
rbfsetv2its
rbfsetv2supportr
rbfsetv3tol
rbfsetzeroterm
rbftscalcbuf
rbftsdiffbuf
rbftshessbuf
rbfunpack
rbfunserialize

Examples

rbf_d_hrbf		Simple model built with HRBF algorithm
rbf_d_polterm		RBF models - working with polynomial term
rbf_d_serialize		Serialization/unserialization
rbf_d_vector		Working with vector functions

`rbfcalcbuffer` class

/*************************************************************************
Buffer object which is used  to  perform  RBF  model  calculation  in  the
multithreaded mode (multiple threads working with same RBF object).

This object should be created with RBFCreateCalcBuffer().
*************************************************************************/
class rbfcalcbuffer
{
public:
    rbfcalcbuffer();
    rbfcalcbuffer(const rbfcalcbuffer &rhs);
    rbfcalcbuffer& operator=(const rbfcalcbuffer &rhs);
    virtual ~rbfcalcbuffer();
};

`rbfmodel` class

/*************************************************************************
RBF model.

Never try to directly work with fields of this object - always use  ALGLIB
functions to use this object.
*************************************************************************/
class rbfmodel
{
public:
    rbfmodel();
    rbfmodel(const rbfmodel &rhs);
    rbfmodel& operator=(const rbfmodel &rhs);
    virtual ~rbfmodel();
};

`rbfreport` class

/*************************************************************************
RBF solution report:
* TerminationType   -   termination type, positive values - success,
                        non-positive - failure.

Fields which are set by modern RBF solvers (hierarchical):
* RMSError          -   root-mean-square error; NAN for old solvers (ML, QNN)
* MaxError          -   maximum error; NAN for old solvers (ML, QNN)
*************************************************************************/
class rbfreport
{
public:
    rbfreport();
    rbfreport(const rbfreport &rhs);
    rbfreport& operator=(const rbfreport &rhs);
    virtual ~rbfreport();
    double rmserror;
    double maxerror;
    ae_int_t arows;
    ae_int_t acols;
    ae_int_t annz;
    ae_int_t iterationscount;
    ae_int_t nmv;
    ae_int_t terminationtype;
};

`rbfbuildmodel` function

/*************************************************************************
This   function  builds  RBF  model  and  returns  report  (contains  some
information which can be used for evaluation of the algorithm properties).

Call to this function modifies RBF model by calculating its centers/radii/
weights  and  saving  them  into  RBFModel  structure.  Initially RBFModel
contain zero coefficients, but after call to this function  we  will  have
coefficients which were calculated in order to fit our dataset.

After you called this function you can call RBFCalc(),  RBFGridCalc()  and
other model calculation functions.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    Rep     -   report:
                * Rep.TerminationType:
                  * -5 - non-distinct basis function centers were detected,
                         interpolation  aborted;  only  QNN  returns  this
                         error   code, other  algorithms  can  handle non-
                         distinct nodes.
                  * -4 - nonconvergence of the internal SVD solver
                  * -3   incorrect model construction algorithm was chosen:
                         QNN or RBF-ML, combined with one of the incompatible
                         features:
                         * NX=1 or NX>3
                         * points with per-dimension scales.
                  *  1 - successful termination
                  *  8 - a termination request was submitted via
                         rbfrequesttermination() function.

                Fields which are set only by modern RBF solvers (hierarchical
                or nonnegative; older solvers like QNN and ML initialize these
                fields by NANs):
                * rep.rmserror - root-mean-square error at nodes
                * rep.maxerror - maximum error at nodes

                Fields are used for debugging purposes:
                * Rep.IterationsCount - iterations count of the LSQR solver
                * Rep.NMV - number of matrix-vector products
                * Rep.ARows - rows count for the system matrix
                * Rep.ACols - columns count for the system matrix
                * Rep.ANNZ - number of significantly non-zero elements
                  (elements above some algorithm-determined threshold)

NOTE:  failure  to  build  model will leave current state of the structure
unchanged.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfbuildmodel(rbfmodel &s, rbfreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`rbfcalc` function

/*************************************************************************
This function calculates values of the RBF model at the given point.

This is general function which can be used for arbitrary NX (dimension  of
the space of arguments) and NY (dimension of the function itself). However
when  you  have  NY=1  you  may  find more convenient to use rbfcalc2() or
rbfcalc3().

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when model is not initialized.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is out-parameter and
                reallocated after call to this function. In case you  want
                to reuse previously allocated Y, you may use RBFCalcBuf(),
                which reallocates Y only when it is too small.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcalc(rbfmodel &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rbfcalc1` function

/*************************************************************************
This function calculates values of the 1-dimensional RBF model with scalar
output (NY=1) at the given point.

IMPORTANT: this function works only with modern  (hierarchical)  RBFs.  It
           can not be used with legacy (version 1) RBFs because older  RBF
           code does not support 1-dimensional models.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when:
* the model is not initialized
* NX<>1
* NY<>1

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   X-coordinate, finite number

RESULT:
    value of the model or 0.0 (as defined above)

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc1(rbfmodel &s, const double x0, const xparams _xparams = alglib::xdefault);

`rbfcalc2` function

/*************************************************************************
This function calculates values of the 2-dimensional RBF model with scalar
output (NY=1) at the given point.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when:
* model is not initialized
* NX<>2
 *NY<>1

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number
    X1      -   second coordinate, finite number

RESULT:
    value of the model or 0.0 (as defined above)

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc2(rbfmodel &s, const double x0, const double x1, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`rbfcalc3` function

/*************************************************************************
This function calculates values of the 3-dimensional RBF model with scalar
output (NY=1) at the given point.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 when:
* model is not initialized
* NX<>3
 *NY<>1

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number
    X1      -   second coordinate, finite number
    X2      -   third coordinate, finite number

RESULT:
    value of the model or 0.0 (as defined above)

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
double rbfcalc3(rbfmodel &s, const double x0, const double x1, const double x2, const xparams _xparams = alglib::xdefault);

`rbfcalcbuf` function

/*************************************************************************
This function calculates values of the RBF model at the given point.

Same as rbfcalc(), but does not reallocate Y when in is large enough to
store function values.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y       -   possibly preallocated array

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is not reallocated when it
                is larger than NY.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfcalcbuf(rbfmodel &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbfcreate` function

/*************************************************************************
This function creates RBF  model  for  a  scalar (NY=1)  or  vector (NY>1)
function in a NX-dimensional space (NX>=1).

Newly created model is empty. It can be used for interpolation right after
creation, but it just returns zeros. You have to add points to the  model,
tune interpolation settings, and then  call  model  construction  function
rbfbuildmodel() which will update model according to your specification.

USAGE:
1. User creates model with rbfcreate()
2. User adds dataset with rbfsetpoints() or rbfsetpointsandscales()
3. User selects RBF solver by calling:
   * rbfsetalgohierarchical() - for a HRBF solver,  a  hierarchical large-
     scale Gaussian RBFs  (works  well  for  uniformly  distributed  point
     clouds, but may fail when the data are non-uniform; use other solvers
     below in such cases)
   * rbfsetalgothinplatespline() - for a large-scale DDM-RBF  solver  with
     thin plate spline basis function being used
   * rbfsetalgobiharmonic() -  for  a  large-scale  DDM-RBF  solver   with
     biharmonic basis function being used
   * rbfsetalgomultiquadricauto() -  for a large-scale DDM-RBF solver with
     multiquadric basis function being used (automatic  selection  of  the
     scale parameter Alpha)
   * rbfsetalgomultiquadricmanual() -  for a  large-scale  DDM-RBF  solver
     with multiquadric basis function being used (manual selection  of the
     scale parameter Alpha)
4. (OPTIONAL) User chooses polynomial term by calling:
   * rbflinterm() to set linear term (default)
   * rbfconstterm() to set constant term
   * rbfzeroterm() to set zero term
5. User calls rbfbuildmodel() function which rebuilds model  according  to
   the specification

INPUT PARAMETERS:
    NX      -   dimension of the space, NX>=1
    NY      -   function dimension, NY>=1

OUTPUT PARAMETERS:
    S       -   RBF model (initially equals to zero)

NOTE 1: memory requirements. RBF models require amount of memory  which is
        proportional  to the number of data points. Some additional memory
        is allocated during model construction, but most of this memory is
        freed after the model  coefficients  are   calculated.  Amount  of
        this additional memory depends  on  model  construction  algorithm
        being used.

  -- ALGLIB --
     Copyright 13.12.2011, 20.06.2016 by Bochkanov Sergey
*************************************************************************/
void rbfcreate(const ae_int_t nx, const ae_int_t ny, rbfmodel &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3] [4]

`rbfcreatecalcbuffer` function

/*************************************************************************
This function creates buffer  structure  which  can  be  used  to  perform
parallel  RBF  model  evaluations  (with  one  RBF  model  instance  being
used from multiple threads, as long as  different  threads  use  different
instances of the buffer).

This buffer object can be used with  rbftscalcbuf()  function  (here  "ts"
stands for "thread-safe", "buf" is a suffix which denotes  function  which
reuses previously allocated output space).

A buffer creation function (this function) is also thread-safe.  I.e.  you
may safely create multiple buffers for the same  RBF  model  from multiple
threads.

NOTE: the  buffer  object  is  just  a  collection of several preallocated
      dynamic arrays and precomputed values. If you  delete  its  "parent"
      RBF model when the buffer is still alive, nothing  bad  will  happen
      (no dangling pointers or resource leaks).  The  buffer  will  simply
      become useless.

How to use it:
* create RBF model structure with rbfcreate()
* load data, tune parameters
* call rbfbuildmodel()
* call rbfcreatecalcbuffer(), once per thread working with RBF model  (you
  should call this function only AFTER call to rbfbuildmodel(), see  below
  for more information)
* call rbftscalcbuf() from different threads,  with  each  thread  working
  with its own copy of buffer object.
* it is recommended to reuse buffer as much  as  possible  because  buffer
  creation involves allocation of several large dynamic arrays.  It  is  a
  huge waste of resource to use it just once.

INPUT PARAMETERS
    S           -   RBF model

OUTPUT PARAMETERS
    Buf         -   external buffer.

IMPORTANT: buffer object should be used only with  RBF model object  which
           was used to initialize buffer. Any attempt to use buffer   with
           different object is dangerous - you may  get  memory  violation
           error because sizes of internal arrays do not fit to dimensions
           of RBF structure.

IMPORTANT: you  should  call  this function only for model which was built
           with rbfbuildmodel() function, after successful  invocation  of
           rbfbuildmodel().  Sizes   of   some   internal  structures  are
           determined only after model is built, so buffer object  created
           before model  construction  stage  will  be  useless  (and  any
           attempt to use it will result in exception).

  -- ALGLIB --
     Copyright 02.04.2016 by Sergey Bochkanov
*************************************************************************/
void rbfcreatecalcbuffer(const rbfmodel &s, rbfcalcbuffer &buf, const xparams _xparams = alglib::xdefault);

`rbfdiff` function

/*************************************************************************
This function calculates values of the RBF model and  its  derivatives  at
the given point.

This is general function which can be used for arbitrary NX (dimension  of
the space of arguments) and NY (dimension of the function itself). However
if you have NX=3 and NY=1, you may find more convenient to use rbfdiff3().

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).

           If you want to perform parallel model evaluation  from multiple
           threads, use rbftsdiffbuf() with per-thread buffer object.

This function returns 0.0 in Y and/or DY in the following cases:
* the model is not initialized (Y=0, DY=0)
* the gradient is undefined at the trial point. Some basis  functions have
  discontinuous derivatives at the interpolation nodes:
  * biharmonic splines f=r have no Hessian and no gradient at the nodes
  In these cases only DY is set to zero (Y is still returned)

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is out-parameter and
                reallocated after call to this function. In case you  want
                to reuse previously allocated Y, you may use RBFDiffBuf(),
                which reallocates Y only when it is too small.
    DY      -   derivatives, array[NX*NY]:
                * Y[I*NX+J] with 0<=I<NY and 0<=J<NX  stores derivative of
                  function component I with respect to input J.
                * for NY=1 it is simply NX-dimensional gradient of the
                  scalar NX-dimensional function
                DY is out-parameter and reallocated  after  call  to  this
                function. In case you want to reuse  previously  allocated
                DY, you may use RBFDiffBuf(), which  reallocates  DY  only
                when it is too small to store the result.

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfdiff(rbfmodel &s, const real_1d_array &x, real_1d_array &y, real_1d_array &dy, const xparams _xparams = alglib::xdefault);

`rbfdiff1` function

/*************************************************************************
This function calculates value and derivatives of  the  1-dimensional  RBF
model with scalar output (NY=1) at the given point.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 in Y and/or DY in the following cases:
* the model is not initialized (Y=0, DY=0)
* NX<>1 or NY<>1 (Y=0, DY=0)
* the gradient is undefined at the trial point. Some basis  functions have
  discontinuous derivatives at the interpolation nodes:
  * biharmonic splines f=r have no Hessian and no gradient at the nodes
  In these cases only DY is set to zero (Y is still returned)

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number

OUTPUT PARAMETERS:
    Y       -   value of the model or 0.0 (as defined above)
    DY0     -   derivative with respect to X0

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfdiff1(rbfmodel &s, const double x0, double &y, double &dy0, const xparams _xparams = alglib::xdefault);

`rbfdiff2` function

/*************************************************************************
This function calculates value and derivatives of  the  2-dimensional  RBF
model with scalar output (NY=1) at the given point.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 in Y and/or DY in the following cases:
* the model is not initialized (Y=0, DY=0)
* NX<>2 or NY<>1 (Y=0, DY=0)
* the gradient is undefined at the trial point. Some basis  functions have
  discontinuous derivatives at the interpolation nodes:
  * biharmonic splines f=r have no Hessian and no gradient at the nodes
  In these cases only DY is set to zero (Y is still returned)

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number
    X1      -   second coordinate, finite number

OUTPUT PARAMETERS:
    Y       -   value of the model or 0.0 (as defined above)
    DY0     -   derivative with respect to X0
    DY1     -   derivative with respect to X1

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfdiff2(rbfmodel &s, const double x0, const double x1, double &y, double &dy0, double &dy1, const xparams _xparams = alglib::xdefault);

`rbfdiff3` function

/*************************************************************************
This function calculates value and derivatives of  the  3-dimensional  RBF
model with scalar output (NY=1) at the given point.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with per-thread buffer object.

This function returns 0.0 in Y and/or DY in the following cases:
* the model is not initialized (Y=0, DY=0)
* NX<>3 or NY<>1 (Y=0, DY=0)
* the gradient is undefined at the trial point. Some basis  functions have
  discontinuous derivatives at the interpolation nodes:
  * biharmonic splines f=r have no Hessian and no gradient at the nodes
  In these cases only DY is set to zero (Y is still returned)

INPUT PARAMETERS:
    S       -   RBF model
    X0      -   first coordinate, finite number
    X1      -   second coordinate, finite number
    X2      -   third coordinate, finite number

OUTPUT PARAMETERS:
    Y       -   value of the model or 0.0 (as defined above)
    DY0     -   derivative with respect to X0
    DY1     -   derivative with respect to X1
    DY2     -   derivative with respect to X2

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfdiff3(rbfmodel &s, const double x0, const double x1, const double x2, double &y, double &dy0, double &dy1, double &dy2, const xparams _xparams = alglib::xdefault);

`rbfdiffbuf` function

/*************************************************************************
This function calculates values of the RBF model and  its  derivatives  at
the given point. It is a buffered version of the RBFDiff() which tries  to
reuse possibly preallocated output arrays Y/DY as much as possible.

This is general function which can be used for arbitrary NX (dimension  of
the space of arguments) and NY (dimension of the function itself). However
if you have NX=1, 2 or 3 and NY=1, you may find  more  convenient  to  use
rbfdiff1(), rbfdiff2() or rbfdiff3().

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).

           If you want to perform parallel model evaluation  from multiple
           threads, use rbftsdiffbuf() with per-thread buffer object.

This function returns 0.0 in Y and/or DY in the following cases:
* the model is not initialized (Y=0, DY=0)
* the gradient is undefined at the trial point. Some basis  functions have
  discontinuous derivatives at the interpolation nodes:
  * biharmonic splines f=r have no Hessian and no gradient at the nodes
  In these cases only DY is set to zero (Y is still returned)

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y, DY   -   possibly preallocated arrays; if array size is large enough
                to store results, this function does not  reallocate  array
                to fit output size exactly.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY].
    DY      -   derivatives, array[NX*NY]:
                * Y[I*NX+J] with 0<=I<NY and 0<=J<NX  stores derivative of
                  function component I with respect to input J.
                * for NY=1 it is simply NX-dimensional gradient of the
                  scalar NX-dimensional function

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfdiffbuf(rbfmodel &s, const real_1d_array &x, real_1d_array &y, real_1d_array &dy, const xparams _xparams = alglib::xdefault);

`rbffastcalc` function

/*************************************************************************
This function calculates values of the RBF model at the given point  using
a fast approximate algorithm whenever possible. If no  fast  algorithm  is
available for a given model type, traditional O(N) approach is used.

Presently, fast evaluation is implemented only for biharmonic splines.

The absolute approximation accuracy is controlled by the rbfsetfastevaltol()
function.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).
           If you want to perform parallel model evaluation  from multiple
           threads, use rbftscalcbuf() with a per-thread buffer object.

This function returns 0.0 when model is not initialized.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is out-parameter and
                reallocated after call to this function. In case you  want
                to reuse previously allocated Y, you may use RBFCalcBuf(),
                which reallocates Y only when it is too small.

  -- ALGLIB --
     Copyright 19.09.2022 by Bochkanov Sergey
*************************************************************************/
void rbffastcalc(rbfmodel &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbfgetmodelversion` function

/*************************************************************************
This function returns model version.

INPUT PARAMETERS:
    S       -   RBF model

RESULT:
    * 1 - for models created by QNN and RBF-ML algorithms,
      compatible with ALGLIB 3.10 or earlier.
    * 2 - for models created by HierarchicalRBF, requires
      ALGLIB 3.11 or later

  -- ALGLIB --
     Copyright 06.07.2016 by Bochkanov Sergey
*************************************************************************/
ae_int_t rbfgetmodelversion(rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfgridcalc2` function

/*************************************************************************
This is legacy function for gridded calculation of RBF model.

It is superseded by rbfgridcalc2v() and  rbfgridcalc2vsubset()  functions.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc2(rbfmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, real_2d_array &y, const xparams _xparams = alglib::xdefault);

`rbfgridcalc2v` function

/*************************************************************************
This function calculates values of the RBF  model  at  the  regular  grid,
which  has  N0*N1 points, with Point[I,J] = (X0[I], X1[J]).  Vector-valued
RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1], where NY is a  number of
                "output" vector values (this  function   supports  vector-
                valued RBF models). Y is out-variable and  is  reallocated
                by this function.
                Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]), for:
                *  K=0...NY-1
                * I0=0...N0-1
                * I1=0...N1-1

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

NOTE: if you need function values on some subset  of  regular  grid, which
      may be described as "several compact and  dense  islands",  you  may
      use rbfgridcalc2vsubset().

  -- ALGLIB --
     Copyright 27.01.2017 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc2v(const rbfmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbfgridcalc2vsubset` function

/*************************************************************************
This function calculates values of the RBF model at some subset of regular
grid:
* grid has N0*N1 points, with Point[I,J] = (X0[I], X1[J])
* only values at some subset of this grid are required
Vector-valued RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>2

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

    FlagY   -   array[N0*N1]:
                * Y[I0+I1*N0] corresponds to node (X0[I0],X1[I1])
                * it is a "bitmap" array which contains  False  for  nodes
                  which are NOT calculated, and True for nodes  which  are
                  required.

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued RBF models):
                * Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]),
                  for K=0...NY-1, I0=0...N0-1, I1=0...N1-1.
                * elements of Y[] which correspond  to  FlagY[]=True   are
                  loaded by model values (which may be  exactly  zero  for
                  some nodes).
                * elements of Y[] which correspond to FlagY[]=False MAY be
                  initialized by zeros OR may be calculated. This function
                  processes  grid  as  a  hierarchy  of  nested blocks and
                  micro-rows. If just one element of micro-row is required,
                  entire micro-row (up to 8 nodes in the current  version,
                  but no promises) is calculated.

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

  -- ALGLIB --
     Copyright 04.03.2016 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc2vsubset(const rbfmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, const boolean_1d_array &flagy, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbfgridcalc3v` function

/*************************************************************************
This function calculates values of the RBF  model  at  the  regular  grid,
which  has  N0*N1*N2  points,  with  Point[I,J,K] = (X0[I], X1[J], X2[K]).
Vector-valued RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>3

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

    X2      -   array of grid nodes, third coordinates, array[N2]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N2      -   grid size (number of nodes) in the third dimension

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued RBF models). Y is out-variable and  is  reallocated
                by this function.
                Y[K+NY*(I0+I1*N0+I2*N0*N1)]=F_k(X0[I0],X1[I1],X2[I2]), for:
                *  K=0...NY-1
                * I0=0...N0-1
                * I1=0...N1-1
                * I2=0...N2-1

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

NOTE: if you need function values on some subset  of  regular  grid, which
      may be described as "several compact and  dense  islands",  you  may
      use rbfgridcalc3vsubset().

  -- ALGLIB --
     Copyright 04.03.2016 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc3v(const rbfmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbfgridcalc3vsubset` function

/*************************************************************************
This function calculates values of the RBF model at some subset of regular
grid:
* grid has N0*N1*N2 points, with Point[I,J,K] = (X0[I], X1[J], X2[K])
* only values at some subset of this grid are required
Vector-valued RBF models are supported.

This function returns 0.0 when:
* model is not initialized
* NX<>3

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

NOTE: Parallel  processing  is  implemented only for modern (hierarchical)
      RBFs. Legacy version 1 RBFs (created  by  QNN  or  RBF-ML) are still
      processed serially.

INPUT PARAMETERS:
    S       -   RBF model, used in read-only mode, can be  shared  between
                multiple   invocations  of  this  function  from  multiple
                threads.

    X0      -   array of grid nodes, first coordinates, array[N0].
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N0      -   grid size (number of nodes) in the first dimension

    X1      -   array of grid nodes, second coordinates, array[N1]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N1      -   grid size (number of nodes) in the second dimension

    X2      -   array of grid nodes, third coordinates, array[N2]
                Must be ordered by ascending. Exception is generated
                if the array is not correctly ordered.
    N2      -   grid size (number of nodes) in the third dimension

    FlagY   -   array[N0*N1*N2]:
                * Y[I0+I1*N0+I2*N0*N1] corresponds to node (X0[I0],X1[I1],X2[I2])
                * it is a "bitmap" array which contains  False  for  nodes
                  which are NOT calculated, and True for nodes  which  are
                  required.

OUTPUT PARAMETERS:
    Y       -   function values, array[NY*N0*N1*N2], where NY is a  number
                of "output" vector values (this function  supports vector-
                valued RBF models):
                * Y[K+NY*(I0+I1*N0+I2*N0*N1)]=F_k(X0[I0],X1[I1],X2[I2]),
                  for K=0...NY-1, I0=0...N0-1, I1=0...N1-1, I2=0...N2-1.
                * elements of Y[] which correspond  to  FlagY[]=True   are
                  loaded by model values (which may be  exactly  zero  for
                  some nodes).
                * elements of Y[] which correspond to FlagY[]=False MAY be
                  initialized by zeros OR may be calculated. This function
                  processes  grid  as  a  hierarchy  of  nested blocks and
                  micro-rows. If just one element of micro-row is required,
                  entire micro-row (up to 8 nodes in the current  version,
                  but no promises) is calculated.

NOTE: this function supports weakly ordered grid nodes, i.e. you may  have
      X[i]=X[i+1] for some i. It does  not  provide  you  any  performance
      benefits  due  to   duplication  of  points,  just  convenience  and
      flexibility.

NOTE: this  function  is  re-entrant,  i.e.  you  may  use  same  rbfmodel
      structure in multiple threads calling  this function  for  different
      grids.

  -- ALGLIB --
     Copyright 04.03.2016 by Bochkanov Sergey
*************************************************************************/
void rbfgridcalc3vsubset(const rbfmodel &s, const real_1d_array &x0, const ae_int_t n0, const real_1d_array &x1, const ae_int_t n1, const real_1d_array &x2, const ae_int_t n2, const boolean_1d_array &flagy, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbfhess` function

/*************************************************************************
This function calculates values of the RBF model and  its first and second
derivatives (Hessian matrix) at the given point.

This function supports both scalar (NY=1) and vector-valued (NY>1) RBFs.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).

           If you want to perform parallel model evaluation  from multiple
           threads, use rbftshessbuf() with per-thread buffer object.

This function returns 0 in Y and/or DY and/or D2Y in the following cases:
* the model is not initialized (Y=0, DY=0, D2Y=0)
* the gradient and/or Hessian is undefined at the trial point.  Some basis
  functions have discontinuous derivatives at the interpolation nodes:
  * thin plate splines have no Hessian at the nodes
  * biharmonic splines f=r have no Hessian and no gradient at the  nodes
  In these cases only corresponding derivative is set  to  zero,  and  the
  rest of the derivatives is still returned.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY].
                Y is out-parameter and  reallocated  after  call  to  this
                function. In case you  want to reuse previously  allocated
                Y, you may use RBFHessBuf(), which reallocates Y only when
                it is too small.
    DY      -   first derivatives, array[NY*NX]:
                * Y[I*NX+J] with 0<=I<NY and 0<=J<NX  stores derivative of
                  function component I with respect to input J.
                * for NY=1 it is simply NX-dimensional gradient of the
                  scalar NX-dimensional function
                DY is out-parameter and reallocated  after  call  to  this
                function. In case you want to reuse  previously  allocated
                DY, you may use RBFHessBuf(), which  reallocates  DY  only
                when it is too small to store the result.
    D2Y     -   second derivatives, array[NY*NX*NX]:
                * for NY=1 it is NX*NX array that stores  Hessian  matrix,
                  with Y[I*NX+J]=Y[J*NX+I].
                * for  a  vector-valued  RBF  with  NY>1  it  contains  NY
                  subsequently stored Hessians: an element Y[K*NX*NX+I*NX+J]
                  with  0<=K<NY,  0<=I<NX  and  0<=J<NX    stores   second
                  derivative of the function #K  with  respect  to  inputs
                  #I and #J.
                D2Y is out-parameter and reallocated  after  call  to this
                function. In case you want to reuse  previously  allocated
                D2Y, you may use RBFHessBuf(), which  reallocates D2Y only
                when it is too small to store the result.

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfhess(rbfmodel &s, const real_1d_array &x, real_1d_array &y, real_1d_array &dy, real_1d_array &d2y, const xparams _xparams = alglib::xdefault);

`rbfhessbuf` function

/*************************************************************************
This function calculates values of the RBF model and  its first and second
derivatives (Hessian matrix) at the given point. It is a buffered  version
that reuses memory  allocated  in  output  buffers  Y/DY/D2Y  as  much  as
possible.

This function supports both scalar (NY=1) and vector-valued (NY>1) RBFs.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. It uses fields of  rbfmodel  as
           temporary arrays, i.e. it is  impossible  to  perform  parallel
           evaluation on the same rbfmodel object (parallel calls of  this
           function for independent rbfmodel objects are safe).

           If you want to perform parallel model evaluation  from multiple
           threads, use rbftshessbuf() with per-thread buffer object.

This function returns 0 in Y and/or DY and/or D2Y in the following cases:
* the model is not initialized (Y=0, DY=0, D2Y=0)
* the gradient and/or Hessian is undefined at the trial point.  Some basis
  functions have discontinuous derivatives at the interpolation nodes:
  * thin plate splines have no Hessian at the nodes
  * biharmonic splines f=r have no Hessian and no gradient at the  nodes
  In these cases only corresponding derivative is set  to  zero,  and  the
  rest of the derivatives is still returned.

INPUT PARAMETERS:
    S       -   RBF model
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y,DY,D2Y-   possible preallocated output arrays. If these  arrays  are
                smaller than  required  to  store  the  result,  they  are
                automatically reallocated. If array is large enough, it is
                not resized.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY].
    DY      -   first derivatives, array[NY*NX]:
                * Y[I*NX+J] with 0<=I<NY and 0<=J<NX  stores derivative of
                  function component I with respect to input J.
                * for NY=1 it is simply NX-dimensional gradient of the
                  scalar NX-dimensional function
    D2Y     -   second derivatives, array[NY*NX*NX]:
                * for NY=1 it is NX*NX array that stores  Hessian  matrix,
                  with Y[I*NX+J]=Y[J*NX+I].
                * for  a  vector-valued  RBF  with  NY>1  it  contains  NY
                  subsequently stored Hessians: an element Y[K*NX*NX+I*NX+J]
                  with  0<=K<NY,  0<=I<NX  and  0<=J<NX    stores   second
                  derivative of the function #K  with  respect  to  inputs
                  #I and #J.

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfhessbuf(rbfmodel &s, const real_1d_array &x, real_1d_array &y, real_1d_array &dy, real_1d_array &d2y, const xparams _xparams = alglib::xdefault);

`rbfpeekprogress` function

/*************************************************************************
This function is used to peek into hierarchical RBF  construction  process
from  some  other  thread  and  get current progress indicator. It returns
value in [0,1].

IMPORTANT: only HRBFs (hierarchical RBFs) support  peeking  into  progress
           indicator. Legacy RBF-ML and RBF-QNN do  not  support  it.  You
           will always get 0 value.

INPUT PARAMETERS:
    S           -   RBF model object

RESULT:
    progress value, in [0,1]

  -- ALGLIB --
     Copyright 17.11.2018 by Bochkanov Sergey
*************************************************************************/
double rbfpeekprogress(const rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfrequesttermination` function

/*************************************************************************
This function  is  used  to  submit  a  request  for  termination  of  the
hierarchical RBF construction process from some other thread.  As  result,
RBF construction is terminated smoothly (with proper deallocation  of  all
necessary resources) and resultant model is filled by zeros.

A rep.terminationtype=8 will be returned upon receiving such request.

IMPORTANT: only  HRBFs  (hierarchical  RBFs) support termination requests.
           Legacy RBF-ML and RBF-QNN do not  support  it.  An  attempt  to
           terminate their construction will be ignored.

IMPORTANT: termination request flag is cleared when the model construction
           starts. Thus, any pre-construction termination requests will be
           silently ignored - only ones submitted AFTER  construction  has
           actually began will be handled.

INPUT PARAMETERS:
    S           -   RBF model object

  -- ALGLIB --
     Copyright 17.11.2018 by Bochkanov Sergey
*************************************************************************/
void rbfrequesttermination(rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void rbfserialize(const rbfmodel &obj, std::string &s_out);
void rbfserialize(const rbfmodel &obj, std::ostream &s_out);

`rbfsetalgobiharmonic` function

/*************************************************************************
This  function  chooses  a  biharmonic DDM-RBF solver, a fast  RBF  solver
with f(r)=r as a basis function.

This algorithm has following important features:
* no tunable parameters
* C0 continuous RBF model (the model has discontinuous derivatives at  the
  interpolation nodes)
* fast model construction algorithm with O(N) memory and O(N*logN) running
  time requirements. Hundreds of thousands of points can be  handled  with
  this algorithm.
* accelerated evaluation using far field expansions  (aka  fast multipoles
  method) is supported. See rbffastcalc() for more information.
* controllable smoothing via optional nonlinearity penalty

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call
    LambdaV -   smoothing parameter, LambdaV>=0, defaults to 0.0:
                * LambdaV=0 means that no smoothing is applied,  i.e.  the
                  spline tries to pass through all dataset points exactly
                * LambdaV>0 means that a multiquadric spline is built with
                  larger  LambdaV   corresponding   to  models  with  less
                  nonlinearities.  Smoothing   spline   reproduces  target
                  values at nodes with small error; from the  other  side,
                  it is much more stable.
                  Recommended values:
                  * 1.0E-6 for minimal stability improving smoothing
                  * 1.0E-3 a good value to start experiments; first results
                    are visible
                  * 1.0 for strong smoothing

IMPORTANT: this model construction algorithm was introduced in ALGLIB 3.19
           and  produces  models  which  are  INCOMPATIBLE  with  previous
           versions of ALGLIB. You can  not  unserialize  models  produced
           with this function in ALGLIB 3.18 or earlier.

NOTE:      polyharmonic RBFs, including thin plate splines,  are  somewhat
           slower than compactly supported RBFs built with  HRBF algorithm
           due to the fact that non-compact basis function does not vanish
           far away from the nodes. From the other side, polyharmonic RBFs
           often produce much better results than HRBFs.

NOTE:      this algorithm supports specification of per-dimensional  radii
           via scale vector, which is set by means of rbfsetpointsandscales()
           function. This feature is useful if  you solve  spatio-temporal
           interpolation problems where different radii are  required  for
           spatial and temporal dimensions.

  -- ALGLIB --
     Copyright 12.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgobiharmonic(rbfmodel &s, const double lambdav, const xparams _xparams = alglib::xdefault);
void rbfsetalgobiharmonic(rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfsetalgohierarchical` function

/*************************************************************************
This function chooses HRBF solver, a 2nd version of ALGLIB RBFs.

This  algorithm is called Hierarchical RBF. It  similar  to  its  previous
incarnation, RBF-ML, i.e.  it  also  builds  a  sequence  of  models  with
decreasing radii. However, it uses more economical way of  building  upper
layers (ones with large radii), which results in faster model construction
and evaluation, as well as smaller memory footprint during construction.

This algorithm has following important features:
* ability to handle millions of points
* controllable smoothing via nonlinearity penalization
* support for specification of per-dimensional  radii  via  scale  vector,
  which is set by means of rbfsetpointsandscales() function. This  feature
  is useful if you solve  spatio-temporal  interpolation  problems,  where
  different radii are required for spatial and temporal dimensions.

Running times are roughly proportional to:
* N*log(N)*NLayers - for the model construction
* N*NLayers - for the model evaluation
You may see that running time does not depend on search radius  or  points
density, just on the number of layers in the hierarchy.

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call
    RBase   -   RBase parameter, RBase>0
    NLayers -   NLayers parameter, NLayers>0, recommended value  to  start
                with - about 5.
    LambdaNS-   >=0, nonlinearity penalty coefficient, negative values are
                not allowed. This parameter adds controllable smoothing to
                the problem, which may reduce noise. Specification of non-
                zero lambda means that in addition to fitting error solver
                will  also  minimize   LambdaNS*|S''(x)|^2  (appropriately
                generalized to multiple dimensions.

                Specification of exactly zero value means that no  penalty
                is added  (we  do  not  even  evaluate  matrix  of  second
                derivatives which is necessary for smoothing).

                Calculation of nonlinearity penalty is costly - it results
                in  several-fold  increase  of  model  construction  time.
                Evaluation time remains the same.

                Optimal  lambda  is  problem-dependent and requires  trial
                and  error.  Good  value to  start  from  is  1e-5...1e-6,
                which corresponds to slightly noticeable smoothing  of the
                function.  Value  1e-2  usually  means  that  quite  heavy
                smoothing is applied.

TUNING ALGORITHM

In order to use this algorithm you have to choose three parameters:
* initial radius RBase
* number of layers in the model NLayers
* penalty coefficient LambdaNS

Initial radius is easy to choose - you can pick any number  several  times
larger  than  the  average  distance between points. Algorithm won't break
down if you choose radius which is too large (model construction time will
increase, but model will be built correctly).

Choose such number of layers that RLast=RBase/2^(NLayers-1)  (radius  used
by  the  last  layer)  will  be  smaller than the typical distance between
points.  In  case  model  error  is  too large, you can increase number of
layers.  Having  more  layers  will make model construction and evaluation
proportionally slower, but it will allow you to have model which precisely
fits your data. From the other side, if you want to  suppress  noise,  you
can DECREASE number of layers to make your model less flexible (or specify
non-zero LambdaNS).

TYPICAL ERRORS

1. Using too small number of layers - RBF models with large radius are not
   flexible enough to reproduce small variations in the  target  function.
   You  need  many  layers  with  different radii, from large to small, in
   order to have good model.

2. Using  initial  radius  which  is  too  small.  You will get model with
   "holes" in the areas which are too far away from interpolation centers.
   However, algorithm will work correctly (and quickly) in this case.

  -- ALGLIB --
     Copyright 20.06.2016 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgohierarchical(rbfmodel &s, const double rbase, const ae_int_t nlayers, const double lambdans, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`rbfsetalgomultilayer` function

/*************************************************************************
DEPRECATED: this function is deprecated. ALGLIB  includes  new  RBF  model
            construction algorithms: DDM-RBF (since version 3.19) and HRBF
            (since version 3.11).

  -- ALGLIB --
     Copyright 02.03.2012 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgomultilayer(rbfmodel &s, const double rbase, const ae_int_t nlayers, const double lambdav, const xparams _xparams = alglib::xdefault);
void rbfsetalgomultilayer(rbfmodel &s, const double rbase, const ae_int_t nlayers, const xparams _xparams = alglib::xdefault);

`rbfsetalgomultiquadricauto` function

/*************************************************************************
This function chooses a multiquadric DDM-RBF solver,  a  fast  RBF  solver
with f(r)=sqrt(r^2+Alpha^2)  as  a  basis  function,  with   Alpha   being
automatically determined.

This algorithm has following important features:
* easy setup - no need to tune Alpha, good value is automatically assigned
* C2 continuous RBF model
* fast  model construction algorithm with O(N) memory and  O(N^2)  running
  time requirements. Hundreds of thousands of points can be  handled  with
  this algorithm.
* controllable smoothing via optional nonlinearity penalty

This algorithm automatically selects Alpha  as  a  mean  distance  to  the
nearest neighbor (ignoring neighbors that are too close).

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call
    LambdaV -   smoothing parameter, LambdaV>=0, defaults to 0.0:
                * LambdaV=0 means that no smoothing is applied,  i.e.  the
                  spline tries to pass through all dataset points exactly
                * LambdaV>0 means that a multiquadric spline is built with
                  larger  LambdaV   corresponding   to  models  with  less
                  nonlinearities.  Smoothing   spline   reproduces  target
                  values at nodes with small error; from the  other  side,
                  it is much more stable.
                  Recommended values:
                  * 1.0E-6 for minimal stability improving smoothing
                  * 1.0E-3 a good value to start experiments; first results
                    are visible
                  * 1.0 for strong smoothing

IMPORTANT: this model construction algorithm was introduced in ALGLIB 3.19
           and  produces  models  which  are  INCOMPATIBLE  with  previous
           versions of ALGLIB. You can  not  unserialize  models  produced
           with this function in ALGLIB 3.18 or earlier.

NOTE:      polyharmonic RBFs, including thin plate splines,  are  somewhat
           slower than compactly supported RBFs built with  HRBF algorithm
           due to the fact that non-compact basis function does not vanish
           far away from the nodes. From the other side, polyharmonic RBFs
           often produce much better results than HRBFs.

NOTE:      this algorithm supports specification of per-dimensional  radii
           via scale vector, which is set by means of rbfsetpointsandscales()
           function. This feature is useful if  you solve  spatio-temporal
           interpolation problems where different radii are  required  for
           spatial and temporal dimensions.

  -- ALGLIB --
     Copyright 12.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgomultiquadricauto(rbfmodel &s, const double lambdav, const xparams _xparams = alglib::xdefault);
void rbfsetalgomultiquadricauto(rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfsetalgomultiquadricmanual` function

/*************************************************************************
This function chooses a multiquadric DDM-RBF solver,  a  fast  RBF  solver
with f(r)=sqrt(r^2+Alpha^2) as a basis function,  with  manual  choice  of
the scale parameter Alpha.

This algorithm has following important features:
* C2 continuous RBF model (when Alpha>0 is used; for Alpha=0 the model  is
  merely C0 continuous)
* fast  model construction algorithm with O(N) memory and  O(N^2)  running
  time requirements. Hundreds of thousands of points can be  handled  with
  this algorithm.
* controllable smoothing via optional nonlinearity penalty

One important point is that  this  algorithm  includes  tunable  parameter
Alpha, which should be carefully chosen. Selecting too  large  value  will
result in extremely badly  conditioned  problems  (interpolation  accuracy
may degrade up to complete breakdown) whilst selecting too small value may
produce models that are precise but nearly nonsmooth at the nodes.

Good value to  start  from  is  mean  distance  between  nodes. Generally,
choosing too small Alpha is better than choosing too large - in the former
case you still have model that reproduces target values at the nodes.

In most cases, better option is to choose good Alpha automatically - it is
done by another version of the same algorithm that is activated by calling
rbfsetalgomultiquadricauto() method.

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call
    Alpha   -   basis function parameter, Alpha>=0:
                * Alpha>0  means that multiquadric algorithm is used which
                  produces C2-continuous RBF model
                * Alpha=0  means that the multiquadric kernel  effectively
                  becomes a biharmonic one: f=r. As a  result,  the  model
                  becomes nonsmooth at nodes, and hence is C0 continuous
    LambdaV -   smoothing parameter, LambdaV>=0, defaults to 0.0:
                * LambdaV=0 means that no smoothing is applied,  i.e.  the
                  spline tries to pass through all dataset points exactly
                * LambdaV>0 means that a multiquadric spline is built with
                  larger  LambdaV   corresponding   to  models  with  less
                  nonlinearities.  Smoothing   spline   reproduces  target
                  values at nodes with small error; from the  other  side,
                  it is much more stable.
                  Recommended values:
                  * 1.0E-6 for minimal stability improving smoothing
                  * 1.0E-3 a good value to start experiments; first results
                    are visible
                  * 1.0 for strong smoothing

IMPORTANT: this model construction algorithm was introduced in ALGLIB 3.19
           and  produces  models  which  are  INCOMPATIBLE  with  previous
           versions of ALGLIB. You can  not  unserialize  models  produced
           with this function in ALGLIB 3.18 or earlier.

NOTE:      polyharmonic RBFs, including thin plate splines,  are  somewhat
           slower than compactly supported RBFs built with  HRBF algorithm
           due to the fact that non-compact basis function does not vanish
           far away from the nodes. From the other side, polyharmonic RBFs
           often produce much better results than HRBFs.

NOTE:      this algorithm supports specification of per-dimensional  radii
           via scale vector, which is set by means of rbfsetpointsandscales()
           function. This feature is useful if  you solve  spatio-temporal
           interpolation problems where different radii are  required  for
           spatial and temporal dimensions.

  -- ALGLIB --
     Copyright 12.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgomultiquadricmanual(rbfmodel &s, const double alpha, const double lambdav, const xparams _xparams = alglib::xdefault);
void rbfsetalgomultiquadricmanual(rbfmodel &s, const double alpha, const xparams _xparams = alglib::xdefault);

`rbfsetalgoqnn` function

/*************************************************************************
DEPRECATED: this function is deprecated. ALGLIB  includes  new  RBF  model
            construction algorithms: DDM-RBF (since version 3.19) and HRBF
            (since version 3.11).

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgoqnn(rbfmodel &s, const double q, const double z, const xparams _xparams = alglib::xdefault);
void rbfsetalgoqnn(rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfsetalgothinplatespline` function

/*************************************************************************
This function chooses a thin plate  spline  DDM-RBF  solver,  a  fast  RBF
solver with f(r)=r^2*ln(r) basis function.

This algorithm has following important features:
* easy setup - no tunable parameters
* C1 continuous RBF model (gradient is defined everywhere, but Hessian  is
  undefined at nodes), high-quality interpolation
* fast  model construction algorithm with O(N) memory and  O(N^2)  running
  time requirements. Hundreds of thousands of points can be  handled  with
  this algorithm.
* controllable smoothing via optional nonlinearity penalty

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call
    LambdaV -   smoothing parameter, LambdaV>=0, defaults to 0.0:
                * LambdaV=0 means that no smoothing is applied,  i.e.  the
                  spline tries to pass through all dataset points exactly
                * LambdaV>0 means that a smoothing thin  plate  spline  is
                  built, with larger LambdaV corresponding to models  with
                  less nonlinearities. Smoothing spline reproduces  target
                  values at nodes with small error; from the  other  side,
                  it is much more stable.
                  Recommended values:
                  * 1.0E-6 for minimal stability improving smoothing
                  * 1.0E-3 a good value to start experiments; first results
                    are visible
                  * 1.0 for strong smoothing

IMPORTANT: this model construction algorithm was introduced in ALGLIB 3.19
           and  produces  models  which  are  INCOMPATIBLE  with  previous
           versions of ALGLIB. You can  not  unserialize  models  produced
           with this function in ALGLIB 3.18 or earlier.

NOTE:      polyharmonic RBFs, including thin plate splines,  are  somewhat
           slower than compactly supported RBFs built with  HRBF algorithm
           due to the fact that non-compact basis function does not vanish
           far away from the nodes. From the other side, polyharmonic RBFs
           often produce much better results than HRBFs.

NOTE:      this algorithm supports specification of per-dimensional  radii
           via scale vector, which is set by means of rbfsetpointsandscales()
           function. This feature is useful if  you solve  spatio-temporal
           interpolation problems where different radii are  required  for
           spatial and temporal dimensions.

  -- ALGLIB --
     Copyright 12.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbfsetalgothinplatespline(rbfmodel &s, const double lambdav, const xparams _xparams = alglib::xdefault);
void rbfsetalgothinplatespline(rbfmodel &s, const xparams _xparams = alglib::xdefault);

`rbfsetconstterm` function

/*************************************************************************
This function sets constant term (model is a sum of radial basis functions
plus constant).  This  function  won't  have  effect  until  next  call to
RBFBuildModel().

IMPORTANT: thin plate splines require  polynomial term to be  linear,  not
           constant,  in  order  to  provide   interpolation   guarantees.
           Although  failures  are  exceptionally  rare,  some  small  toy
           problems may result in degenerate linear systems. Thus,  it  is
           advised to use linear term when one fits data with TPS.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetconstterm(rbfmodel &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rbfsetfastevaltol` function

/*************************************************************************
This function sets absolute accuracy of a  fast evaluation  algorithm used
by rbffastcalc() and other fast evaluation functions.

A fast evaluation algorithm is model-dependent and is available  only  for
some RBF models. Usually it utilizes far field expansions (a generalization
of the fast multipoles  method).  If  no  approximate  fast  evaluator  is
available for the  current RBF model type, this function has no effect.

NOTE: this function can be called before or after the model was built. The
      result will be the same.

NOTE: this  function  has  O(N) running time, where N is a  points  count.
      Most fast evaluators work by aggregating influence of  point groups,
      i.e. by computing so called far field. Changing evaluator  tolerance
      means that far field radii have to  be  recomputed  for  each  point
      cluster, and we have O(N) such clusters.

      This function is still very fast, but  it  should  not be called too
      often, e.g. every time you call rbffastcalc() in a loop.

NOTE: the tolerance  set  by this function is an accuracy of an  evaluator
      which computes the value of the model. It is  NOT  accuracy  of  the
      model itself.

      E.g., if you set evaluation accuracy to 1E-12, the model value  will
      be computed with required precision. However, the model itself is an
      approximation of the target (the default requirement is to fit model
      with ~6 digits of precision) and THIS accuracy can  not  be  changed
      after the model was built.

IMPORTANT: THIS FUNCTION IS THREAD-UNSAFE. Calling it while another thread
           tries to use rbffastcalc() is unsafe because it means that  the
           accuracy requirements will change in the middle of computations.
           The algorithm may behave unpredictably.

INPUT PARAMETERS:
    S       -   RBF model
    TOL     -   TOL>0, desired evaluation tolerance:
                * should be somewhere between 1E-3 and 1E-6
                * values outside of this range will cause no problems (the
                  evaluator will do the job anyway). However,  too  strict
                  precision requirements may mean  that  no  approximation
                  speed-up will be achieved.

  -- ALGLIB --
     Copyright 19.09.2022 by Bochkanov Sergey
*************************************************************************/
void rbfsetfastevaltol(rbfmodel &s, const double tol, const xparams _xparams = alglib::xdefault);

`rbfsetlinterm` function

/*************************************************************************
This function sets linear term (model is a sum of radial  basis  functions
plus linear polynomial). This function won't have effect until  next  call
to RBFBuildModel().

Using linear term is a default option and it is the best one - it provides
best convergence guarantees for all RBF model  types: legacy  RBF-QNN  and
RBF-ML, Gaussian HRBFs and all types of DDM-RBF models.

Other options, like constant or zero term, work for HRBFs,  almost  always
work for DDM-RBFs but provide no stability  guarantees  in the latter case
(e.g. the solver may fail on some carefully prepared problems).

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetlinterm(rbfmodel &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rbfsetpoints` function

/*************************************************************************
This function adds dataset.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

IMPORTANT: ALGLIB version 3.11 and later allows you to specify  a  set  of
           per-dimension scales. Interpolation radii are multiplied by the
           scale vector. It may be useful if you have mixed spatio-temporal
           data (say, a set of 3D slices recorded at different times).
           You should call rbfsetpointsandscales() function  to  use  this
           feature.

INPUT PARAMETERS:
    S       -   RBF model, initialized by rbfcreate() call.
    XY      -   points, array[N,NX+NY]. One row corresponds to  one  point
                in the dataset. First NX elements  are  coordinates,  next
                NY elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset

After you've added dataset and (optionally) tuned algorithm  settings  you
should call rbfbuildmodel() in order to build a model for you.

NOTE: dataset added by this function is not saved during model serialization.
      MODEL ITSELF is serialized, but data used to build it are not.

      So, if you 1) add dataset to  empty  RBF  model,  2)  serialize  and
      unserialize it, then you will get an empty RBF model with no dataset
      being attached.

      From the other side, if you call rbfbuildmodel() between (1) and (2),
      then after (2) you will get your fully constructed RBF model  -  but
      again with no dataset attached, so subsequent calls to rbfbuildmodel()
      will produce empty model.


  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetpoints(rbfmodel &s, const real_2d_array &xy, const ae_int_t n, const xparams _xparams = alglib::xdefault);
void rbfsetpoints(rbfmodel &s, const real_2d_array &xy, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`rbfsetpointsandscales` function

/*************************************************************************
This function adds dataset and a vector of per-dimension scales.

It may be useful if you have mixed spatio-temporal data - say, a set of 3D
slices recorded at different times. Such data typically require  different
RBF radii for spatial and temporal dimensions. ALGLIB solves this  problem
by specifying single RBF radius, which is (optionally) multiplied  by  the
scale vector.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

IMPORTANT: only modern RBF algorithms  support  variable  scaling.  Legacy
           algorithms like RBF-ML or QNN algorithms  will  result  in   -3
           completion code being returned (incorrect algorithm).

INPUT PARAMETERS:
    R       -   RBF model, initialized by rbfcreate() call.
    XY      -   points, array[N,NX+NY]. One row corresponds to  one  point
                in the dataset. First NX elements  are  coordinates,  next
                NY elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset
    S       -   array[NX], scale vector, S[i]>0.

After you've added dataset and (optionally) tuned algorithm  settings  you
should call rbfbuildmodel() in order to build a model for you.

NOTE: dataset added by this function is not saved during model serialization.
      MODEL ITSELF is serialized, but data used to build it are not.

      So, if you 1) add dataset to  empty  RBF  model,  2)  serialize  and
      unserialize it, then you will get an empty RBF model with no dataset
      being attached.

      From the other side, if you call rbfbuildmodel() between (1) and (2),
      then after (2) you will get your fully constructed RBF model  -  but
      again with no dataset attached, so subsequent calls to rbfbuildmodel()
      will produce empty model.


  -- ALGLIB --
     Copyright 20.06.2016 by Bochkanov Sergey
*************************************************************************/
void rbfsetpointsandscales(rbfmodel &r, const real_2d_array &xy, const ae_int_t n, const real_1d_array &s, const xparams _xparams = alglib::xdefault);
void rbfsetpointsandscales(rbfmodel &r, const real_2d_array &xy, const real_1d_array &s, const xparams _xparams = alglib::xdefault);

`rbfsetv2bf` function

/*************************************************************************
This function sets basis function type, which can be:
* 0 for classic Gaussian
* 1 for fast and compact bell-like basis function, which  becomes  exactly
  zero at distance equal to 3*R (default option).

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    BF      -   basis function type:
                * 0 - classic Gaussian
                * 1 - fast and compact one

  -- ALGLIB --
     Copyright 01.02.2017 by Bochkanov Sergey
*************************************************************************/
void rbfsetv2bf(rbfmodel &s, const ae_int_t bf, const xparams _xparams = alglib::xdefault);

`rbfsetv2its` function

/*************************************************************************
This function sets stopping criteria of the underlying linear  solver  for
hierarchical (version 2) RBF constructor.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    MaxIts  -   this criterion will stop algorithm after MaxIts iterations.
                Typically a few hundreds iterations is required,  with 400
                being a good default value to start experimentation.
                Zero value means that default value will be selected.

  -- ALGLIB --
     Copyright 01.02.2017 by Bochkanov Sergey
*************************************************************************/
void rbfsetv2its(rbfmodel &s, const ae_int_t maxits, const xparams _xparams = alglib::xdefault);

`rbfsetv2supportr` function

/*************************************************************************
This function sets support radius parameter  of  hierarchical  (version 2)
RBF constructor.

Hierarchical RBF model achieves great speed-up  by removing from the model
excessive (too dense) nodes. Say, if you have RBF radius equal to 1 meter,
and two nodes are just 1 millimeter apart, you  may  remove  one  of  them
without reducing model quality.

Support radius parameter is used to justify which points need removal, and
which do not. If two points are less than  SUPPORT_R*CUR_RADIUS  units  of
distance apart, one of them is removed from the model. The larger  support
radius  is, the faster model  construction  AND  evaluation are.  However,
too large values result in "bumpy" models.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    R       -   support radius coefficient, >=0.
                Recommended values are [0.1,0.4] range, with 0.1 being
                default value.

  -- ALGLIB --
     Copyright 01.02.2017 by Bochkanov Sergey
*************************************************************************/
void rbfsetv2supportr(rbfmodel &s, const double r, const xparams _xparams = alglib::xdefault);

`rbfsetv3tol` function

/*************************************************************************
This function sets desired accuracy for a version 3 RBF model.

As of ALGLIB 3.20.0, version 3 models include biharmonic RBFs, thin  plate
splines, multiquadrics.

Version 3 models are fit  with  specialized  domain  decomposition  method
which splits problem into smaller  chunks.  Models  with  size  less  than
the DDM chunk size are computed nearly exactly in one step. Larger  models
are built with an iterative linear solver. This function controls accuracy
of the solver.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call
    TOL     -   desired precision:
                * must be non-negative
                * should be somewhere between 0.001 and 0.000001
                * values higher than 0.001 make little sense   -  you  may
                  lose a lot of precision with no performance gains.
                * values below 1E-6 usually require too much time to converge,
                  so they are silenly replaced by a 1E-6 cutoff value. Thus,
                  zero can be used to denote 'maximum precision'.

  -- ALGLIB --
     Copyright 01.10.2022 by Bochkanov Sergey
*************************************************************************/
void rbfsetv3tol(rbfmodel &s, const double tol, const xparams _xparams = alglib::xdefault);

`rbfsetzeroterm` function

/*************************************************************************
This  function  sets  zero  term (model is a sum of radial basis functions
without polynomial term). This function won't have effect until next  call
to RBFBuildModel().

IMPORTANT: only  Gaussian  RBFs  (HRBF  algorithm)  provide  interpolation
           guarantees when no polynomial term is used.  Most  other  RBFs,
           including   biharmonic  splines,   thin   plate   splines   and
           multiquadrics, require at least constant term  (biharmonic  and
           multiquadric) or linear one (thin plate splines)  in  order  to
           guarantee non-degeneracy of linear systems being solved.

           Although  failures  are  exceptionally  rare,  some  small  toy
           problems still may result in degenerate linear systems. Thus,it
           is advised to use constant/linear term, unless one is 100% sure
           that he needs zero term.

INPUT PARAMETERS:
    S       -   RBF model, initialized by RBFCreate() call

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfsetzeroterm(rbfmodel &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rbftscalcbuf` function

/*************************************************************************
This function calculates values of the RBF model at the given point, using
external  buffer  object  (internal  temporaries  of  RBF  model  are  not
modified).

This function allows to use same RBF model object  in  different  threads,
assuming  that  different   threads  use  different  instances  of  buffer
structure.

INPUT PARAMETERS:
    S       -   RBF model, may be shared between different threads
    Buf     -   buffer object created for this particular instance of  RBF
                model with rbfcreatecalcbuffer().
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y       -   possibly preallocated array

OUTPUT PARAMETERS:
    Y       -   function value, array[NY]. Y is not reallocated when it
                is larger than NY.

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbftscalcbuf(const rbfmodel &s, rbfcalcbuffer &buf, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`rbftsdiffbuf` function

/*************************************************************************
This function calculates values of the RBF model and  its  derivatives  at
the given point, using external buffer object (internal temporaries of the
RBF model are not modified).

This function allows to use same RBF model object  in  different  threads,
assuming  that  different   threads  use different instances of the buffer
structure.

This function returns 0.0 in Y and/or DY in the following cases:
* the model is not initialized (Y=0, DY=0)
* the gradient is undefined at the trial point. Some basis  functions have
  discontinuous derivatives at the interpolation nodes:
  * biharmonic splines f=r have no Hessian and no gradient at the nodes
  In these cases only DY is set to zero (Y is still returned)

INPUT PARAMETERS:
    S       -   RBF model, may be shared between different threads
    Buf     -   buffer object created for this particular instance of  RBF
                model with rbfcreatecalcbuffer().
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y, DY   -   possibly preallocated arrays; if array size is large enough
                to store results, this function does not  reallocate  array
                to fit output size exactly.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY].
    DY      -   derivatives, array[NX*NY]:
                * Y[I*NX+J] with 0<=I<NY and 0<=J<NX  stores derivative of
                  function component I with respect to input J.
                * for NY=1 it is simply NX-dimensional gradient of the
                  scalar NX-dimensional function
                Zero is returned when the first derivative is undefined.

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbftsdiffbuf(const rbfmodel &s, rbfcalcbuffer &buf, const real_1d_array &x, real_1d_array &y, real_1d_array &dy, const xparams _xparams = alglib::xdefault);

`rbftshessbuf` function

/*************************************************************************
This function calculates values of the RBF model and  its first and second
derivatives (Hessian matrix) at the given  point,  using  external  buffer
object (internal temporaries of the RBF  model  are  not  modified).

This function allows to use same RBF model object  in  different  threads,
assuming  that  different   threads  use different instances of the buffer
structure.

This function returns 0 in Y and/or DY and/or D2Y in the following cases:
* the model is not initialized (Y=0, DY=0, D2Y=0)
* the gradient and/or Hessian is undefined at the trial point.  Some basis
  functions have discontinuous derivatives at the interpolation nodes:
  * thin plate splines have no Hessian at the nodes
  * biharmonic splines f=r have no Hessian and no gradient at the  nodes
  In these cases only corresponding derivative is set  to  zero,  and  the
  rest of the derivatives is still returned.

INPUT PARAMETERS:
    S       -   RBF model, may be shared between different threads
    Buf     -   buffer object created for this particular instance of  RBF
                model with rbfcreatecalcbuffer().
    X       -   coordinates, array[NX].
                X may have more than NX elements, in this case only
                leading NX will be used.
    Y,DY,D2Y-   possible preallocated output arrays. If these  arrays  are
                smaller than  required  to  store  the  result,  they  are
                automatically reallocated. If array is large enough, it is
                not resized.

OUTPUT PARAMETERS:
    Y       -   function value, array[NY].
    DY      -   first derivatives, array[NY*NX]:
                * Y[I*NX+J] with 0<=I<NY and 0<=J<NX  stores derivative of
                  function component I with respect to input J.
                * for NY=1 it is simply NX-dimensional gradient of the
                  scalar NX-dimensional function
                Zero is returned when the first derivative is undefined.
    D2Y     -   second derivatives, array[NY*NX*NX]:
                * for NY=1 it is NX*NX array that stores  Hessian  matrix,
                  with Y[I*NX+J]=Y[J*NX+I].
                * for  a  vector-valued  RBF  with  NY>1  it  contains  NY
                  subsequently stored Hessians: an element Y[K*NX*NX+I*NX+J]
                  with  0<=K<NY,  0<=I<NX  and  0<=J<NX    stores   second
                  derivative of the function #K  with  respect  to  inputs
                  #I and #J.
                Zero is returned when the second derivative is undefined.

  -- ALGLIB --
     Copyright 13.12.2021 by Bochkanov Sergey
*************************************************************************/
void rbftshessbuf(const rbfmodel &s, rbfcalcbuffer &buf, const real_1d_array &x, real_1d_array &y, real_1d_array &dy, real_1d_array &d2y, const xparams _xparams = alglib::xdefault);

`rbfunpack` function

/*************************************************************************
This function "unpacks" RBF model by extracting its coefficients.

INPUT PARAMETERS:
    S       -   RBF model

OUTPUT PARAMETERS:
    NX      -   dimensionality of argument
    NY      -   dimensionality of the target function
    XWR     -   model  information ,  2D  array.  One  row  of  the  array
                corresponds to one basis function.

                For ModelVersion=1 we have NX+NY+1 columns:
                * first NX columns  - coordinates of the center
                * next  NY columns  - weights, one per dimension of the
                                      function being modeled
                * last column       - radius, same for all dimensions of
                                      the function being modeled

                For ModelVersion=2 we have NX+NY+NX columns:
                * first NX columns  - coordinates of the center
                * next  NY columns  - weights, one per dimension of the
                                      function being modeled
                * last NX columns   - radii, one per dimension

                For ModelVersion=3 we have NX+NY+NX+3 columns:
                * first NX columns  - coordinates of the center
                * next  NY columns  - weights, one per dimension of the
                                      function being modeled
                * next NX columns   - radii, one per dimension
                * next column       - basis function type:
                                      * 1  for f=r
                                      * 2  for f=r^2*ln(r)
                                      * 10 for multiquadric f=sqrt(r^2+alpha^2)
                * next column       - basis function parameter:
                                      * alpha, for basis function type 10
                                      * ignored (zero) for other basis function types
                * next column       - point index in the original dataset,
                                      or -1 for an artificial node created
                                      by the solver. The algorithm may reorder
                                      the nodes, drop some nodes or add
                                      artificial nodes. Thus, one parsing
                                      this column should expect all these
                                      kinds of alterations in the dataset.

    NC      -   number of the centers
    V       -   polynomial  term , array[NY,NX+1]. One row per one
                dimension of the function being modelled. First NX
                elements are linear coefficients, V[NX] is equal to the
                constant part.
    ModelVersion-version of the RBF model:
                * 1 - for models created by QNN and RBF-ML algorithms,
                  compatible with ALGLIB 3.10 or earlier.
                * 2 - for models created by HierarchicalRBF, requires
                  ALGLIB 3.11 or later
                * 3 - for models created by DDM-RBF, requires
                  ALGLIB 3.19 or later

  -- ALGLIB --
     Copyright 13.12.2011 by Bochkanov Sergey
*************************************************************************/
void rbfunpack(rbfmodel &s, ae_int_t &nx, ae_int_t &ny, real_2d_array &xwr, ae_int_t &nc, real_2d_array &v, ae_int_t &modelversion, const xparams _xparams = alglib::xdefault);

Examples: [1]

`rbfunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void rbfunserialize(const std::string &s_in, rbfmodel &obj);
void rbfunserialize(const std::istream &s_in, rbfmodel &obj);

rbf_d_hrbf example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example illustrates basic concepts of the RBF models: creation, modification,
        // evaluation.
        // 
        // Suppose that we have set of 2-dimensional points with associated
        // scalar function values, and we want to build a RBF model using
        // our data.
        // 
        // NOTE: we can work with 3D models too :)
        // 
        // Typical sequence of steps is given below:
        // 1. we create RBF model object
        // 2. we attach our dataset to the RBF model and tune algorithm settings
        // 3. we rebuild RBF model using QNN algorithm on new data
        // 4. we use RBF model (evaluate, serialize, etc.)
        //
        double v;

        //
        // Step 1: RBF model creation.
        //
        // We have to specify dimensionality of the space (2 or 3) and
        // dimensionality of the function (scalar or vector).
        //
        // New model is empty - it can be evaluated,
        // but we just get zero value at any point.
        //
        rbfmodel model;
        rbfcreate(2, 1, model);

        v = rbfcalc2(model, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 0.000

        //
        // Step 2: we add dataset.
        //
        // XY contains two points - x0=(-1,0) and x1=(+1,0) -
        // and two function values f(x0)=2, f(x1)=3.
        //
        // We added points, but model was not rebuild yet.
        // If we call rbfcalc2(), we still will get 0.0 as result.
        //
        real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
        rbfsetpoints(model, xy);

        v = rbfcalc2(model, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 0.000

        //
        // Step 3: rebuild model
        //
        // After we've configured model, we should rebuild it -
        // it will change coefficients stored internally in the
        // rbfmodel structure.
        //
        // We use hierarchical RBF algorithm with following parameters:
        // * RBase - set to 1.0
        // * NLayers - three layers are used (although such simple problem
        //   does not need more than 1 layer)
        // * LambdaReg - is set to zero value, no smoothing is required
        //
        rbfreport rep;
        rbfsetalgohierarchical(model, 1.0, 3, 0.0);
        rbfbuildmodel(model, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

        //
        // Step 4: model was built
        //
        // After call of rbfbuildmodel(), rbfcalc2() will return
        // value of the new model.
        //
        v = rbfcalc2(model, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.500
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

rbf_d_polterm example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example show how to work with polynomial term
        // 
        // Suppose that we have set of 2-dimensional points with associated
        // scalar function values, and we want to build a RBF model using
        // our data.
        //
        // We use hierarchical RBF algorithm with following parameters:
        // * RBase - set to 1.0
        // * NLayers - three layers are used (although such simple problem
        //   does not need more than 1 layer)
        // * LambdaReg - is set to zero value, no smoothing is required
        //
        double v;
        rbfmodel model;
        real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
        rbfreport rep;

        rbfcreate(2, 1, model);
        rbfsetpoints(model, xy);
        rbfsetalgohierarchical(model, 1.0, 3, 0.0);

        //
        // By default, RBF model uses linear term. It means that model
        // looks like
        //     f(x,y) = SUM(RBF[i]) + a*x + b*y + c
        // where RBF[i] is I-th radial basis function and a*x+by+c is a
        // linear term. Having linear terms in a model gives us:
        // (1) improved extrapolation properties
        // (2) linearity of the model when data can be perfectly fitted
        //     by the linear function
        // (3) linear asymptotic behavior
        //
        // Our simple dataset can be modelled by the linear function
        //     f(x,y) = 0.5*x + 2.5
        // and rbfbuildmodel() with default settings should preserve this
        // linearity.
        //
        ae_int_t nx;
        ae_int_t ny;
        ae_int_t nc;
        ae_int_t modelversion;
        real_2d_array xwr;
        real_2d_array c;
        rbfbuildmodel(model, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
        printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.500,0.000,2.500]]

        // asymptotic behavior of our function is linear
        v = rbfcalc2(model, 1000.0, 0.0);
        printf("%.1f\n", double(v)); // EXPECTED: 502.50

        //
        // Instead of linear term we can use constant term. In this case
        // we will get model which has form
        //     f(x,y) = SUM(RBF[i]) + c
        // where RBF[i] is I-th radial basis function and c is a constant,
        // which is equal to the average function value on the dataset.
        //
        // Because we've already attached dataset to the model the only
        // thing we have to do is to call rbfsetconstterm() and then
        // rebuild model with rbfbuildmodel().
        //
        rbfsetconstterm(model);
        rbfbuildmodel(model, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
        printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.000,0.000,2.500]]

        // asymptotic behavior of our function is constant
        v = rbfcalc2(model, 1000.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.500

        //
        // Finally, we can use zero term. Just plain RBF without polynomial
        // part:
        //     f(x,y) = SUM(RBF[i])
        // where RBF[i] is I-th radial basis function.
        //
        rbfsetzeroterm(model);
        rbfbuildmodel(model, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
        rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
        printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.000,0.000,0.000]]

        // asymptotic behavior of our function is just zero constant
        v = rbfcalc2(model, 1000.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 0.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

rbf_d_serialize example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example show how to serialize and unserialize RBF model
        // 
        // Suppose that we have set of 2-dimensional points with associated
        // scalar function values, and we want to build a RBF model using
        // our data. Then we want to serialize it to string and to unserialize
        // from string, loading to another instance of RBF model.
        //
        // Here we assume that you already know how to create RBF models.
        //
        std::string s;
        double v;
        rbfmodel model0;
        rbfmodel model1;
        real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
        rbfreport rep;

        // model initialization
        rbfcreate(2, 1, model0);
        rbfsetpoints(model0, xy);
        rbfsetalgohierarchical(model0, 1.0, 3, 0.0);
        rbfbuildmodel(model0, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

        //
        // Serialization - it looks easy,
        // but you should carefully read next section.
        //
        alglib::rbfserialize(model0, s);
        alglib::rbfunserialize(s, model1);

        // both models return same value
        v = rbfcalc2(model0, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.500
        v = rbfcalc2(model1, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.500

        //
        // Previous section shows that model state is saved/restored during
        // serialization. However, some properties are NOT serialized.
        //
        // Serialization saves/restores RBF model, but it does NOT saves/restores
        // settings which were used to build current model. In particular, dataset
        // which was used to build model, is not preserved.
        //
        // What does it mean in for us?
        //
        // Do you remember this sequence: rbfcreate-rbfsetpoints-rbfbuildmodel?
        // First step creates model, second step adds dataset and tunes model
        // settings, third step builds model using current dataset and model
        // construction settings.
        //
        // If you call rbfbuildmodel() without calling rbfsetpoints() first, you
        // will get empty (zero) RBF model. In our example, model0 contains
        // dataset which was added by rbfsetpoints() call. However, model1 does
        // NOT contain dataset - because dataset is NOT serialized.
        //
        // This, if we call rbfbuildmodel(model0,rep), we will get same model,
        // which returns 2.5 at (x,y)=(0,0). However, after same call model1 will
        // return zero - because it contains RBF model (coefficients), but does NOT
        // contain dataset which was used to build this model.
        //
        // Basically, it means that:
        // * serialization of the RBF model preserves anything related to the model
        //   EVALUATION
        // * but it does NOT creates perfect copy of the original object.
        //
        rbfbuildmodel(model0, rep);
        v = rbfcalc2(model0, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.500

        rbfbuildmodel(model1, rep);
        v = rbfcalc2(model1, 0.0, 0.0);
        printf("%.2f\n", double(v)); // EXPECTED: 0.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

rbf_d_vector example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Suppose that we have set of 2-dimensional points with associated VECTOR
        // function values, and we want to build a RBF model using our data.
        // 
        // Typical sequence of steps is given below:
        // 1. we create RBF model object
        // 2. we attach our dataset to the RBF model and tune algorithm settings
        // 3. we rebuild RBF model using new data
        // 4. we use RBF model (evaluate, serialize, etc.)
        //
        real_1d_array x;
        real_1d_array y;

        //
        // Step 1: RBF model creation.
        //
        // We have to specify dimensionality of the space (equal to 2) and
        // dimensionality of the function (2-dimensional vector function).
        //
        // New model is empty - it can be evaluated,
        // but we just get zero value at any point.
        //
        rbfmodel model;
        rbfcreate(2, 2, model);

        x = "[+1,+1]";
        rbfcalc(model, x, y);
        printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,0.000]

        //
        // Step 2: we add dataset.
        //
        // XY arrays containt four points:
        // * (x0,y0) = (+1,+1), f(x0,y0)=(0,-1)
        // * (x1,y1) = (+1,-1), f(x1,y1)=(-1,0)
        // * (x2,y2) = (-1,-1), f(x2,y2)=(0,+1)
        // * (x3,y3) = (-1,+1), f(x3,y3)=(+1,0)
        //
        real_2d_array xy = "[[+1,+1,0,-1],[+1,-1,-1,0],[-1,-1,0,+1],[-1,+1,+1,0]]";
        rbfsetpoints(model, xy);

        // We added points, but model was not rebuild yet.
        // If we call rbfcalc(), we still will get 0.0 as result.
        rbfcalc(model, x, y);
        printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,0.000]

        //
        // Step 3: rebuild model
        //
        // We use hierarchical RBF algorithm with following parameters:
        // * RBase - set to 1.0
        // * NLayers - three layers are used (although such simple problem
        //   does not need more than 1 layer)
        // * LambdaReg - is set to zero value, no smoothing is required
        //
        // After we've configured model, we should rebuild it -
        // it will change coefficients stored internally in the
        // rbfmodel structure.
        //
        rbfreport rep;
        rbfsetalgohierarchical(model, 1.0, 3, 0.0);
        rbfbuildmodel(model, rep);
        printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1

        //
        // Step 4: model was built
        //
        // After call of rbfbuildmodel(), rbfcalc() will return
        // value of the new model.
        //
        rbfcalc(model, x, y);
        printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,-1.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`rcond` subpackage

Functions

cmatrixlurcond1
cmatrixlurcondinf
cmatrixrcond1
cmatrixrcondinf
cmatrixtrrcond1
cmatrixtrrcondinf
hpdmatrixcholeskyrcond
hpdmatrixrcond
rmatrixlurcond1
rmatrixlurcondinf
rmatrixrcond1
rmatrixrcond2
rmatrixrcond2rect
rmatrixrcondinf
rmatrixtrrcond1
rmatrixtrrcond2
rmatrixtrrcondinf
spdmatrixcholeskyrcond
spdmatrixrcond
spdmatrixrcond2

Examples

`cmatrixlurcond1` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA         -   LU decomposition of a matrix in compact form. Output of
                    the CMatrixLU subroutine.
    N           -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double cmatrixlurcond1(const complex_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`cmatrixlurcondinf` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition
(infinity norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA     -   LU decomposition of a matrix in compact form. Output of
                the CMatrixLU subroutine.
    N       -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double cmatrixlurcondinf(const complex_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`cmatrixrcond1` function

/*************************************************************************
Estimate of a matrix condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double cmatrixrcond1(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`cmatrixrcondinf` function

/*************************************************************************
Estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double cmatrixrcondinf(const complex_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`cmatrixtrrcond1` function

/*************************************************************************
Triangular matrix: estimate of a condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A       -   matrix. Array[0..N-1, 0..N-1].
    N       -   size of A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double cmatrixtrrcond1(const complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);

`cmatrixtrrcondinf` function

/*************************************************************************
Triangular matrix: estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double cmatrixtrrcondinf(const complex_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);

`hpdmatrixcholeskyrcond` function

/*************************************************************************
Condition number estimate of a Hermitian positive definite matrix given by
Cholesky decomposition.

The algorithm calculates a lower bound of the condition number. In this
case, the algorithm does not return a lower bound of the condition number,
but an inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    CD  - Cholesky decomposition of matrix A,
          output of SMatrixCholesky subroutine.
    N   - size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double hpdmatrixcholeskyrcond(const complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`hpdmatrixrcond` function

/*************************************************************************
Condition number estimate of a Hermitian positive definite matrix.

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm of condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    A       -   Hermitian positive definite matrix which is given by its
                upper or lower triangle depending on the value of
                IsUpper. Array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format.

Result:
    1/LowerBound(cond(A)), if matrix A is positive definite,
   -1, if matrix A is not positive definite, and its condition number
    could not be found by this algorithm.

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double hpdmatrixrcond(const complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`rmatrixlurcond1` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA         -   LU decomposition of a matrix in compact form. Output of
                    the RMatrixLU subroutine.
    N           -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixlurcond1(const real_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixlurcondinf` function

/*************************************************************************
Estimate of the condition number of a matrix given by its LU decomposition
(infinity norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    LUA     -   LU decomposition of a matrix in compact form. Output of
                the RMatrixLU subroutine.
    N       -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixlurcondinf(const real_2d_array &lua, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixrcond1` function

/*************************************************************************
Estimate of a matrix condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcond1(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixrcond2` function

/*************************************************************************
Estimate of a matrix condition number (2-norm)

The algorithm calculates exact 2-norm reciprocal condition number using SVD.

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/cond2(A)

NOTE:
    if k(A) is very large, then the matrix is  assumed to be degenerate,
    k(A)=INF, 0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcond2(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixrcond2rect` function

/*************************************************************************
Estimate of a matrix condition number (2-norm) for a rectangular matrix.

The algorithm calculates exact 2-norm reciprocal condition number using SVD.

Input parameters:
    A   -   matrix. Array[M,N]
    M, N-   rows and columns count, >=1

Result: 1/cond2(A)

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcond2rect(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixrcondinf` function

/*************************************************************************
Estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixrcondinf(const real_2d_array &a, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`rmatrixtrrcond1` function

/*************************************************************************
Triangular matrix: estimate of a condition number (1-norm)

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A       -   matrix. Array[0..N-1, 0..N-1].
    N       -   size of A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixtrrcond1(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);

`rmatrixtrrcond2` function

/*************************************************************************
Triangular matrix: reciprocal 2-norm condition number

The algorithm calculates a reciprocal 2-norm condition number using SVD.

Input parameters:
    A       -   matrix. Array[0..N-1, 0..N-1].
    N       -   size of A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/cond(A)

NOTE:
    if k(A) is very large, then matrix is assumed to be degenerate,
    k(A)=INF, 0.0 is returned in such cases.
*************************************************************************/
double rmatrixtrrcond2(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);

`rmatrixtrrcondinf` function

/*************************************************************************
Triangular matrix: estimate of a matrix condition number (infinity-norm).

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

Input parameters:
    A   -   matrix. Array whose indexes range within [0..N-1, 0..N-1].
    N   -   size of matrix A.
    IsUpper -   True, if the matrix is upper triangular.
    IsUnit  -   True, if the matrix has a unit diagonal.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double rmatrixtrrcondinf(const real_2d_array &a, const ae_int_t n, const bool isupper, const bool isunit, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskyrcond` function

/*************************************************************************
Condition number estimate of a symmetric positive definite matrix given by
Cholesky decomposition.

The algorithm calculates a lower bound of the condition number. In this
case, the algorithm does not return a lower bound of the condition number,
but an inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    CD  - Cholesky decomposition of matrix A,
          output of SMatrixCholesky subroutine.
    N   - size of matrix A.

Result: 1/LowerBound(cond(A))

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double spdmatrixcholeskyrcond(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`spdmatrixrcond` function

/*************************************************************************
Condition number estimate of a symmetric positive definite matrix.

The algorithm calculates a lower bound of the condition number. In this case,
the algorithm does not return a lower bound of the condition number, but an
inverse number (to avoid an overflow in case of a singular matrix).

It should be noted that 1-norm and inf-norm of condition numbers of symmetric
matrices are equal, so the algorithm doesn't take into account the
differences between these types of norms.

Input parameters:
    A       -   symmetric positive definite matrix which is given by its
                upper or lower triangle depending on the value of
                IsUpper. Array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   storage format.

Result:
    1/LowerBound(cond(A)), if matrix A is positive definite,
   -1, if matrix A is not positive definite, and its condition number
    could not be found by this algorithm.

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double spdmatrixrcond(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`spdmatrixrcond2` function

/*************************************************************************
2-norm condition number of a symmetric positive definite matrix using EVD.

Input parameters:
    A       -   symmetric positive definite matrix which is given by its
                upper or lower triangle depending on the value of
                IsUpper. Array[N,N]
    N       -   size of matrix A.
    IsUpper -   storage format.

Result:
    1/cond(A), if matrix A is positive definite,
    0, if matrix A is not positive definite

NOTE:
    if k(A) is very large, then matrix is  assumed  degenerate,  k(A)=INF,
    0.0 is returned in such cases.
*************************************************************************/
double spdmatrixrcond2(const real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`schur` subpackage

Functions

rmatrixschur

Examples

`rmatrixschur` function

/*************************************************************************
Subroutine performing the Schur decomposition of a general matrix by using
the QR algorithm with multiple shifts.

COMMERCIAL EDITION OF ALGLIB:

! Commercial version of ALGLIB includes one important improvement of
! this function, which can be used from C++ and C#:
! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB)
!
! Intel MKL gives approximately constant (with respect to number of
! worker threads) acceleration factor which depends on CPU being used,
! problem size and "baseline" ALGLIB edition which is used for
! comparison.
!
! Multithreaded acceleration is NOT supported for this function.
!
! We recommend you to read 'Working with commercial version' section of
! ALGLIB Reference Manual in order to find out how to use performance-
! related features provided by commercial edition of ALGLIB.

The source matrix A is represented as S'*A*S = T, where S is an orthogonal
matrix (Schur vectors), T - upper quasi-triangular matrix (with blocks of
sizes 1x1 and 2x2 on the main diagonal).

Input parameters:
A - matrix to be decomposed.
Array whose indexes range within [0..N-1, 0..N-1].
N - size of A, N>=0.

Output parameters:
A - contains matrix T.
Array whose indexes range within [0..N-1, 0..N-1].
S - contains Schur vectors.
Array whose indexes range within [0..N-1, 0..N-1].

Note 1:
The block structure of matrix T can be easily recognized: since all
the elements below the blocks are zeros, the elements a[i+1,i] which
are equal to 0 show the block border.

Note 2:
The algorithm performance depends on the value of the internal parameter
NS of the InternalSchurDecomposition subroutine which defines the number
of shifts in the QR algorithm (similarly to the block width in block-matrix
algorithms in linear algebra). If you require maximum performance on
your machine, it is recommended to adjust this parameter manually.

Result:
True,
if the algorithm has converged and parameters A and S contain the result.
False,
if the algorithm has not converged.

Algorithm implemented on the basis of the DHSEQR subroutine (LAPACK 3.0 library).
*************************************************************************/
bool rmatrixschur(real_2d_array &a, const ae_int_t n, real_2d_array &s, const xparams _xparams = alglib::xdefault);

`sparse` subpackage

Classes

sparsebuffers
sparsematrix

Functions

sparseadd
sparseappendcompressedrow
sparseappendelement
sparseappendemptyrow
sparseappendmatrix
sparseconvertto
sparseconverttocrs
sparseconverttohash
sparseconverttosks
sparsecopy
sparsecopybuf
sparsecopytobuf
sparsecopytocrs
sparsecopytocrsbuf
sparsecopytohash
sparsecopytohashbuf
sparsecopytosks
sparsecopytosksbuf
sparsecopytransposecrs
sparsecopytransposecrsbuf
sparsecreate
sparsecreatebuf
sparsecreatecrs
sparsecreatecrsbuf
sparsecreatecrsempty
sparsecreatecrsemptybuf
sparsecreatecrsfromdense
sparsecreatecrsfromdensebuf
sparsecreatecrsfromdensev
sparsecreatecrsfromdensevbuf
sparsecreatesks
sparsecreatesksband
sparsecreatesksbandbuf
sparsecreatesksbuf
sparseenumerate
sparseexists
sparsefree
sparsegemv
sparseget
sparsegetcompressedrow
sparsegetdiagonal
sparsegetlowercount
sparsegetmatrixtype
sparsegetncols
sparsegetnrows
sparsegetrow
sparsegetuppercount
sparseiscrs
sparseishash
sparseissks
sparsemm
sparsemm2
sparsemtm
sparsemtv
sparsemultiplycolsby
sparsemultiplyrowsby
sparsemv
sparsemv2
sparseresizematrix
sparserewriteexisting
sparsescale
sparseserialize
sparseset
sparsesmm
sparsesmv
sparseswap
sparsesymmpermtbl
sparsesymmpermtblbuf
sparsesymmpermtbltranspose
sparsesymmpermtbltransposebuf
sparsetransposecrs
sparsetransposesks
sparsetrmv
sparsetrsv
sparseunserialize
sparsevsmv

Examples

sparse_d_1		Basic operations with sparse matrices
sparse_d_crs		Advanced topic: creation in the CRS format.

`sparsebuffers` class

/*************************************************************************
Temporary buffers for sparse matrix operations.

You should pass an instance of this structure to factorization  functions.
It allows to reuse memory during repeated sparse  factorizations.  You  do
not have to call some initialization function - simply passing an instance
to factorization function is enough.
*************************************************************************/
class sparsebuffers
{
public:
    sparsebuffers();
    sparsebuffers(const sparsebuffers &rhs);
    sparsebuffers& operator=(const sparsebuffers &rhs);
    virtual ~sparsebuffers();
};

`sparsematrix` class

/*************************************************************************
Sparse matrix structure.

You should use ALGLIB functions to work with sparse matrix. Never  try  to
access its fields directly!

NOTES ON THE SPARSE STORAGE FORMATS

Sparse matrices can be stored using several formats:
* Hash-Table representation
* Compressed Row Storage (CRS)
* Skyline matrix storage (SKS)

Each of the formats has benefits and drawbacks:
* Hash-table is good for dynamic operations (insertion of new elements),
  but does not support linear algebra operations
* CRS is good for operations like matrix-vector or matrix-matrix products,
  but its initialization is less convenient - you have to tell row   sizes
  at the initialization, and you have to fill  matrix  only  row  by  row,
  from left to right.
* SKS is a special format which is used to store triangular  factors  from
  Cholesky factorization. It does not support  dynamic  modification,  and
  support for linear algebra operations is very limited.

Tables below outline information about these two formats:

    OPERATIONS WITH MATRIX      HASH        CRS         SKS
    creation                    +           +           +
    SparseGet                   +           +           +
    SparseExists                +           +           +
    SparseRewriteExisting       +           +           +
    SparseSet                   +           +           +
    SparseAdd                   +
    SparseGetRow                            +           +
    SparseGetCompressedRow                  +           +
    SparseAppendCompressedRow               +
    sparse-dense linear algebra             +           +
*************************************************************************/
class sparsematrix
{
public:
    sparsematrix();
    sparsematrix(const sparsematrix &rhs);
    sparsematrix& operator=(const sparsematrix &rhs);
    virtual ~sparsematrix();
};

`sparseadd` function

/*************************************************************************
This function adds value to S[i,j] - element of the sparse matrix. Matrix
must be in a Hash-Table mode.

In case S[i,j] already exists in the table, V i added to  its  value.  In
case  S[i,j]  is  non-existent,  it  is  inserted  in  the  table.  Table
automatically grows when necessary.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N
    V           -   value to add, must be finite number

OUTPUT PARAMETERS
    S           -   modified matrix

NOTE 1:  when  S[i,j]  is exactly zero after modification, it is  deleted
from the table.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseadd(sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparseappendcompressedrow` function

/*************************************************************************
This function appends a compressed sparse row to a CRS matrix,  increasing
its row count by 1.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format, including one created
                    with sparsecreatecrsempty().
    ColIdx      -   array[NZ], column indexes, values  in  [0,N-1]  range.
                    ColIdx[] can store non-distinct  values;  elements  of
                    Vals[] corresponding to duplicate column indexes  will
                    be summed up.
    Vals        -   array[NZ], element values.
    NZ          -   nonzeros count, NZ>=0. Both ColIdx[]  and  Vals[]  can
                    be longer than NZ, in   which  case  only  leading  NZ
                    elements are used.

OUTPUT PARAMETERS:
    S           -   (M+1)*N matrix in the CRS format.

NOTE: this function has amortized O(NZ*logNZ) cost.

  -- ALGLIB PROJECT --
     Copyright 2024.02.19 by Bochkanov Sergey
*************************************************************************/
void sparseappendcompressedrow(sparsematrix &s, const integer_1d_array &colidx, const real_1d_array &vals, const ae_int_t nz, const xparams _xparams = alglib::xdefault);

`sparseappendelement` function

/*************************************************************************
This function appends an element to the last row  of  a  CRS  matrix.  New
elements can be added ONLY from left to right (column indexes are strictly
increasing).

INPUT PARAMETERS:
    S           -   a fully initialized sparse M*N matrix in CRS format, M>0
    K           -   column index, 0<=K<N, must be  strictly  greater  than
                    the last element in the last row.
    V           -   element value

OUTPUT PARAMETERS:
    S           -   M*N matrix in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 2024.02.19 by Bochkanov Sergey
*************************************************************************/
void sparseappendelement(sparsematrix &s, const ae_int_t k, const double v, const xparams _xparams = alglib::xdefault);

`sparseappendemptyrow` function

/*************************************************************************
This function appends an empty row to a CRS matrix,  increasing  its  rows
count by 1. The newly added row can be modified with sparseappendelement().
The matrix is a valid CRS matrix at any moment of the process.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format, including one created
                    with sparsecreatecrsempty().

OUTPUT PARAMETERS:
    S           -   (M+1)*N matrix in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 2024.02.19 by Bochkanov Sergey
*************************************************************************/
void sparseappendemptyrow(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparseappendmatrix` function

/*************************************************************************
This function appends from below a  sparse  CRS-based  matrix  to  another
sparse CRS-based matrix. The matrix  being  appended  must  be  completely
initialized CRS matrix.

INPUT PARAMETERS:
    SDst        -   sparse X*N matrix in CRS format, including one created
                    with sparsecreatecrsempty (in the latter case, X=0).
    SSrc        -   sparse M*N matrix in the CRS format

OUTPUT PARAMETERS:
    SDst        -   (X+M)*N matrix in the CRS format, SSrc appended from
                    below

NOTE: this  function  has  amortized  O(MSrc+NZCnt) cost, where NZCnt is a
      total number of nonzero elements in SSrc.

  -- ALGLIB PROJECT --
     Copyright 2024.03.23 by Bochkanov Sergey
*************************************************************************/
void sparseappendmatrix(sparsematrix &sdst, const sparsematrix &ssrc, const xparams _xparams = alglib::xdefault);

`sparseconvertto` function

/*************************************************************************
This  function  performs  in-place  conversion  to  desired sparse storage
format.

INPUT PARAMETERS
    S0      -   sparse matrix in any format.
    Fmt     -   desired storage format  of  the  output,  as  returned  by
                SparseGetMatrixType() function:
                * 0 for hash-based storage
                * 1 for CRS
                * 2 for SKS

OUTPUT PARAMETERS
    S0          -   sparse matrix in requested format.

NOTE: in-place conversion wastes a lot of memory which is  used  to  store
      temporaries.  If  you  perform  a  lot  of  repeated conversions, we
      recommend to use out-of-place buffered  conversion  functions,  like
      SparseCopyToBuf(), which can reuse already allocated memory.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseconvertto(sparsematrix &s0, const ae_int_t fmt, const xparams _xparams = alglib::xdefault);

`sparseconverttocrs` function

/*************************************************************************
This function converts matrix to CRS format.

Some  algorithms  (linear  algebra ones, for example) require matrices in
CRS format. This function allows to perform in-place conversion.

INPUT PARAMETERS
    S           -   sparse M*N matrix in any format

OUTPUT PARAMETERS
    S           -   matrix in CRS format

NOTE: this   function  has  no  effect  when  called with matrix which is
      already in CRS mode.

NOTE: this function allocates temporary memory to store a   copy  of  the
      matrix. If you perform a lot of repeated conversions, we  recommend
      you  to  use  SparseCopyToCRSBuf()  function,   which   can   reuse
      previously allocated memory.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseconverttocrs(sparsematrix &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparseconverttohash` function

/*************************************************************************
This function performs in-place conversion to Hash table storage.

INPUT PARAMETERS
    S           -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S           -   sparse matrix in Hash table format.

NOTE: this  function  has   no  effect  when  called with matrix which  is
      already in Hash table mode.

NOTE: in-place conversion involves allocation of temporary arrays. If  you
      perform a lot of repeated in- place  conversions,  it  may  lead  to
      memory fragmentation. Consider using out-of-place SparseCopyToHashBuf()
      function in this case.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparseconverttohash(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparseconverttosks` function

/*************************************************************************
This function performs in-place conversion to SKS format.

INPUT PARAMETERS
    S           -   sparse matrix in any format.

OUTPUT PARAMETERS
    S           -   sparse matrix in SKS format.

NOTE: this  function  has   no  effect  when  called with matrix which  is
      already in SKS mode.

NOTE: in-place conversion involves allocation of temporary arrays. If  you
      perform a lot of repeated in- place  conversions,  it  may  lead  to
      memory fragmentation. Consider using out-of-place SparseCopyToSKSBuf()
      function in this case.

  -- ALGLIB PROJECT --
     Copyright 15.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseconverttosks(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecopy` function

/*************************************************************************
This function copies S0 to S1.
This function completely deallocates memory owned by S1 before creating a
copy of S0. If you want to reuse memory, use SparseCopyBuf.

NOTE:  this  function  does  not verify its arguments, it just copies all
fields of the structure.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecopy(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopybuf` function

/*************************************************************************
This function copies S0 to S1.
Memory already allocated in S1 is reused as much as possible.

NOTE:  this  function  does  not verify its arguments, it just copies all
fields of the structure.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecopybuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytobuf` function

/*************************************************************************
This  function  performs out-of-place conversion to desired sparse storage
format. S0 is copied to S1 and converted on-the-fly. Memory  allocated  in
S1 is reused to maximum extent possible.

INPUT PARAMETERS
    S0      -   sparse matrix in any format.
    Fmt     -   desired storage format  of  the  output,  as  returned  by
                SparseGetMatrixType() function:
                * 0 for hash-based storage
                * 1 for CRS
                * 2 for SKS

OUTPUT PARAMETERS
    S1          -   sparse matrix in requested format.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecopytobuf(const sparsematrix &s0, const ae_int_t fmt, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytocrs` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  CRS format.  S0 is
copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in CRS format.

NOTE: if S0 is stored as CRS, it is just copied without conversion.

NOTE: this function de-allocates memory occupied by S1 before starting CRS
      conversion. If you perform a lot of repeated CRS conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToCRSBuf() function which re-uses memory in S1 as much  as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytocrs(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytocrsbuf` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  CRS format.  S0 is
copied to S1 and converted on-the-fly. Memory allocated in S1 is reused to
maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.
    S1          -   matrix which may contain some pre-allocated memory, or
                    can be just uninitialized structure.

OUTPUT PARAMETERS
    S1          -   sparse matrix in CRS format.

NOTE: if S0 is stored as CRS, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytocrsbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytohash` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  Hash table storage
format. S0 is copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in Hash table format.

NOTE: if S0 is stored as Hash-table, it is just copied without conversion.

NOTE: this function de-allocates memory  occupied  by  S1 before  starting
      conversion. If you perform a  lot  of  repeated  conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToHashBuf() function which re-uses memory in S1 as much as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytohash(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytohashbuf` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to  Hash table storage
format. S0 is copied to S1 and converted on-the-fly. Memory  allocated  in
S1 is reused to maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in Hash table format.

NOTE: if S0 is stored as Hash-table, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytohashbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytosks` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to SKS storage format.
S0 is copied to S1 and converted on-the-fly.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in SKS format.

NOTE: if S0 is stored as SKS, it is just copied without conversion.

NOTE: this function de-allocates memory  occupied  by  S1 before  starting
      conversion. If you perform a  lot  of  repeated  conversions, it may
      lead to memory fragmentation. In this case we recommend you  to  use
      SparseCopyToSKSBuf() function which re-uses memory in S1 as much  as
      possible.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytosks(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytosksbuf` function

/*************************************************************************
This  function  performs  out-of-place  conversion  to SKS format.  S0  is
copied to S1 and converted on-the-fly. Memory  allocated  in S1 is  reused
to maximum extent possible.

INPUT PARAMETERS
    S0          -   sparse matrix in any format.

OUTPUT PARAMETERS
    S1          -   sparse matrix in SKS format.

NOTE: if S0 is stored as SKS, it is just copied without conversion.

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsecopytosksbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytransposecrs` function

/*************************************************************************
This function performs copying with transposition of CRS matrix.

INPUT PARAMETERS
    S0      -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S1      -   sparse matrix, transposed

  -- ALGLIB PROJECT --
     Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrs(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecopytransposecrsbuf` function

/*************************************************************************
This function performs copying with transposition of CRS matrix  (buffered
version which reuses memory already allocated by  the  target as  much  as
possible).

INPUT PARAMETERS
    S0      -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S1      -   sparse matrix, transposed; previously allocated memory  is
                reused if possible.

  -- ALGLIB PROJECT --
     Copyright 23.07.2018 by Bochkanov Sergey
*************************************************************************/
void sparsecopytransposecrsbuf(const sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsecreate` function

/*************************************************************************
This function creates sparse matrix in a Hash-Table format.

This function creates Hast-Table matrix, which can be  converted  to  CRS
format after its initialization is over. Typical  usage  scenario  for  a
sparse matrix is:
1. creation in a Hash-Table format
2. insertion of the matrix elements
3. conversion to the CRS representation
4. matrix is passed to some linear algebra algorithm

Some  information  about  different matrix formats can be found below, in
the "NOTES" section.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    K           -   K>=0, expected number of non-zero elements in a matrix.
                    K can be inexact approximation, can be less than actual
                    number  of  elements  (table will grow when needed) or
                    even zero).
                    It is important to understand that although hash-table
                    may grow automatically, it is better to  provide  good
                    estimate of data size.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    All elements of the matrix are zero.

NOTE 1

Hash-tables use memory inefficiently, and they have to keep  some  amount
of the "spare memory" in order to have good performance. Hash  table  for
matrix with K non-zero elements will  need  C*K*(8+2*sizeof(int))  bytes,
where C is a small constant, about 1.5-2 in magnitude.

CRS storage, from the other side, is  more  memory-efficient,  and  needs
just K*(8+sizeof(int))+M*sizeof(int) bytes, where M is a number  of  rows
in a matrix.

When you convert from the Hash-Table to CRS  representation, all unneeded
memory will be freed.

NOTE 2

Comments of SparseMatrix structure outline  information  about  different
sparse storage formats. We recommend you to read them before starting  to
use ALGLIB sparse matrices.

NOTE 3

This function completely  overwrites S with new sparse matrix. Previously
allocated storage is NOT reused. If you  want  to reuse already allocated
memory, call SparseCreateBuf function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreate(const ae_int_t m, const ae_int_t n, const ae_int_t k, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreate(const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparsecreatebuf` function

/*************************************************************************
This version of SparseCreate function creates sparse matrix in Hash-Table
format, reusing previously allocated storage as much  as  possible.  Read
comments for SparseCreate() for more information.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    K           -   K>=0, expected number of non-zero elements in a matrix.
                    K can be inexact approximation, can be less than actual
                    number  of  elements  (table will grow when needed) or
                    even zero).
                    It is important to understand that although hash-table
                    may grow automatically, it is better to  provide  good
                    estimate of data size.
    S           -   SparseMatrix structure which MAY contain some  already
                    allocated storage.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    All elements of the matrix are zero.
                    Previously allocated storage is reused, if  its  size
                    is compatible with expected number of non-zeros K.

  -- ALGLIB PROJECT --
     Copyright 14.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, const ae_int_t k, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatebuf(const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrs` function

/*************************************************************************
This function creates sparse matrix in a CRS format - the least  flexible
but the most efficient format implemented in ALGLIB.

This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell the number of non-zero elements at each row
   at this moment)
2. initialization of the matrix elements (row by row, from left to right)
3. the matrix is passed to some linear algebra algorithm

This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some  information about  different matrix formats can be found
in comments on SparseMatrix structure.  We recommend  you  to  read  them
before starting to use ALGLIB sparse matrices.

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    NER         -   number of elements at each row, array[M], NER[I]>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in CRS representation.
                    You have to fill ALL non-zero elements by calling
                    SparseSet() BEFORE you try to use this matrix.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSBuf function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrs(const ae_int_t m, const ae_int_t n, const integer_1d_array &ner, sparsematrix &s, const xparams _xparams = alglib::xdefault);

Examples: [1]

`sparsecreatecrsbuf` function

/*************************************************************************
This function creates sparse matrix in a CRS format (expert function  for
situations when you are running out  of  memory).  This  version  of  CRS
matrix creation function may reuse memory already allocated in S.

This function creates CRS matrix. Typical usage scenario for a CRS matrix
is:
1. creation (you have to tell number of non-zero elements at each row  at
   this moment)
2. insertion of the matrix elements (row by row, from left to right)
3. matrix is passed to some linear algebra algorithm

This function is a memory-efficient alternative to SparseCreate(), but it
is more complex because it requires you to know in advance how large your
matrix is. Some  information about  different matrix formats can be found
in comments on SparseMatrix structure.  We recommend  you  to  read  them
before starting to use ALGLIB sparse matrices..

INPUT PARAMETERS
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    NER         -   number of elements at each row, array[M], NER[I]>=0
    S           -   sparse matrix structure with possibly preallocated
                    memory.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in CRS representation.
                    You have to fill ALL non-zero elements by calling
                    SparseSet() BEFORE you try to use this matrix.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsbuf(const ae_int_t m, const ae_int_t n, const integer_1d_array &ner, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrsempty` function

/*************************************************************************
This function creates an EMPTY sparse matrix stored in the CRS format.

The empty matrix is a degenerate 0*N-dimensional matrix which can be used
ONLY for:
* appending rows with sparseappendcompressedrow()
* appending non-degenerate CRS matrices with sparseappendmatrix()
Before the first row is appended, the matrix is in a special intermediate
state. After the first append it becomes a standard CRS matrix.

The main purpose of this function is to simplify step-by-step initialization
of CRS matrices.

INPUT PARAMETERS
    N           -   number of columns in a matrix, N>=1

OUTPUT PARAMETERS
    S           -   sparse 0*N matrix in a partially initialized state

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSEmptyBuf function.

  -- ALGLIB PROJECT --
     Copyright 20.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsempty(const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrsemptybuf` function

/*************************************************************************
This function creates an EMPTY sparse matrix stored in the CRS format. It
is a buffered version of the function which  reuses  previosly  allocated
space as much as possible.

INPUT PARAMETERS
    N           -   number of columns in a matrix, N>=1

OUTPUT PARAMETERS
    S           -   sparse 0*N matrix in a partially initialized state

  -- ALGLIB PROJECT --
     Copyright 20.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsemptybuf(const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrsfromdense` function

/*************************************************************************
This function creates a CRS-based sparse matrix from  the  dense  matrix.

This function is intended for situations when you already  have  a  dense
matrix and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M,N]. If larger, only leading MxN submatrix
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSFromDenseBuf function.

  -- ALGLIB PROJECT --
     Copyright 16.06.2023 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdense(const real_2d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatecrsfromdense(const real_2d_array &a, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrsfromdensebuf` function

/*************************************************************************
This function creates a CRS-based sparse matrix from  the  dense  matrix.
A buffered version which reused memory already allocated in S as much  as
possible.

This function is intended for situations when you already  have  a  dense
matrix and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M,N]. If larger, only leading MxN submatrix
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    S           -   an already allocated structure; if it already has
                    enough memory to store the matrix, no new memory
                    will be allocated.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 16.06.2023 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdensebuf(const real_2d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);
void sparsecreatecrsfromdensebuf(const real_2d_array &a, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrsfromdensev` function

/*************************************************************************
This function creates a  CRS-based  sparse  matrix  from  a  dense  vector
which stores a dense 1-dimensional representation of a dense M*N matrix.

This function is intended for situations when you already  have  a  dense
vector and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M*N]. If larger, only leading M*N elements
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateCRSFromDenseBuf function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdensev(const real_1d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatecrsfromdensevbuf` function

/*************************************************************************
This function creates a  CRS-based  sparse  matrix  from  a  dense vector
which stores a dense 1-dimensional representation of a dense M*N  matrix.
A buffered version which reused memory already allocated in S as much  as
possible.

This function is intended for situations when you already  have  a  dense
vector and need a convenient way of converting it to the CRS format.

INPUT PARAMETERS
    A           -   array[M*N]. If larger, only leading M*N elements
                    will be used.
    M           -   number of rows in a matrix, M>=1
    N           -   number of columns in a matrix, N>=1
    S           -   an already allocated structure; if it already has
                    enough memory to store the matrix, no new memory
                    will be allocated.

OUTPUT PARAMETERS
    S           -   sparse M*N matrix A in the CRS format.

  -- ALGLIB PROJECT --
     Copyright 16.06.2023 by Bochkanov Sergey
*************************************************************************/
void sparsecreatecrsfromdensevbuf(const real_1d_array &a, const ae_int_t m, const ae_int_t n, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatesks` function

/*************************************************************************
This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). In most cases you do not need this function - CRS format  better
suits most use cases.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    D           -   "bottom" bandwidths, array[M], D[I]>=0.
                    I-th element stores number of non-zeros at I-th  row,
                    below the diagonal (diagonal itself is not  included)
    U           -   "top" bandwidths, array[N], U[I]>=0.
                    I-th element stores number of non-zeros  at I-th row,
                    above the diagonal (diagonal itself  is not included)

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call SparseCreateSKSBuf function.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesks(const ae_int_t m, const ae_int_t n, const integer_1d_array &d, const integer_1d_array &u, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatesksband` function

/*************************************************************************
This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). Unlike more general  sparsecreatesks(),  this  function  creates
sparse matrix with constant bandwidth.

You may want to use this function instead of sparsecreatesks() when  your
matrix has  constant  or  nearly-constant  bandwidth,  and  you  want  to
simplify source code.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    BW          -   matrix bandwidth, BW>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to  change  their values.

NOTE: this function completely  overwrites  S  with  new  sparse  matrix.
      Previously allocated storage is NOT reused. If you  want  to  reuse
      already allocated memory, call sparsecreatesksbandbuf function.

  -- ALGLIB PROJECT --
     Copyright 25.12.2017 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksband(const ae_int_t m, const ae_int_t n, const ae_int_t bw, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatesksbandbuf` function

/*************************************************************************
This is "buffered" version  of  sparsecreatesksband() which reuses memory
previously allocated in S (of course, memory is reallocated if needed).

You may want to use this function instead  of  sparsecreatesksbuf()  when
your matrix has  constant or nearly-constant  bandwidth,  and you want to
simplify source code.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    BW          -   bandwidth, BW>=0

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksbandbuf(const ae_int_t m, const ae_int_t n, const ae_int_t bw, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsecreatesksbuf` function

/*************************************************************************
This is "buffered"  version  of  SparseCreateSKS()  which  reuses  memory
previously allocated in S (of course, memory is reallocated if needed).

This function creates sparse matrix in  a  SKS  format  (skyline  storage
format). In most cases you do not need this function - CRS format  better
suits most use cases.

INPUT PARAMETERS
    M, N        -   number of rows(M) and columns (N) in a matrix:
                    * M=N (as for now, ALGLIB supports only square SKS)
                    * N>=1
                    * M>=1
    D           -   "bottom" bandwidths, array[M], 0<=D[I]<=I.
                    I-th element stores number of non-zeros at I-th row,
                    below the diagonal (diagonal itself is not included)
    U           -   "top" bandwidths, array[N], 0<=U[I]<=I.
                    I-th element stores number of non-zeros at I-th row,
                    above the diagonal (diagonal itself is not included)

OUTPUT PARAMETERS
    S           -   sparse M*N matrix in SKS representation.
                    All elements are filled by zeros.
                    You may use sparseset() to change their values.

  -- ALGLIB PROJECT --
     Copyright 13.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsecreatesksbuf(const ae_int_t m, const ae_int_t n, const integer_1d_array &d, const integer_1d_array &u, sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparseenumerate` function

/*************************************************************************
This  function  is  used  to enumerate all elements of the sparse matrix.
Before  first  call  user  initializes  T0 and T1 counters by zero. These
counters are used to remember current position in a  matrix;  after  each
call they are updated by the function.

Subsequent calls to this function return non-zero elements of the  sparse
matrix, one by one. If you enumerate CRS matrix, matrix is traversed from
left to right, from top to bottom. In case you enumerate matrix stored as
Hash table, elements are returned in random order.

EXAMPLE
    > T0=0
    > T1=0
    > while SparseEnumerate(S,T0,T1,I,J,V) do
    >     ....do something with I,J,V

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table or CRS representation.
    T0          -   internal counter
    T1          -   internal counter

OUTPUT PARAMETERS
    T0          -   new value of the internal counter
    T1          -   new value of the internal counter
    I           -   row index of non-zero element, 0<=I<M.
    J           -   column index of non-zero element, 0<=J<N
    V           -   value of the T-th element

RESULT
    True in case of success (next non-zero element was retrieved)
    False in case all non-zero elements were enumerated

NOTE: you may call SparseRewriteExisting() during enumeration, but it  is
      THE  ONLY  matrix  modification  function  you  can  call!!!  Other
      matrix modification functions should not be called during enumeration!

  -- ALGLIB PROJECT --
     Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseenumerate(const sparsematrix &s, ae_int_t &t0, ae_int_t &t1, ae_int_t &i, ae_int_t &j, double &v, const xparams _xparams = alglib::xdefault);

`sparseexists` function

/*************************************************************************
This function checks whether S[i,j] is present in the sparse  matrix.  It
returns True even for elements  that  are  numerically  zero  (but  still
have place allocated for them).

The matrix  can be in any mode (Hash-Table, CRS, SKS), but this  function
is less efficient for CRS matrices. Hash-Table and SKS matrices can  find
element in O(1) time, while  CRS  matrices need O(log(RS)) time, where RS
is an number of non-zero elements in a row.

INPUT PARAMETERS
    S           -   sparse M*N matrix
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N

RESULT
    whether S[I,J] is present in the data structure or not

  -- ALGLIB PROJECT --
     Copyright 14.10.2020 by Bochkanov Sergey
*************************************************************************/
bool sparseexists(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const xparams _xparams = alglib::xdefault);

`sparsefree` function

/*************************************************************************
The function frees all memory occupied by  sparse  matrix.  Sparse  matrix
structure becomes unusable after this call.

OUTPUT PARAMETERS
    S   -   sparse matrix to delete

  -- ALGLIB PROJECT --
     Copyright 24.07.2012 by Bochkanov Sergey
*************************************************************************/
void sparsefree(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsegemv` function

/*************************************************************************
This function calculates generalized sparse matrix-vector product

    y := alpha*op(S)*x + beta*y

Matrix S must be stored in CRS or SKS format (exception  will  be  thrown
otherwise). op(S) can be either S or S^T.

NOTE: this  function  expects  Y  to  be  large enough to store result. No
      automatic preallocation happens for smaller arrays.

INPUT PARAMETERS
    S           -   sparse matrix in CRS or SKS format.
    Alpha       -   source coefficient
    OpS         -   operation type:
                    * OpS=0     =>  op(S) = S
                    * OpS=1     =>  op(S) = S^T
    X           -   input vector, must have at least Cols(op(S))+IX elements
    IX          -   subvector offset
    Beta        -   destination coefficient
    Y           -   preallocated output array, must have at least Rows(op(S))+IY elements
    IY          -   subvector offset

OUTPUT PARAMETERS
    Y           -   elements [IY...IY+Rows(op(S))-1] are replaced by result,
                    other elements are not modified

HANDLING OF SPECIAL CASES:
* below M=Rows(op(S)) and N=Cols(op(S)). Although current  ALGLIB  version
  does not allow you to  create  zero-sized  sparse  matrices,  internally
  ALGLIB  can  deal  with  such matrices. So, comments for M or N equal to
  zero are for internal use only.
* if M=0, then subroutine does nothing. It does not even touch arrays.
* if N=0 or Alpha=0.0, then:
  * if Beta=0, then Y is filled by zeros. S and X are  not  referenced  at
    all. Initial values of Y are ignored (we do not  multiply  Y by  zero,
    we just rewrite it by zeros)
  * if Beta<>0, then Y is replaced by Beta*Y
* if M>0, N>0, Alpha<>0, but  Beta=0, then  Y is replaced by alpha*op(S)*x
  initial state of Y  is ignored (rewritten without initial multiplication
  by zeros).

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 10.12.2019 by Bochkanov Sergey
*************************************************************************/
void sparsegemv(const sparsematrix &s, const double alpha, const ae_int_t ops, const real_1d_array &x, const ae_int_t ix, const double beta, real_1d_array &y, const ae_int_t iy, const xparams _xparams = alglib::xdefault);

`sparseget` function

/*************************************************************************
This function returns S[i,j] - element of the sparse matrix.  Matrix  can
be in any mode (Hash-Table, CRS, SKS), but this function is less efficient
for CRS matrices. Hash-Table and SKS matrices can find  element  in  O(1)
time, while  CRS  matrices need O(log(RS)) time, where RS is an number of
non-zero elements in a row.

INPUT PARAMETERS
    S           -   sparse M*N matrix
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N

RESULT
    value of S[I,J] or zero (in case no element with such index is found)

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
double sparseget(const sparsematrix &s, const ae_int_t i, const ae_int_t j, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`sparsegetcompressedrow` function

/*************************************************************************
This function returns I-th row of the sparse matrix IN COMPRESSED FORMAT -
only non-zero elements are returned (with their indexes). Matrix  must  be
stored in CRS or SKS format.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format
    I           -   row index, 0<=I<M
    ColIdx      -   output buffer for column indexes, can be preallocated.
                    In case buffer size is too small to store I-th row, it
                    is automatically reallocated.
    Vals        -   output buffer for values, can be preallocated. In case
                    buffer size is too small to  store  I-th  row,  it  is
                    automatically reallocated.

OUTPUT PARAMETERS:
    ColIdx      -   column   indexes   of  non-zero  elements,  sorted  by
                    ascending. Symbolically non-zero elements are  counted
                    (i.e. if you allocated place for element, but  it  has
                    zero numerical value - it is counted).
    Vals        -   values. Vals[K] stores value of  matrix  element  with
                    indexes (I,ColIdx[K]). Symbolically non-zero  elements
                    are counted (i.e. if you allocated place for  element,
                    but it has zero numerical value - it is counted).
    NZCnt       -   number of symbolically non-zero elements per row.

NOTE: when  incorrect  I  (outside  of  [0,M-1]) or  matrix (non  CRS/SKS)
      is passed, this function throws exception.

NOTE: this function may allocate additional, unnecessary place for  ColIdx
      and Vals arrays. It is dictated by  performance  reasons  -  on  SKS
      matrices it is faster  to  allocate  space  at  the  beginning  with
      some "extra"-space, than performing two passes over matrix  -  first
      time to calculate exact space required for data, second  time  -  to
      store data itself.

  -- ALGLIB PROJECT --
     Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void sparsegetcompressedrow(const sparsematrix &s, const ae_int_t i, integer_1d_array &colidx, real_1d_array &vals, ae_int_t &nzcnt, const xparams _xparams = alglib::xdefault);

`sparsegetdiagonal` function

/*************************************************************************
This function returns I-th diagonal element of the sparse matrix.

Matrix can be in any mode (Hash-Table or CRS storage), but this  function
is most efficient for CRS matrices - it requires less than 50 CPU  cycles
to extract diagonal element. For Hash-Table matrices we still  have  O(1)
query time, but function is many times slower.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table representation.
                    Exception will be thrown for CRS matrix.
    I           -   index of the element to modify, 0<=I<min(M,N)

RESULT
    value of S[I,I] or zero (in case no element with such index is found)

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
double sparsegetdiagonal(const sparsematrix &s, const ae_int_t i, const xparams _xparams = alglib::xdefault);

`sparsegetlowercount` function

/*************************************************************************
The function returns number of strictly lower triangular non-zero elements
in  the  matrix.  It  counts  SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element  has  zero  numerical
value, it is still counted.

This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
  with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).

RESULT: number of non-zero elements strictly below main diagonal

  -- ALGLIB PROJECT --
     Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetlowercount(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsegetmatrixtype` function

/*************************************************************************
This function returns type of the matrix storage format.

INPUT PARAMETERS:
    S           -   sparse matrix.

RESULT:
    sparse storage format used by matrix:
        0   -   Hash-table
        1   -   CRS (compressed row storage)
        2   -   SKS (skyline)

NOTE: future  versions  of  ALGLIB  may  include additional sparse storage
      formats.


  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetmatrixtype(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsegetncols` function

/*************************************************************************
The function returns number of columns of a sparse matrix.

RESULT: number of columns of a sparse matrix.

  -- ALGLIB PROJECT --
     Copyright 23.08.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetncols(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsegetnrows` function

/*************************************************************************
The function returns number of rows of a sparse matrix.

RESULT: number of rows of a sparse matrix.

  -- ALGLIB PROJECT --
     Copyright 23.08.2012 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetnrows(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsegetrow` function

/*************************************************************************
This function returns I-th row of the sparse matrix. Matrix must be stored
in CRS or SKS format.

INPUT PARAMETERS:
    S           -   sparse M*N matrix in CRS format
    I           -   row index, 0<=I<M
    IRow        -   output buffer, can be  preallocated.  In  case  buffer
                    size  is  too  small  to  store  I-th   row,   it   is
                    automatically reallocated.

OUTPUT PARAMETERS:
    IRow        -   array[M], I-th row.

NOTE: this function has O(N) running time, where N is a  column  count. It
      allocates and fills N-element  array,  even  although  most  of  its
      elemets are zero.

NOTE: If you have O(non-zeros-per-row) time and memory  requirements,  use
      SparseGetCompressedRow() function. It  returns  data  in  compressed
      format.

NOTE: when  incorrect  I  (outside  of  [0,M-1]) or  matrix (non  CRS/SKS)
      is passed, this function throws exception.

  -- ALGLIB PROJECT --
     Copyright 10.12.2014 by Bochkanov Sergey
*************************************************************************/
void sparsegetrow(const sparsematrix &s, const ae_int_t i, real_1d_array &irow, const xparams _xparams = alglib::xdefault);

`sparsegetuppercount` function

/*************************************************************************
The function returns number of strictly upper triangular non-zero elements
in  the  matrix.  It  counts  SYMBOLICALLY non-zero elements, i.e. entries
in the sparse matrix data structure. If some element  has  zero  numerical
value, it is still counted.

This function has different cost for different types of matrices:
* for hash-based matrices it involves complete pass over entire hash-table
  with O(NNZ) cost, where NNZ is number of non-zero elements
* for CRS and SKS matrix types cost of counting is O(N) (N - matrix size).

RESULT: number of non-zero elements strictly above main diagonal

  -- ALGLIB PROJECT --
     Copyright 12.02.2014 by Bochkanov Sergey
*************************************************************************/
ae_int_t sparsegetuppercount(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparseiscrs` function

/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using CRS representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is CRS
    False if matrix type is not CRS

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseiscrs(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparseishash` function

/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using Hash table representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is Hash table
    False if matrix type is not Hash table

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseishash(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparseissks` function

/*************************************************************************
This function checks matrix storage format and returns True when matrix is
stored using SKS representation.

INPUT PARAMETERS:
    S   -   sparse matrix.

RESULT:
    True if matrix type is SKS
    False if matrix type is not SKS

  -- ALGLIB PROJECT --
     Copyright 20.07.2012 by Bochkanov Sergey
*************************************************************************/
bool sparseissks(const sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsemm` function

/*************************************************************************
This function calculates matrix-matrix product  S*A.  Matrix  S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    A           -   array[N][K], input dense matrix. For  performance reasons
                    we make only quick checks - we check that array size
                    is at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[M][K], S*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemm(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);

`sparsemm2` function

/*************************************************************************
This function simultaneously calculates two matrix-matrix products:
    S*A and S^T*A.
S  must  be  square (non-rectangular) matrix stored in CRS or  SKS  format
(exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse N*N matrix in CRS or SKS format.
    A           -   array[N][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B0          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.
    B1          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B0          -   array[N][K], S*A
    B1          -   array[N][K], S^T*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemm2(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b0, real_2d_array &b1, const xparams _xparams = alglib::xdefault);

`sparsemtm` function

/*************************************************************************
This function calculates matrix-matrix product  S^T*A. Matrix S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    A           -   array[M][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size  is
                    at least M, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[N][K], S^T*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemtm(const sparsematrix &s, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);

`sparsemtv` function

/*************************************************************************
This function calculates matrix-vector product  S^T*x. Matrix S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    X           -   array[M], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least M, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[N], S^T*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemtv(const sparsematrix &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`sparsemultiplycolsby` function

/*************************************************************************
This function perform in-place multiplication of the matrix columns by  a
user-supplied vector X. The matrix S must be stored in CRS format.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    X           -   array[N], coefficients vector.

OUTPUT PARAMETERS
    S           -   in-place multiplied by diag(X) from the right

NOTE: this function throws exception when called for  a  non-CRS  matrix.
You must convert your matrix with SparseConvertToCRS() before using  this
function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsemultiplycolsby(sparsematrix &s, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`sparsemultiplyrowsby` function

/*************************************************************************
This function perform in-place multiplication of the matrix rows by  a
user-supplied vector X. The matrix S must be stored in CRS format.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    X           -   array[M], coefficients vector.

OUTPUT PARAMETERS
    S           -   in-place multiplied by diag(X) from the left

NOTE: this function throws exception when called for  a  non-CRS  matrix.
You must convert your matrix with SparseConvertToCRS() before using  this
function.

  -- ALGLIB PROJECT --
     Copyright 17.02.2024 by Bochkanov Sergey
*************************************************************************/
void sparsemultiplyrowsby(sparsematrix &s, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

`sparsemv` function

/*************************************************************************
This function calculates matrix-vector product  S*x.  Matrix  S  must  be
stored in CRS or SKS format (exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS or SKS format.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[M], S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemv(const sparsematrix &s, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`sparsemv2` function

/*************************************************************************
This function simultaneously calculates two matrix-vector products:
    S*x and S^T*x.
S must be square (non-rectangular) matrix stored in  CRS  or  SKS  format
(exception will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse N*N matrix in CRS or SKS format.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y0          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.
    Y1          -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y0          -   array[N], S*x
    Y1          -   array[N], S^T*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsemv2(const sparsematrix &s, const real_1d_array &x, real_1d_array &y0, real_1d_array &y1, const xparams _xparams = alglib::xdefault);

`sparseresizematrix` function

/*************************************************************************
This procedure resizes Hash-Table matrix. It can be called when you  have
deleted too many elements from the matrix, and you want to  free unneeded
memory.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseresizematrix(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparserewriteexisting` function

/*************************************************************************
This function rewrites existing (non-zero) element. It  returns  True   if
element  exists  or  False,  when  it  is  called for non-existing  (zero)
element.

This function works with any kind of the matrix.

The purpose of this function is to provide convenient thread-safe  way  to
modify  sparse  matrix.  Such  modification  (already  existing element is
rewritten) is guaranteed to be thread-safe without any synchronization, as
long as different threads modify different elements.

INPUT PARAMETERS
    S           -   sparse M*N matrix in any kind of representation
                    (Hash, SKS, CRS).
    I           -   row index of non-zero element to modify, 0<=I<M
    J           -   column index of non-zero element to modify, 0<=J<N
    V           -   value to rewrite, must be finite number

OUTPUT PARAMETERS
    S           -   modified matrix
RESULT
    True in case when element exists
    False in case when element doesn't exist or it is zero

  -- ALGLIB PROJECT --
     Copyright 14.03.2012 by Bochkanov Sergey
*************************************************************************/
bool sparserewriteexisting(sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);

`sparsescale` function

/*************************************************************************
This function performs an in-place matrix conditioning scaling  such  that

    A = R*Z*C

where A is an original matrix, R and C are diagonal scaling  matrices, and
Z is a scaled matrix. Z replaces A, R and C are returned as 1D arrays.

INPUT PARAMETERS
    S           -   sparse M*N matrix in CRS format.
    SclType     -   scaling type:
                    * 0     for automatically chosen scaling
                    * 1     for equilibration scaling
    ScaleRows   -   if False, rows are not scaled (R=identity)
    ScaleCols   -   if False, cols are not scaled (C=identity)
    ColsFirst   -   scale columns first. If False, rows are  scaled  prior
                    to scaling columns. Ignored for ScaleCols=False.

OUTPUT PARAMETERS
    R           -   array[M], row scales, R[i]>0
    C           -   array[N], col scales, C[i]>0

NOTE: this function throws exception when called  for  a  non-CRS  matrix.
      You must convert your matrix with SparseConvertToCRS()  before using
      this function.

NOTE: this  function  works  with  general  (nonsymmetric)  matrices.  See
      sparsesymmscale() for a symmetric version. See sparsescalebuf()  for
      a version which reuses space already present in output arrays R/C.

NOTE: if both ScaleRows=False and ScaleCols=False, this  function  returns
      an identity scaling.

NOTE: R[] and C[] are guaranteed to be strictly positive. When the  matrix
      has zero rows/cols, corresponding elements of R/C are set to 1.

  -- ALGLIB PROJECT --
     Copyright 12.11.2023 by Bochkanov Sergey
*************************************************************************/
void sparsescale(sparsematrix &s, const ae_int_t scltype, const bool scalerows, const bool scalecols, const bool colsfirst, real_1d_array &r, real_1d_array &c, const xparams _xparams = alglib::xdefault);

`sparseserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void sparseserialize(const sparsematrix &obj, std::string &s_out);
void sparseserialize(const sparsematrix &obj, std::ostream &s_out);

`sparseset` function

/*************************************************************************
This function modifies S[i,j] - element of the sparse matrix.

For Hash-based storage format:
* this function can be called at any moment - during matrix initialization
  or later
* new value can be zero or non-zero.  In case new value of S[i,j] is zero,
  this element is deleted from the table.
* this  function  has  no  effect when called with zero V for non-existent
  element.

For CRS-bases storage format:
* this function can be called ONLY DURING MATRIX INITIALIZATION
* zero values are stored in the matrix similarly to non-zero ones
* elements must be initialized in correct order -  from top row to bottom,
  within row - from left to right.

For SKS storage:
* this function can be called at any moment - during matrix initialization
  or later
* zero values are stored in the matrix similarly to non-zero ones
* this function CAN NOT be called for non-existent (outside  of  the  band
  specified during SKS matrix creation) elements. Say, if you created  SKS
  matrix  with  bandwidth=2  and  tried to call sparseset(s,0,10,VAL),  an
  exception will be generated.

INPUT PARAMETERS
    S           -   sparse M*N matrix in Hash-Table, SKS or CRS format.
    I           -   row index of the element to modify, 0<=I<M
    J           -   column index of the element to modify, 0<=J<N
    V           -   value to set, must be finite number, can be zero

OUTPUT PARAMETERS
    S           -   modified matrix

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparseset(sparsematrix &s, const ae_int_t i, const ae_int_t j, const double v, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`sparsesmm` function

/*************************************************************************
This function calculates matrix-matrix product  S*A, when S  is  symmetric
matrix. Matrix S must be stored in CRS or SKS format  (exception  will  be
thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    A           -   array[N][K], input dense matrix. For performance reasons
                    we make only quick checks - we check that array size is
                    at least N, but we do not check for NAN's or INF's.
    K           -   number of columns of matrix (A).
    B           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    B           -   array[M][K], S*A

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesmm(const sparsematrix &s, const bool isupper, const real_2d_array &a, const ae_int_t k, real_2d_array &b, const xparams _xparams = alglib::xdefault);

`sparsesmv` function

/*************************************************************************
This function calculates matrix-vector product  S*x, when S is  symmetric
matrix. Matrix S  must be stored in CRS or SKS format  (exception will be
thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.
    Y           -   output buffer, possibly preallocated. In case  buffer
                    size is too small to store  result,  this  buffer  is
                    automatically resized.

OUTPUT PARAMETERS
    Y           -   array[M], S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 14.10.2011 by Bochkanov Sergey
*************************************************************************/
void sparsesmv(const sparsematrix &s, const bool isupper, const real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`sparseswap` function

/*************************************************************************
This function efficiently swaps contents of S0 and S1.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparseswap(sparsematrix &s0, sparsematrix &s1, const xparams _xparams = alglib::xdefault);

`sparsesymmpermtbl` function

/*************************************************************************
This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

This function allocates completely new instance of B. Use buffered version
SparseSymmPermTblBuf() if you want to reuse already allocated structure.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only upper or lower triangle (depending
                    on IsUpper) is stored.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 05.10.2020 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtbl(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);

`sparsesymmpermtblbuf` function

/*************************************************************************
This function is a buffered version  of  SparseSymmPermTbl()  that  reuses
previously allocated storage in B as much as possible.

This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.
    B           -   sparse matrix object that will hold the result.
                    Previously allocated memory will be reused as much  as
                    possible.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only upper or lower triangle (depending
                    on IsUpper) is stored.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 05.10.2020 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtblbuf(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);

`sparsesymmpermtbltranspose` function

/*************************************************************************
This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

It outputs TRANSPOSED matrix, i.e. if A is given  by  the  lower  triangle
then B is given by the upper one, and vice versa.

This function allocates completely new instance of B. Use buffered version
SparseSymmPermTblTransposeBuf() if you want to reuse an already  allocated
structure.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only triangle OPPOSITE to that of A is
                    returned: a lower one if IsUpper=True,  and  an  upper
                    one otherwise.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 24.080.2024 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtbltranspose(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);

`sparsesymmpermtbltransposebuf` function

/*************************************************************************
This function applies permutation given by permutation table P (as opposed
to product form of permutation) to sparse symmetric  matrix  A,  given  by
either upper or lower triangle: B := P*A*P'.

It outputs TRANSPOSED matrix, i.e. if A is given  by  the  lower  triangle
then B is given by the upper one, and vice versa.

This function reuses memory already allocated in B as much as possible.

INPUT PARAMETERS
    A           -   sparse square matrix in CRS format.
    IsUpper     -   whether upper or lower triangle of A is used:
                    * if upper triangle is given,  only   A[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   A[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    P           -   array[N] which stores permutation table;  P[I]=J means
                    that I-th row/column of matrix  A  is  moved  to  J-th
                    position. For performance reasons we do NOT check that
                    P[] is  a   correct   permutation  (that there  is  no
                    repetitions, just that all its elements  are  in [0,N)
                    range.
    B           -   sparse matrix object that will hold the result.
                    Previously allocated memory will be reused as much  as
                    possible.

OUTPUT PARAMETERS
    B           -   permuted matrix.  Permutation  is  applied  to A  from
                    the both sides, only triangle OPPOSITE to that of A is
                    returned: a lower one if IsUpper=True,  and  an  upper
                    one otherwise.

NOTE: this function throws exception when called for non-CRS  matrix.  You
      must convert your matrix with SparseConvertToCRS() before using this
      function.

  -- ALGLIB PROJECT --
     Copyright 24.080.2024 by Bochkanov Sergey.
*************************************************************************/
void sparsesymmpermtbltransposebuf(const sparsematrix &a, const bool isupper, const integer_1d_array &p, sparsematrix &b, const xparams _xparams = alglib::xdefault);

`sparsetransposecrs` function

/*************************************************************************
This function performs transpose of CRS matrix.

INPUT PARAMETERS
    S       -   sparse matrix in CRS format.

OUTPUT PARAMETERS
    S           -   sparse matrix, transposed.

NOTE: internal  temporary  copy  is  allocated   for   the   purposes   of
      transposition. It is deallocated after transposition.

  -- ALGLIB PROJECT --
     Copyright 30.01.2018 by Bochkanov Sergey
*************************************************************************/
void sparsetransposecrs(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsetransposesks` function

/*************************************************************************
This function performs efficient in-place  transpose  of  SKS  matrix.  No
additional memory is allocated during transposition.

This function supports only skyline storage format (SKS).

INPUT PARAMETERS
    S       -   sparse matrix in SKS format.

OUTPUT PARAMETERS
    S           -   sparse matrix, transposed.

  -- ALGLIB PROJECT --
     Copyright 16.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetransposesks(sparsematrix &s, const xparams _xparams = alglib::xdefault);

`sparsetrmv` function

/*************************************************************************
This function calculates matrix-vector product op(S)*x, when x is  vector,
S is symmetric triangular matrix, op(S) is transposition or no  operation.
Matrix S must be stored in CRS or SKS format  (exception  will  be  thrown
otherwise).

INPUT PARAMETERS
    S           -   sparse square matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is used:
                    * if upper triangle is given,  only   S[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    IsUnit      -   unit or non-unit diagonal:
                    * if True, diagonal elements of triangular matrix  are
                      considered equal to 1.0. Actual elements  stored  in
                      S are not referenced at all.
                    * if False, diagonal stored in S is used
    OpType      -   operation type:
                    * if 0, S*x is calculated
                    * if 1, (S^T)*x is calculated (transposition)
    X           -   array[N] which stores input  vector.  For  performance
                    reasons we make only quick  checks  -  we  check  that
                    array  size  is  at  least  N, but we do not check for
                    NAN's or INF's.
    Y           -   possibly  preallocated  input   buffer.  Automatically
                    resized if its size is too small.

OUTPUT PARAMETERS
    Y           -   array[N], op(S)*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetrmv(const sparsematrix &s, const bool isupper, const bool isunit, const ae_int_t optype, real_1d_array &x, real_1d_array &y, const xparams _xparams = alglib::xdefault);

`sparsetrsv` function

/*************************************************************************
This function solves linear system op(S)*y=x  where  x  is  vector,  S  is
symmetric  triangular  matrix,  op(S)  is  transposition  or no operation.
Matrix S must be stored in CRS or SKS format  (exception  will  be  thrown
otherwise).

INPUT PARAMETERS
    S           -   sparse square matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is used:
                    * if upper triangle is given,  only   S[i,j] for  j>=i
                      are used, and lower triangle is  ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for  j<=i
                      are used, and upper triangle is ignored.
    IsUnit      -   unit or non-unit diagonal:
                    * if True, diagonal elements of triangular matrix  are
                      considered equal to 1.0. Actual elements  stored  in
                      S are not referenced at all.
                    * if False, diagonal stored in S is used. It  is  your
                      responsibility  to  make  sure  that   diagonal   is
                      non-zero.
    OpType      -   operation type:
                    * if 0, S*x is calculated
                    * if 1, (S^T)*x is calculated (transposition)
    X           -   array[N] which stores input  vector.  For  performance
                    reasons we make only quick  checks  -  we  check  that
                    array  size  is  at  least  N, but we do not check for
                    NAN's or INF's.

OUTPUT PARAMETERS
    X           -   array[N], inv(op(S))*x

NOTE: this function throws exception when called for  non-CRS/SKS  matrix.
      You must convert your matrix  with  SparseConvertToCRS/SKS()  before
      using this function.

NOTE: no assertion or tests are done during algorithm  operation.   It  is
      your responsibility to provide invertible matrix to algorithm.

  -- ALGLIB PROJECT --
     Copyright 20.01.2014 by Bochkanov Sergey
*************************************************************************/
void sparsetrsv(const sparsematrix &s, const bool isupper, const bool isunit, const ae_int_t optype, real_1d_array &x, const xparams _xparams = alglib::xdefault);

`sparseunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void sparseunserialize(const std::string &s_in, sparsematrix &obj);
void sparseunserialize(const std::istream &s_in, sparsematrix &obj);

`sparsevsmv` function

/*************************************************************************
This function calculates vector-matrix-vector product x'*S*x, where  S is
symmetric matrix. Matrix S must be stored in CRS or SKS format (exception
will be thrown otherwise).

INPUT PARAMETERS
    S           -   sparse M*M matrix in CRS or SKS format.
    IsUpper     -   whether upper or lower triangle of S is given:
                    * if upper triangle is given,  only   S[i,j] for j>=i
                      are used, and lower triangle is ignored (it can  be
                      empty - these elements are not referenced at all).
                    * if lower triangle is given,  only   S[i,j] for j<=i
                      are used, and upper triangle is ignored.
    X           -   array[N], input vector. For  performance  reasons  we
                    make only quick checks - we check that array size  is
                    at least N, but we do not check for NAN's or INF's.

RESULT
    x'*S*x

NOTE: this function throws exception when called for non-CRS/SKS  matrix.
You must convert your matrix with SparseConvertToCRS/SKS()  before  using
this function.

  -- ALGLIB PROJECT --
     Copyright 27.01.2014 by Bochkanov Sergey
*************************************************************************/
double sparsevsmv(const sparsematrix &s, const bool isupper, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

sparse_d_1 example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates creation/initialization of the sparse matrix
        // and matrix-vector multiplication.
        //
        // First, we have to create matrix and initialize it. Matrix is initially created
        // in the Hash-Table format, which allows convenient initialization. We can modify
        // Hash-Table matrix with sparseset() and sparseadd() functions.
        //
        // NOTE: Unlike CRS format, Hash-Table representation allows you to initialize
        // elements in the arbitrary order. You may see that we initialize a[0][0] first,
        // then move to the second row, and then move back to the first row.
        //
        sparsematrix s;
        sparsecreate(2, 2, s);
        sparseset(s, 0, 0, 2.0);
        sparseset(s, 1, 1, 1.0);
        sparseset(s, 0, 1, 1.0);

        sparseadd(s, 1, 1, 4.0);

        //
        // Now S is equal to
        //   [ 2 1 ]
        //   [   5 ]
        // Lets check it by reading matrix contents with sparseget().
        // You may see that with sparseget() you may read both non-zero
        // and zero elements.
        //
        double v;
        v = sparseget(s, 0, 0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.0000
        v = sparseget(s, 0, 1);
        printf("%.2f\n", double(v)); // EXPECTED: 1.0000
        v = sparseget(s, 1, 0);
        printf("%.2f\n", double(v)); // EXPECTED: 0.0000
        v = sparseget(s, 1, 1);
        printf("%.2f\n", double(v)); // EXPECTED: 5.0000

        //
        // After successful creation we can use our matrix for linear operations.
        //
        // However, there is one more thing we MUST do before using S in linear
        // operations: we have to convert it from HashTable representation (used for
        // initialization and dynamic operations) to CRS format with sparseconverttocrs()
        // call. If you omit this call, ALGLIB will generate exception on the first
        // attempt to use S in linear operations. 
        //
        sparseconverttocrs(s);

        //
        // Now S is in the CRS format and we are ready to do linear operations.
        // Lets calculate A*x for some x.
        //
        real_1d_array x = "[1,-1]";
        real_1d_array y = "[]";
        sparsemv(s, x, y);
        printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [1.000,-5.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

sparse_d_crs example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // This example demonstrates creation/initialization of the sparse matrix in the
        // CRS format.
        //
        // Hash-Table format used by default is very convenient (it allows easy
        // insertion of elements, automatic memory reallocation), but has
        // significant memory and performance overhead. Insertion of one element 
        // costs hundreds of CPU cycles, and memory consumption is several times
        // higher than that of CRS.
        //
        // When you work with really large matrices and when you can tell in 
        // advance how many elements EXACTLY you need, it can be beneficial to 
        // create matrix in the CRS format from the very beginning.
        //
        // If you want to create matrix in the CRS format, you should:
        // * use sparsecreatecrs() function
        // * know row sizes in advance (number of non-zero entries in the each row)
        // * initialize matrix with sparseset() - another function, sparseadd(), is not allowed
        // * initialize elements from left to right, from top to bottom, each
        //   element is initialized only once.
        //
        sparsematrix s;
        integer_1d_array row_sizes = "[2,2,2,1]";
        sparsecreatecrs(4, 4, row_sizes, s);
        sparseset(s, 0, 0, 2.0);
        sparseset(s, 0, 1, 1.0);
        sparseset(s, 1, 1, 4.0);
        sparseset(s, 1, 2, 2.0);
        sparseset(s, 2, 2, 3.0);
        sparseset(s, 2, 3, 1.0);
        sparseset(s, 3, 3, 9.0);

        //
        // Now S is equal to
        //   [ 2 1     ]
        //   [   4 2   ]
        //   [     3 1 ]
        //   [       9 ]
        //
        // We should point that we have initialized S elements from left to right,
        // from top to bottom. CRS representation does NOT allow you to do so in
        // the different order. Try to change order of the sparseset() calls above,
        // and you will see that your program generates exception.
        //
        // We can check it by reading matrix contents with sparseget().
        // However, you should remember that sparseget() is inefficient on
        // CRS matrices (it may have to pass through all elements of the row 
        // until it finds element you need).
        //
        double v;
        v = sparseget(s, 0, 0);
        printf("%.2f\n", double(v)); // EXPECTED: 2.0000
        v = sparseget(s, 2, 3);
        printf("%.2f\n", double(v)); // EXPECTED: 1.0000

        // you may see that you can read zero elements (which are not stored) with sparseget()
        v = sparseget(s, 3, 2);
        printf("%.2f\n", double(v)); // EXPECTED: 0.0000

        //
        // After successful creation we can use our matrix for linear operations.
        // Lets calculate A*x for some x.
        //
        real_1d_array x = "[1,-1,1,-1]";
        real_1d_array y = "[]";
        sparsemv(s, x, y);
        printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [1.000,-2.000,2.000,-9]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`spdgevd` subpackage

Functions

smatrixgevd
smatrixgevdreduce

Examples

`smatrixgevd` function

/*************************************************************************
Algorithm for solving the following generalized symmetric positive-definite
eigenproblem:
    A*x = lambda*B*x (1) or
    A*B*x = lambda*x (2) or
    B*A*x = lambda*x (3).
where A is a symmetric matrix, B - symmetric positive-definite matrix.
The problem is solved by reducing it to an ordinary  symmetric  eigenvalue
problem.

Input parameters:
    A           -   symmetric matrix which is given by its upper or lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    N           -   size of matrices A and B.
    IsUpperA    -   storage format of matrix A.
    B           -   symmetric positive-definite matrix which is given by
                    its upper or lower triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    IsUpperB    -   storage format of matrix B.
    ZNeeded     -   if ZNeeded is equal to:
                     * 0, the eigenvectors are not returned;
                     * 1, the eigenvectors are returned.
    ProblemType -   if ProblemType is equal to:
                     * 1, the following problem is solved: A*x = lambda*B*x;
                     * 2, the following problem is solved: A*B*x = lambda*x;
                     * 3, the following problem is solved: B*A*x = lambda*x.

Output parameters:
    D           -   eigenvalues in ascending order.
                    Array whose index ranges within [0..N-1].
    Z           -   if ZNeeded is equal to:
                     * 0, Z hasn't changed;
                     * 1, Z contains eigenvectors.
                    Array whose indexes range within [0..N-1, 0..N-1].
                    The eigenvectors are stored in matrix columns. It should
                    be noted that the eigenvectors in such problems do not
                    form an orthogonal system.

Result:
    True, if the problem was solved successfully.
    False, if the error occurred during the Cholesky decomposition of matrix
    B (the matrix isn't positive-definite) or during the work of the iterative
    algorithm for solving the symmetric eigenproblem.

See also the GeneralizedSymmetricDefiniteEVDReduce subroutine.

  -- ALGLIB --
     Copyright 1.28.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixgevd(const real_2d_array &a, const ae_int_t n, const bool isuppera, const real_2d_array &b, const bool isupperb, const ae_int_t zneeded, const ae_int_t problemtype, real_1d_array &d, real_2d_array &z, const xparams _xparams = alglib::xdefault);

`smatrixgevdreduce` function

/*************************************************************************
Algorithm for reduction of the following generalized symmetric positive-
definite eigenvalue problem:
    A*x = lambda*B*x (1) or
    A*B*x = lambda*x (2) or
    B*A*x = lambda*x (3)
to the symmetric eigenvalues problem C*y = lambda*y (eigenvalues of this and
the given problems are the same, and the eigenvectors of the given problem
could be obtained by multiplying the obtained eigenvectors by the
transformation matrix x = R*y).

Here A is a symmetric matrix, B - symmetric positive-definite matrix.

Input parameters:
    A           -   symmetric matrix which is given by its upper or lower
                    triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    N           -   size of matrices A and B.
    IsUpperA    -   storage format of matrix A.
    B           -   symmetric positive-definite matrix which is given by
                    its upper or lower triangular part.
                    Array whose indexes range within [0..N-1, 0..N-1].
    IsUpperB    -   storage format of matrix B.
    ProblemType -   if ProblemType is equal to:
                     * 1, the following problem is solved: A*x = lambda*B*x;
                     * 2, the following problem is solved: A*B*x = lambda*x;
                     * 3, the following problem is solved: B*A*x = lambda*x.

Output parameters:
    A           -   symmetric matrix which is given by its upper or lower
                    triangle depending on IsUpperA. Contains matrix C.
                    Array whose indexes range within [0..N-1, 0..N-1].
    R           -   upper triangular or low triangular transformation matrix
                    which is used to obtain the eigenvectors of a given problem
                    as the product of eigenvectors of C (from the right) and
                    matrix R (from the left). If the matrix is upper
                    triangular, the elements below the main diagonal
                    are equal to 0 (and vice versa). Thus, we can perform
                    the multiplication without taking into account the
                    internal structure (which is an easier though less
                    effective way).
                    Array whose indexes range within [0..N-1, 0..N-1].
    IsUpperR    -   type of matrix R (upper or lower triangular).

Result:
    True, if the problem was reduced successfully.
    False, if the error occurred during the Cholesky decomposition of
        matrix B (the matrix is not positive-definite).

  -- ALGLIB --
     Copyright 1.28.2006 by Bochkanov Sergey
*************************************************************************/
bool smatrixgevdreduce(real_2d_array &a, const ae_int_t n, const bool isuppera, const real_2d_array &b, const bool isupperb, const ae_int_t problemtype, real_2d_array &r, bool &isupperr, const xparams _xparams = alglib::xdefault);

`spline1d` subpackage

Classes

spline1dfitreport
spline1dinterpolant

Functions

spline1dbuildakima
spline1dbuildakimamod
spline1dbuildcatmullrom
spline1dbuildcubic
spline1dbuildhermite
spline1dbuildhermitebuf
spline1dbuildlinear
spline1dbuildlinearbuf
spline1dbuildmonotone
spline1dcalc
spline1dconvcubic
spline1dconvdiff2cubic
spline1dconvdiffcubic
spline1ddiff
spline1dfit
spline1dgriddiff2cubic
spline1dgriddiffcubic
spline1dintegrate
spline1dlintransx
spline1dlintransy
spline1dserialize
spline1dunpack
spline1dunserialize

Examples

spline1d_d_convdiff		Resampling using cubic splines
spline1d_d_cubic		Cubic spline interpolation
spline1d_d_griddiff		Differentiation on the grid using cubic splines
spline1d_d_linear		Piecewise linear spline interpolation
spline1d_d_monotone		Monotone interpolation

`spline1dfitreport` class

/*************************************************************************
Spline fitting report:
    TerminationType completion code:
                    * >0 for success
                    * <0 for failure
    RMSError        RMS error
    AvgError        average error
    AvgRelError     average relative error (for non-zero Y[I])
    MaxError        maximum error

Fields  below are  filled  by   obsolete    functions   (Spline1DFitCubic,
Spline1DFitHermite). Modern fitting functions do NOT fill these fields:
    TaskRCond       reciprocal of task's condition number
*************************************************************************/
class spline1dfitreport
{
public:
    spline1dfitreport();
    spline1dfitreport(const spline1dfitreport &rhs);
    spline1dfitreport& operator=(const spline1dfitreport &rhs);
    virtual ~spline1dfitreport();
    ae_int_t terminationtype;
    double taskrcond;
    double rmserror;
    double avgerror;
    double avgrelerror;
    double maxerror;
};

`spline1dinterpolant` class

/*************************************************************************
1-dimensional spline interpolant
*************************************************************************/
class spline1dinterpolant
{
public:
    spline1dinterpolant();
    spline1dinterpolant(const spline1dinterpolant &rhs);
    spline1dinterpolant& operator=(const spline1dinterpolant &rhs);
    virtual ~spline1dinterpolant();
};

`spline1dbuildakima` function

/*************************************************************************
This subroutine builds Akima spline interpolant

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]
    Y           -   function values, array[0..N-1]
    N           -   points count (optional):
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C           -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildakima(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildakima(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildakimamod` function

/*************************************************************************
This subroutine builds modified Akima spline interpolant, with weights

    W[i]=|Delta[I]-Delta[I-1]|

replaced by

    W[i]=|Delta[I]-Delta[I-1]|+0.5*|Delta[I]+Delta[I-1]|

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]
    Y           -   function values, array[0..N-1]
    N           -   points count (optional):
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C           -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildakimamod(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildakimamod(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildcatmullrom` function

/*************************************************************************
This subroutine builds Catmull-Rom spline interpolant.

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1].
    Y           -   function values, array[0..N-1].

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundType   -   boundary condition type:
                    * -1 for periodic boundary condition
                    *  0 for parabolically terminated spline (default)
    Tension     -   tension parameter:
                    * tension=0   corresponds to classic Catmull-Rom spline (default)
                    * 0<tension<1 corresponds to more general form - cardinal spline

OUTPUT PARAMETERS:
    C           -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcatmullrom(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundtype, const double tension, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildcatmullrom(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildcubic` function

/*************************************************************************
This subroutine builds cubic spline interpolant.

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1].
    Y           -   function values, array[0..N-1].

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)

OUTPUT PARAMETERS:
    C           -   spline interpolant

ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildcubic(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildhermite` function

/*************************************************************************
This subroutine builds Hermite spline interpolant.

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]
    Y           -   function values, array[0..N-1]
    D           -   derivatives, array[0..N-1]
    N           -   points count (optional):
                    * N>=2
                    * if given, only first N points are used to build spline
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C           -   spline interpolant.


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildhermite(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildhermite(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildhermitebuf` function

/*************************************************************************
This subroutine builds Hermite spline interpolant.

Buffered version which reuses memory previously allocated in C as much  as
possible.

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildhermitebuf(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildhermitebuf(const real_1d_array &x, const real_1d_array &y, const real_1d_array &d, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildlinear` function

/*************************************************************************
This subroutine builds linear spline interpolant

INPUT PARAMETERS:
    X   -   spline nodes, array[0..N-1]
    Y   -   function values, array[0..N-1]
    N   -   points count (optional):
            * N>=2
            * if given, only first N points are used to build spline
            * if not given, automatically detected from X/Y sizes
              (len(X) must be equal to len(Y))

OUTPUT PARAMETERS:
    C   -   spline interpolant


ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildlinear(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildlinear(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`spline1dbuildlinearbuf` function

/*************************************************************************
This subroutine builds linear spline interpolant.

Buffered version of Spline1DBuildLinear() which reused  memory  previously
allocated in C as much as possible.

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildlinearbuf(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildlinearbuf(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline1dbuildmonotone` function

/*************************************************************************
This function builds monotone cubic Hermite interpolant. This interpolant
is monotonic in [x(0),x(n-1)] and is constant outside of this interval.

In  case  y[]  form  non-monotonic  sequence,  interpolant  is  piecewise
monotonic.  Say, for x=(0,1,2,3,4)  and  y=(0,1,2,1,0)  interpolant  will
monotonically grow at [0..2] and monotonically decrease at [2..4].

INPUT PARAMETERS:
    X           -   spline nodes, array[0..N-1]. Subroutine automatically
                    sorts points, so caller may pass unsorted array.
    Y           -   function values, array[0..N-1]
    N           -   the number of points(N>=2).

OUTPUT PARAMETERS:
    C           -   spline interpolant.

 -- ALGLIB PROJECT --
     Copyright 21.06.2012 by Bochkanov Sergey
*************************************************************************/
void spline1dbuildmonotone(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline1dbuildmonotone(const real_1d_array &x, const real_1d_array &y, spline1dinterpolant &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dcalc` function

/*************************************************************************
This subroutine calculates the value of the spline at the given point X.

INPUT PARAMETERS:
    C   -   spline interpolant
    X   -   point

Result:
    S(x)

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double spline1dcalc(const spline1dinterpolant &c, const double x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`spline1dconvcubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
function values y2[] (calculated at x2[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DCalc() calls, but it  can  be  several  times  faster,
whilst still having the same O(N*logN) running time.

When called for ordered X[], this function has O(N) running  time  instead
of O(N*logN).

INPUT PARAMETERS:
    X           -   old spline nodes
    Y           -   function values
    X2           -  new spline nodes

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points from X/Y are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)
    N2          -   new points count:
                    * N2>=2
                    * if given, only first N2 points from X2 are used
                    * if not given, automatically detected from X2 size

OUTPUT PARAMETERS:
    F2          -   function values at X2[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller  may pass unsorted array.
Function  values  are correctly reordered on  return, so F2[I]  is  always
equal to S(X2[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y2, const xparams _xparams = alglib::xdefault);
void spline1dconvcubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dconvdiff2cubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
function  values  y2[],  first  and  second  derivatives  d2[]  and  dd2[]
(calculated at x2[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff2() calls, but it  can  be  several  times faster,
whilst still having the same O(N*logN) running time.

When called for ordered X[], this function has O(N) running  time  instead
of O(N*logN).

INPUT PARAMETERS:
    X           -   old spline nodes
    Y           -   function values
    X2           -  new spline nodes

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points from X/Y are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)
    N2          -   new points count:
                    * N2>=2
                    * if given, only first N2 points from X2 are used
                    * if not given, automatically detected from X2 size

OUTPUT PARAMETERS:
    F2          -   function values at X2[]
    D2          -   first derivatives at X2[]
    DD2         -   second derivatives at X2[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller  may pass unsorted array.
Function  values  are correctly reordered on  return, so F2[I]  is  always
equal to S(X2[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiff2cubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y2, real_1d_array &d2, real_1d_array &dd2, const xparams _xparams = alglib::xdefault);
void spline1dconvdiff2cubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2, real_1d_array &d2, real_1d_array &dd2, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dconvdiffcubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at old nodes x[]  and new nodes  x2[],  it calculates and returns table of
function values y2[] and derivatives d2[] (calculated at x2[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it  can  be  several  times  faster,
whilst still having the same O(N*logN) running time.

When called for ordered X[], this function has O(N) running  time  instead
of O(N*logN).

INPUT PARAMETERS:
    X           -   old spline nodes
    Y           -   function values
    X2           -  new spline nodes

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points from X/Y are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)
    N2          -   new points count:
                    * N2>=2
                    * if given, only first N2 points from X2 are used
                    * if not given, automatically detected from X2 size

OUTPUT PARAMETERS:
    F2          -   function values at X2[]
    D2          -   first derivatives at X2[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller  may pass unsorted array.
Function  values  are correctly reordered on  return, so F2[I]  is  always
equal to S(X2[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dconvdiffcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, const real_1d_array &x2, const ae_int_t n2, real_1d_array &y2, real_1d_array &d2, const xparams _xparams = alglib::xdefault);
void spline1dconvdiffcubic(const real_1d_array &x, const real_1d_array &y, const real_1d_array &x2, real_1d_array &y2, real_1d_array &d2, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1ddiff` function

/*************************************************************************
This subroutine differentiates the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X   -   point

Result:
    S   -   S(x)
    DS  -   S'(x)
    D2S -   S''(x)

  -- ALGLIB PROJECT --
     Copyright 24.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1ddiff(const spline1dinterpolant &c, const double x, double &s, double &ds, double &d2s, const xparams _xparams = alglib::xdefault);

`spline1dfit` function

/*************************************************************************
Fitting by the smoothing (penalized) cubic spline.

This function approximates N scattered points (some of X[] may be equal to
each other) by the cubic spline with M equidistant nodes spanning interval
[min(x),max(x)].

The problem is regularized by adding nonlinearity  penalty  to  the  usual
least squares penalty function:

    MERIT_FUNC = F_LS + F_NL

where F_LS is a least squares error  term,  and  F_NL  is  a  nonlinearity
penalty which is roughly proportional to LambdaNS*integral{ S''(x)^2*dx }.
Algorithm applies automatic renormalization of F_NL  which  makes  penalty
term roughly invariant to scaling of X[] and changes in M.

This function is a new edition  of  penalized  regression  spline fitting,
a fast and compact one which needs much less resources that  its  previous
version: just O(maxMN) memory and O(maxMN) time.

NOTE: it is OK to run this function with both M<<N and M>>N;  say,  it  is
      possible to process 100 points with 1000-node spline.

INPUT PARAMETERS:
    X           -   points, array[0..N-1].
    Y           -   function values, array[0..N-1].
    N           -   number of points (optional):
                    * N>0
                    * if given, only first N elements of X/Y are processed
                    * if not given, automatically determined from lengths
    M           -   number of basis functions ( = number_of_nodes), M>=4.
    LambdaNS    -   LambdaNS>=0, regularization  constant  passed by user.
                    It penalizes nonlinearity in the regression spline.
                    Possible values to start from are 0.00001, 0.1, 1

OUTPUT PARAMETERS:
    S   -   spline interpolant.
    Rep -   Following fields are set:
            * TerminationType set to 1
            * RMSError      rms error on the (X,Y).
            * AvgError      average error on the (X,Y).
            * AvgRelError   average relative error on the non-zero Y
            * MaxError      maximum error

  -- ALGLIB PROJECT --
     Copyright 10.04.2023 by Bochkanov Sergey
*************************************************************************/
void spline1dfit(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t m, const double lambdans, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);
void spline1dfit(const real_1d_array &x, const real_1d_array &y, const ae_int_t m, const double lambdans, spline1dinterpolant &s, spline1dfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dgriddiff2cubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at  nodes  x[],  it  calculates  and  returns  tables  of first and second
function derivatives d1[] and d2[] (calculated at the same nodes x[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff2() calls, but it  can  be  several  times faster,
whilst still having the same O(N*logN) running time.

When called for ordered X[], this function has O(N) running  time  instead
of O(N*logN).


INPUT PARAMETERS:
    X           -   spline nodes
    Y           -   function values

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)

OUTPUT PARAMETERS:
    D1          -   S' values at X[]
    D2          -   S'' values at X[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so  D[I]  is  always
equal to S'(X[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiff2cubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, real_1d_array &d1, real_1d_array &d2, const xparams _xparams = alglib::xdefault);
void spline1dgriddiff2cubic(const real_1d_array &x, const real_1d_array &y, real_1d_array &d1, real_1d_array &d2, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dgriddiffcubic` function

/*************************************************************************
This function solves following problem: given table y[] of function values
at nodes x[], it calculates and returns table of function derivatives  d[]
(calculated at the same nodes x[]).

This function yields same result as Spline1DBuildCubic() call followed  by
sequence of Spline1DDiff() calls, but it  can  be  several  times  faster,
whilst still having the same O(N*logN) running time.

When called for ordered X[], this function has O(N) running  time  instead
of O(N*logN).

INPUT PARAMETERS:
    X           -   spline nodes
    Y           -   function values

OPTIONAL PARAMETERS:
    N           -   points count:
                    * N>=2
                    * if given, only first N points are used
                    * if not given, automatically detected from X/Y sizes
                      (len(X) must be equal to len(Y))
    BoundLType  -   boundary condition type for the left boundary
    BoundL      -   left boundary condition (first or second derivative,
                    depending on the BoundLType)
    BoundRType  -   boundary condition type for the right boundary
    BoundR      -   right boundary condition (first or second derivative,
                    depending on the BoundRType)

OUTPUT PARAMETERS:
    D           -   derivative values at X[]

ORDER OF POINTS

Subroutine automatically sorts points, so caller may pass unsorted array.
Derivative values are correctly reordered on return, so  D[I]  is  always
equal to S'(X[I]) independently of points order.

SETTING BOUNDARY VALUES:

The BoundLType/BoundRType parameters can have the following values:
    * -1, which corresonds to the periodic (cyclic) boundary conditions.
          In this case:
          * both BoundLType and BoundRType must be equal to -1.
          * BoundL/BoundR are ignored
          * Y[last] is ignored (it is assumed to be equal to Y[first]).
    *  0, which  corresponds  to  the  parabolically   terminated  spline
          (BoundL and/or BoundR are ignored).
    *  1, which corresponds to the first derivative boundary condition
    *  2, which corresponds to the second derivative boundary condition
    *  by default, BoundType=0 is used

PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS:

Problems with periodic boundary conditions have Y[first_point]=Y[last_point].
However, this subroutine doesn't require you to specify equal  values  for
the first and last points - it automatically forces them  to  be  equal by
copying  Y[first_point]  (corresponds  to the leftmost,  minimal  X[])  to
Y[last_point]. However it is recommended to pass consistent values of Y[],
i.e. to make Y[first_point]=Y[last_point].

  -- ALGLIB PROJECT --
     Copyright 03.09.2010 by Bochkanov Sergey
*************************************************************************/
void spline1dgriddiffcubic(const real_1d_array &x, const real_1d_array &y, const ae_int_t n, const ae_int_t boundltype, const double boundl, const ae_int_t boundrtype, const double boundr, real_1d_array &d, const xparams _xparams = alglib::xdefault);
void spline1dgriddiffcubic(const real_1d_array &x, const real_1d_array &y, real_1d_array &d, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline1dintegrate` function

/*************************************************************************
This subroutine integrates the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X   -   right bound of the integration interval [a, x],
            here 'a' denotes min(x[])
Result:
    integral(S(t)dt,a,x)

  -- ALGLIB PROJECT --
     Copyright 23.06.2007 by Bochkanov Sergey
*************************************************************************/
double spline1dintegrate(const spline1dinterpolant &c, const double x, const xparams _xparams = alglib::xdefault);

`spline1dlintransx` function

/*************************************************************************
This subroutine performs linear transformation of the spline argument.

INPUT PARAMETERS:
    C   -   spline interpolant.
    A, B-   transformation coefficients: x = A*t + B
Result:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dlintransx(spline1dinterpolant &c, const double a, const double b, const xparams _xparams = alglib::xdefault);

`spline1dlintransy` function

/*************************************************************************
This subroutine performs linear transformation of the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    A, B-   transformation coefficients: S2(x) = A*S(x) + B
Result:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dlintransy(spline1dinterpolant &c, const double a, const double b, const xparams _xparams = alglib::xdefault);

`spline1dserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void spline1dserialize(const spline1dinterpolant &obj, std::string &s_out);
void spline1dserialize(const spline1dinterpolant &obj, std::ostream &s_out);

`spline1dunpack` function

/*************************************************************************
This subroutine unpacks the spline into the coefficients table.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X   -   point

OUTPUT PARAMETERS:
    Tbl -   coefficients table, unpacked format, array[0..N-2, 0..5].
            For I = 0...N-2:
                Tbl[I,0] = X[i]
                Tbl[I,1] = X[i+1]
                Tbl[I,2] = C0
                Tbl[I,3] = C1
                Tbl[I,4] = C2
                Tbl[I,5] = C3
            On [x[i], x[i+1]] spline is equals to:
                S(x) = C0 + C1*t + C2*t^2 + C3*t^3
                t = x-x[i]

NOTE:
    You  can rebuild spline with  Spline1DBuildHermite()  function,  which
    accepts as inputs function values and derivatives at nodes, which  are
    easy to calculate when you have coefficients.

  -- ALGLIB PROJECT --
     Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline1dunpack(const spline1dinterpolant &c, ae_int_t &n, real_2d_array &tbl, const xparams _xparams = alglib::xdefault);

`spline1dunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void spline1dunserialize(const std::string &s_in, spline1dinterpolant &obj);
void spline1dunserialize(const std::istream &s_in, spline1dinterpolant &obj);

spline1d_d_convdiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use cubic spline to do resampling, i.e. having
        // values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
        // we calculate values/derivatives of cubic spline on 
        // another grid (equidistant with 9 nodes on [-1,+1])
        // WITHOUT CONSTRUCTION OF SPLINE OBJECT.
        //
        // There are efficient functions spline1dconvcubic(),
        // spline1dconvdiffcubic() and spline1dconvdiff2cubic() 
        // for such calculations.
        //
        // We use default boundary conditions ("parabolically terminated
        // spline") because cubic spline built with such boundary conditions 
        // will exactly reproduce any quadratic f(x).
        //
        // Actually, we could use natural conditions, but we feel that 
        // spline which exactly reproduces f() will show us more 
        // understandable results.
        //
        real_1d_array x_old = "[-1.0,-0.5,0.0,+0.5,+1.0]";
        real_1d_array y_old = "[+1.0,0.25,0.0,0.25,+1.0]";
        real_1d_array x_new = "[-1.00,-0.75,-0.50,-0.25,0.00,+0.25,+0.50,+0.75,+1.00]";
        real_1d_array y_new;
        real_1d_array d1_new;
        real_1d_array d2_new;

        //
        // First, conversion without differentiation.
        //
        //
        spline1dconvcubic(x_old, y_old, x_new, y_new);
        printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]

        //
        // Then, conversion with differentiation (first derivatives only)
        //
        //
        spline1dconvdiffcubic(x_old, y_old, x_new, y_new, d1_new);
        printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
        printf("%s\n", d1_new.tostring(3).c_str()); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]

        //
        // Finally, conversion with first and second derivatives
        //
        //
        spline1dconvdiff2cubic(x_old, y_old, x_new, y_new, d1_new, d2_new);
        printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
        printf("%s\n", d1_new.tostring(3).c_str()); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
        printf("%s\n", d2_new.tostring(3).c_str()); // EXPECTED: [2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline1d_d_cubic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use cubic spline to interpolate f(x)=x^2 sampled 
        // at 5 equidistant nodes on [-1,+1].
        //
        // First, we use default boundary conditions ("parabolically terminated
        // spline") because cubic spline built with such boundary conditions 
        // will exactly reproduce any quadratic f(x).
        //
        // Then we try to use natural boundary conditions
        //     d2S(-1)/dx^2 = 0.0
        //     d2S(+1)/dx^2 = 0.0
        // and see that such spline interpolated f(x) with small error.
        //
        real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
        real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
        double t = 0.25;
        double v;
        spline1dinterpolant s;
        ae_int_t natural_bound_type = 2;
        //
        // Test exact boundary conditions: build S(x), calculare S(0.25)
        // (almost same as original function)
        //
        spline1dbuildcubic(x, y, s);
        v = spline1dcalc(s, t);
        printf("%.4f\n", double(v)); // EXPECTED: 0.0625

        //
        // Test natural boundary conditions: build S(x), calculare S(0.25)
        // (small interpolation error)
        //
        spline1dbuildcubic(x, y, 5, natural_bound_type, 0.0, natural_bound_type, 0.0, s);
        v = spline1dcalc(s, t);
        printf("%.3f\n", double(v)); // EXPECTED: 0.0580
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline1d_d_griddiff example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use cubic spline to do grid differentiation, i.e. having
        // values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
        // we calculate derivatives of cubic spline at nodes WITHOUT
        // CONSTRUCTION OF SPLINE OBJECT.
        //
        // There are efficient functions spline1dgriddiffcubic() and
        // spline1dgriddiff2cubic() for such calculations.
        //
        // We use default boundary conditions ("parabolically terminated
        // spline") because cubic spline built with such boundary conditions 
        // will exactly reproduce any quadratic f(x).
        //
        // Actually, we could use natural conditions, but we feel that 
        // spline which exactly reproduces f() will show us more 
        // understandable results.
        //
        real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
        real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
        real_1d_array d1;
        real_1d_array d2;

        //
        // We calculate first derivatives: they must be equal to 2*x
        //
        spline1dgriddiffcubic(x, y, d1);
        printf("%s\n", d1.tostring(3).c_str()); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]

        //
        // Now test griddiff2, which returns first AND second derivatives.
        // First derivative is 2*x, second is equal to 2.0
        //
        spline1dgriddiff2cubic(x, y, d1, d2);
        printf("%s\n", d1.tostring(3).c_str()); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]
        printf("%s\n", d2.tostring(3).c_str()); // EXPECTED: [ 2.0,  2.0, 2.0,  2.0,  2.0]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline1d_d_linear example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use piecewise linear spline to interpolate f(x)=x^2 sampled 
        // at 5 equidistant nodes on [-1,+1].
        //
        real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
        real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
        double t = 0.25;
        double v;
        spline1dinterpolant s;

        // build spline
        spline1dbuildlinear(x, y, s);

        // calculate S(0.25) - it is quite different from 0.25^2=0.0625
        v = spline1dcalc(s, t);
        printf("%.4f\n", double(v)); // EXPECTED: 0.125
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline1d_d_monotone example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Spline built witn spline1dbuildcubic() can be non-monotone even when
        // Y-values form monotone sequence. Say, for x=[0,1,2] and y=[0,1,1]
        // cubic spline will monotonically grow until x=1.5 and then start
        // decreasing.
        //
        // That's why ALGLIB provides special spline construction function
        // which builds spline which preserves monotonicity of the original
        // dataset.
        //
        // NOTE: in case original dataset is non-monotonic, ALGLIB splits it
        // into monotone subsequences and builds piecewise monotonic spline.
        //
        real_1d_array x = "[0,1,2]";
        real_1d_array y = "[0,1,1]";
        spline1dinterpolant s;

        // build spline
        spline1dbuildmonotone(x, y, s);

        // calculate S at x = [-0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
        // you may see that spline is really monotonic
        double v;
        v = spline1dcalc(s, -0.5);
        printf("%.4f\n", double(v)); // EXPECTED: 0.0000
        v = spline1dcalc(s, 0.0);
        printf("%.4f\n", double(v)); // EXPECTED: 0.0000
        v = spline1dcalc(s, +0.5);
        printf("%.4f\n", double(v)); // EXPECTED: 0.5000
        v = spline1dcalc(s, 1.0);
        printf("%.4f\n", double(v)); // EXPECTED: 1.0000
        v = spline1dcalc(s, 1.5);
        printf("%.4f\n", double(v)); // EXPECTED: 1.0000
        v = spline1dcalc(s, 2.0);
        printf("%.4f\n", double(v)); // EXPECTED: 1.0000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`spline2d` subpackage

Classes

spline2dbuilder
spline2dfitreport
spline2dinterpolant

Functions

spline2dbuildbicubic
spline2dbuildbicubicmissing
spline2dbuildbicubicmissingbuf
spline2dbuildbicubicv
spline2dbuildbicubicvbuf
spline2dbuildbilinear
spline2dbuildbilinearmissing
spline2dbuildbilinearmissingbuf
spline2dbuildbilinearv
spline2dbuildbilinearvbuf
spline2dbuildclampedv
spline2dbuildercreate
spline2dbuildersetalgoblocklls
spline2dbuildersetalgofastddm
spline2dbuildersetalgonaivells
spline2dbuildersetarea
spline2dbuildersetareaauto
spline2dbuildersetconstterm
spline2dbuildersetgrid
spline2dbuildersetlinterm
spline2dbuildersetpoints
spline2dbuildersetuserterm
spline2dbuildersetzeroterm
spline2dbuildhermitev
spline2dcalc
spline2dcalcv
spline2dcalcvbuf
spline2dcalcvi
spline2dcopy
spline2ddiff
spline2ddiff2
spline2ddiff2vi
spline2ddiffvi
spline2dfit
spline2dlintransf
spline2dlintransxy
spline2dresamplebicubic
spline2dresamplebilinear
spline2dserialize
spline2dunpack
spline2dunpackv
spline2dunserialize

Examples

spline2d_bicubic		Bilinear spline interpolation
spline2d_bilinear		Bilinear spline interpolation
spline2d_copytrans		Copy and transform
spline2d_fit_blocklls		Fitting bicubic spline to irregular data
spline2d_unpack		Unpacking bilinear spline
spline2d_vector		Copy and transform

`spline2dbuilder` class

/*************************************************************************
Nonlinear least squares solver used to fit 2D splines to data
*************************************************************************/
class spline2dbuilder
{
public:
    spline2dbuilder();
    spline2dbuilder(const spline2dbuilder &rhs);
    spline2dbuilder& operator=(const spline2dbuilder &rhs);
    virtual ~spline2dbuilder();
};

`spline2dfitreport` class

/*************************************************************************
Spline 2D fitting report:
    rmserror        RMS error
    avgerror        average error
    maxerror        maximum error
    r2              coefficient of determination,  R-squared, 1-RSS/TSS
*************************************************************************/
class spline2dfitreport
{
public:
    spline2dfitreport();
    spline2dfitreport(const spline2dfitreport &rhs);
    spline2dfitreport& operator=(const spline2dfitreport &rhs);
    virtual ~spline2dfitreport();
    double rmserror;
    double avgerror;
    double maxerror;
    double r2;
};

`spline2dinterpolant` class

/*************************************************************************
2-dimensional spline inteprolant
*************************************************************************/
class spline2dinterpolant
{
public:
    spline2dinterpolant();
    spline2dinterpolant(const spline2dinterpolant &rhs);
    spline2dinterpolant& operator=(const spline2dinterpolant &rhs);
    virtual ~spline2dinterpolant();
};

`spline2dbuildbicubic` function

/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0

We recommend you to switch  to  Spline2DBuildBicubicV(),  which  is  more
flexible and accepts its arguments in more convenient order.

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubic(const real_1d_array &x, const real_1d_array &y, const real_2d_array &f, const ae_int_t m, const ae_int_t n, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbicubicmissing` function

/*************************************************************************
This  subroutine builds bicubic vector-valued  spline,  with  some  spline
cells being missing due to missing nodes.

This function produces C2-continuous spline, i.e. the has smooth first and
second derivatives both inside spline cells and at the boundaries.

When the node (i,j) is missing, it means that: a) we don't  have  function
value at this point (elements of F[] are ignored), and  b)  we  don't need
spline value at cells adjacent to the node (i,j), i.e. up to 4 spline cells
will be dropped. An attempt to compute spline value at  the  missing  cell
will return NAN.

It is important to  understand  that  this  subroutine  does  NOT  support
interpolation on scattered grids. It allows us to drop some nodes, but  at
the cost of making a "hole in the spline" around this point. If  you  want
function  that   can   "fill  the  gap",  use  RBF  or  another  scattered
interpolation method.

The  intended  usage  for  this  subroutine  are  regularly  sampled,  but
non-rectangular datasets.

Input parameters:
    X   -   spline abscissas, array[0..N-1]
    Y   -   spline ordinates, array[0..M-1]
    F   -   function values, array[0..M*N*D-1]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
            * missing values are ignored
    Missing array[M*N], Missing[J*N+I]=True means that corresponding entries
            of F[] are missing nodes.
    M,N -   grid size, M>=2, N>=2
    D   -   vector dimension, D>=1

Output parameters:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 27.06.2022 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubicmissing(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const boolean_1d_array &missing, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbicubicmissingbuf` function

/*************************************************************************
This  subroutine builds bicubic vector-valued  spline,  with  some  spline
cells being missing due to missing nodes.

Buffered version  of  Spline2DBuildBicubicMissing()  which  reuses  memory
previously allocated in C as much as possible.

  -- ALGLIB PROJECT --
     Copyright 27.06.2022 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubicmissingbuf(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const boolean_1d_array &missing, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbicubicv` function

/*************************************************************************
This subroutine builds a bicubic vector-valued spline using  parabolically
terminated end conditions.

This function produces a C2-continuous spline, i.e. the  spline has smooth
first and second  derivatives  both  inside  spline  cells  and  at  their
boundaries.

INPUT PARAMETERS:
    X   -   spline abscissas, array[N]
    N   -   N>=2:
            * if not given, automatically determined as len(X)
            * if given, only leading N elements of X are used
    Y   -   spline ordinates, array[M]
    M   -   M>=2:
            * if not given, automatically determined as len(Y)
            * if given, only leading M elements of Y are used
    F   -   function values, array[M*N*D]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    D   -   vector dimension, D>=1:
            * D=1 means scalar-valued bicubic spline
            * D>1 means vector-valued bicubic spline

OUTPUT PARAMETERS:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 2012-2023 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubicv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline2dbuildbicubicv(const real_1d_array &x, const real_1d_array &y, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dbuildbicubicvbuf` function

/*************************************************************************
This subroutine builds bicubic vector-valued spline.

Buffered version of Spline2DBuildBicubicV() which reuses memory previously
allocated in C as much as possible.

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbicubicvbuf(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbilinear` function

/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0

We recommend you to switch  to  Spline2DBuildBilinearV(),  which  is  more
flexible and accepts its arguments in more convenient order.

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinear(const real_1d_array &x, const real_1d_array &y, const real_2d_array &f, const ae_int_t m, const ae_int_t n, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbilinearmissing` function

/*************************************************************************
This subroutine builds bilinear vector-valued  spline,  with  some  spline
cells being missing due to missing nodes.

This function produces C0-continuous spline, i.e.  the  spline  itself  is
continuous, however its first and second  derivatives have discontinuities
at the spline cell boundaries.

When the node (i,j) is missing, it means that: a) we don't  have  function
value at this point (elements of F[] are ignored), and  b)  we  don't need
spline value at cells adjacent to the node (i,j), i.e. up to 4 spline cells
will be dropped. An attempt to compute spline value at  the  missing  cell
will return NAN.

It is important to  understand  that  this  subroutine  does  NOT  support
interpolation on scattered grids. It allows us to drop some nodes, but  at
the cost of making a "hole in the spline" around this point. If  you  want
function  that   can   "fill  the  gap",  use  RBF  or  another  scattered
interpolation method.

The  intended  usage  for  this  subroutine  are  regularly  sampled,  but
non-rectangular datasets.

Input parameters:
    X   -   spline abscissas, array[0..N-1]
    Y   -   spline ordinates, array[0..M-1]
    F   -   function values, array[0..M*N*D-1]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
            * missing values are ignored
    Missing array[M*N], Missing[J*N+I]=True means that corresponding entries
            of F[] are missing nodes.
    M,N -   grid size, M>=2, N>=2
    D   -   vector dimension, D>=1

Output parameters:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 27.06.2022 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinearmissing(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const boolean_1d_array &missing, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbilinearmissingbuf` function

/*************************************************************************
This subroutine builds bilinear vector-valued  spline,  with  some  spline
cells being missing due to missing nodes.

Buffered version of  Spline2DBuildBilinearMissing()  which  reuses  memory
previously allocated in C as much as possible.

  -- ALGLIB PROJECT --
     Copyright 27.06.2022 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinearmissingbuf(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const boolean_1d_array &missing, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildbilinearv` function

/*************************************************************************
This subroutine builds bilinear vector-valued spline.

This function produces C0-continuous spline, i.e.  the  spline  itself  is
continuous, however its first and second  derivatives have discontinuities
at the spline cell boundaries.

Input parameters:
    X   -   spline abscissas, array[0..N-1]
    Y   -   spline ordinates, array[0..M-1]
    F   -   function values, array[0..M*N*D-1]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    M,N -   grid size, M>=2, N>=2
    D   -   vector dimension, D>=1

Output parameters:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinearv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`spline2dbuildbilinearvbuf` function

/*************************************************************************
This subroutine builds bilinear vector-valued spline.

Buffered version of Spline2DBuildBilinearV() which reuses memory previously
allocated in C as much as possible.

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildbilinearvbuf(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildclampedv` function

/*************************************************************************
This subroutine  builds  a  bicubic  vector-valued  spline  using  clamped
boundary conditions:
* spline values at the grid nodes are specified
* boundary conditions for  first,  second  derivatives  or  for  parabolic
  termination at four boundaries (bottom  y=min(Y[]), top y=max(Y[]), left
  x=min(X[]), right x=max(X[])) are specified
* mixed derivatives at corners are specified
* it is possible to  have  different  boundary  conditions  for  different
  boundaries (first derivatives along  one  boundary,  second  derivatives
  along other one, parabolic termination along the rest and so on)
* it is possible to have either a scalar (D=1) or a vector-valued spline

This function produces a C2-continuous spline, i.e. the  spline has smooth
first and second  derivatives  both  inside  spline  cells  and  at  their
boundaries.

INPUT PARAMETERS:
    X           -   spline  abscissas,  array[N].  Can  be  unsorted,  the
                    function will sort it together with boundary conditions
                    and F[] array (the same set of  permutations  will  be
                    applied to X[] and F[]).
    N           -   N>=2:
                    * if not given, automatically determined as len(X)
                    * if given, only leading N elements of X are used
    Y           -   spline ordinates, array[M].  Can   be   unsorted,  the
                    function will sort it together with boundary conditions
                    and F[] array (the same set of  permutations  will  be
                    applied to X[] and F[]).
    M           -   M>=2:
                    * if not given, automatically determined as len(Y)
                    * if given, only leading M elements of Y are used
    BndBtm      -   array[D*N], boundary conditions at the bottom boundary
                    of the interpolation area  (corresponds to y=min(Y[]):
                    * if  BndTypeBtm=0,  the  spline  has   a   'parabolic
                      termination' boundary condition across that specific
                      boundary. In this case BndBtm is not even referenced
                      by the function and can be unallocated.
                    * otherwise contains derivatives with respect to X
                    * if BndTypeBtm=1, first derivatives are given
                    * if BndTypeBtm=2, second derivatives are given
                    * first D entries store derivatives at x=X[0], y=minY,
                      subsequent D entries store  derivatives  at  x=X[1],
                      y=minY and so on
    BndTop      -   array[D*N],  boundary  conditions  at the top boundary
                    of the interpolation area  (corresponds to y=max(Y[]):
                    * if  BndTypeTop=0,  the  spline  has   a   'parabolic
                      termination' boundary condition across that specific
                      boundary. In this case BndTop is not even referenced
                      by the function and can be unallocated.
                    * otherwise contains derivatives with respect to X
                    * if BndTypeTop=1, first derivatives are given
                    * if BndTypeTop=2, second derivatives are given
                    * first D entries store derivatives at x=X[0], y=maxY,
                      subsequent D entries store  derivatives  at  x=X[1],
                      y=maxY and so on
    BndLft      -   array[D*M], boundary conditions at  the  left boundary
                    of the  interpolation area (corresponds to x=min(X[]):
                    * if  BndTypeLft=0,  the  spline  has   a   'parabolic
                      termination' boundary condition across that specific
                      boundary. In this case BndLft is not even referenced
                      by the function and can be unallocated.
                    * otherwise contains derivatives with respect to Y
                    * if BndTypeLft=1, first derivatives are given
                    * if BndTypeLft=2, second derivatives are given
                    * first D entries store derivatives at x=minX, y=Y[0],
                      subsequent D entries store  derivatives  at  x=minX,
                      y=Y[1] and so on
    BndRgt      -   array[D*M], boundary conditions at  the right boundary
                    of the  interpolation area (corresponds to x=max(X[]):
                    * if  BndTypeRgt=0,  the  spline  has   a   'parabolic
                      termination' boundary condition across that specific
                      boundary. In this case BndRgt is not even referenced
                      by the function and can be unallocated.
                    * otherwise contains derivatives with respect to Y
                    * if BndTypeRgt=1, first derivatives are given
                    * if BndTypeRgt=2, second derivatives are given
                    * first D entries store derivatives at x=maxX, y=Y[0],
                      subsequent D entries store  derivatives  at  x=maxX,
                      y=Y[1] and so on
    MixedD      -   array[D*4], mixed derivatives  at  4  corners  of  the
                    interpolation area:
                    * derivative order depends on the order  of   boundary
                      conditions (bottom/top and left/right)  intersecting
                      at that corner:
                      **  for BndType(Btm|Top)=BndType(Lft|Rgt)=1 user has
                          to provide d2S/dXdY
                      **  for BndType(Btm|Top)=BndType(Lft|Rgt)=2 user has
                          to provide d4S/(dX^2*dY^2)
                      **  for BndType(Btm|Top)=1, BndType(Lft|Rgt)=2  user
                          has to provide d3S/(dX^2*dY)
                      **  for BndType(Btm|Top)=2, BndType(Lft|Rgt)=1  user
                          has to provide d3S/(dX*dY^2)
                      **  if one of the intersecting bounds has 'parabolic
                          termination'  condition,   this  specific  mixed
                          derivative is not used
                    * first D entries store derivatives at the bottom left
                      corner x=min(X[]), y=min(Y[])
                    * subsequent D entries store derivatives at the bottom
                      right corner x=max(X[]), y=min(Y[])
                    * subsequent D entries store derivatives  at  the  top
                      left corner  x=min(X[]), y=max(Y[])
                    * subsequent D entries store derivatives  at  the  top
                      right corner x=max(X[]), y=max(Y[])
                    * if all bounds have 'parabolic termination' condition,
                      MixedD[]  is  not  referenced  at  all  and  can  be
                      unallocated.
    F           -   function values, array[M*N*D]:
                    * first D elements store D values at (X[0],Y[0])
                    * next D elements store D values at (X[1],Y[0])
                    * general form - D function values at (X[i],Y[j])  are
                      stored at F[D*(J*N+I)...D*(J*N+I)+D-1].
    D           -   vector dimension, D>=1:
                    * D=1 means scalar-valued bicubic spline
                    * D>1 means vector-valued bicubic spline

OUTPUT PARAMETERS:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 2012-2023 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildclampedv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &bndbtm, const ae_int_t bndtypebtm, const real_1d_array &bndtop, const ae_int_t bndtypetop, const real_1d_array &bndlft, const ae_int_t bndtypelft, const real_1d_array &bndrgt, const ae_int_t bndtypergt, const real_1d_array &mixedd, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline2dbuildclampedv(const real_1d_array &x, const real_1d_array &y, const real_1d_array &bndbtm, const ae_int_t bndtypebtm, const real_1d_array &bndtop, const ae_int_t bndtypetop, const real_1d_array &bndlft, const ae_int_t bndtypelft, const real_1d_array &bndrgt, const ae_int_t bndtypergt, const real_1d_array &mixedd, const real_1d_array &f, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dbuildercreate` function

/*************************************************************************
This subroutine creates least squares solver used to  fit  2D  splines  to
irregularly sampled (scattered) data.

Solver object is used to perform spline fits as follows:
* solver object is created with spline2dbuildercreate() function
* dataset is added with spline2dbuildersetpoints() function
* fit area is chosen:
  * spline2dbuildersetarea()     - for user-defined area
  * spline2dbuildersetareaauto() - for automatically chosen area
* number of grid nodes is chosen with spline2dbuildersetgrid()
* prior term is chosen with one of the following functions:
  * spline2dbuildersetlinterm()   to set linear prior
  * spline2dbuildersetconstterm() to set constant prior
  * spline2dbuildersetzeroterm()  to set zero prior
  * spline2dbuildersetuserterm()  to set user-defined constant prior
* solver algorithm is chosen with either:
  * spline2dbuildersetalgoblocklls() - BlockLLS algorithm, medium-scale problems
  * spline2dbuildersetalgofastddm()  - FastDDM algorithm, large-scale problems
* finally, fitting itself is performed with spline2dfit() function.

Most of the steps above can be omitted,  solver  is  configured with  good
defaults. The minimum is to call:
* spline2dbuildercreate() to create solver object
* spline2dbuildersetpoints() to specify dataset
* spline2dbuildersetgrid() to tell how many nodes you need
* spline2dfit() to perform fit

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    D   -   positive number, number of Y-components: D=1 for simple scalar
            fit, D>1 for vector-valued spline fitting.

OUTPUT PARAMETERS:
    S   -   solver object

  -- ALGLIB PROJECT --
     Copyright 29.01.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildercreate(const ae_int_t d, spline2dbuilder &state, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dbuildersetalgoblocklls` function

/*************************************************************************
This  function  allows  you to choose least squares solver used to perform
fitting. This function sets solver algorithm to "BlockLLS", which performs
least squares fitting  with  fast  sparse  direct  solver,  with  optional
nonsmoothness penalty being applied.

This solver produces C2-continuous spline.

Nonlinearity penalty has the following form:

                          [                                            ]
    P() ~ Lambda* integral[ (d2S/dx2)^2 + 2*(d2S/dxdy)^2 + (d2S/dy2)^2 ]dxdy
                          [                                            ]

here integral is calculated over entire grid, and "~" means "proportional"
because integral is normalized after calcilation. Extremely  large  values
of Lambda result in linear fit being performed.

NOTE: this algorithm is the most robust and controllable one,  but  it  is
      limited by 512x512 grids and (say) up to 1.000.000 points.  However,
      ALGLIB has one more  spline  solver:  FastDDM  algorithm,  which  is
      intended for really large-scale problems (in 10M-100M range). FastDDM
      algorithm also has better parallelism properties.

More information on BlockLLS solver:
* memory requirements: ~[32*K^3+256*NPoints]  bytes  for  KxK  grid   with
  NPoints-sized dataset
* serial running time: O(K^4+NPoints)
* parallelism potential: limited. You may get some sublinear gain when
  working with large grids (K's in 256..512 range)

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    LambdaNS-   non-negative value:
                * positive value means that some smoothing is applied
                * zero value means  that  no  smoothing  is  applied,  and
                  corresponding entries of design matrix  are  numerically
                  zero and dropped from consideration.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetalgoblocklls(spline2dbuilder &state, const double lambdans, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetalgofastddm` function

/*************************************************************************
This  function  allows  you to choose least squares solver used to perform
fitting. This function sets solver algorithm to "FastDDM", which  performs
fast parallel fitting by splitting problem into smaller chunks and merging
results together.

Unlike BlockLLS, this solver produces merely C1 continuous models  because
domain decomposition part disrupts C2 continuity.

This solver is optimized for large-scale problems, starting  from  256x256
grids, and up to 10000x10000 grids. Of course, it will  work  for  smaller
grids too.

More detailed description of the algorithm is given below:
* algorithm generates hierarchy  of  nested  grids,  ranging  from  ~16x16
  (topmost "layer" of the model) to ~KX*KY one (final layer). Upper layers
  model global behavior of the function, lower layers are  used  to  model
  fine details. Moving from layer to layer doubles grid density.
* fitting  is  started  from  topmost  layer, subsequent layers are fitted
  using residuals from previous ones.
* user may choose to skip generation of upper layers and generate  only  a
  few bottom ones, which  will  result  in  much  better  performance  and
  parallelization efficiency, at the cost of algorithm inability to "patch"
  large holes in the dataset.
* every layer is regularized using progressively increasing regularization
  coefficient; thus, increasing  LambdaV  penalizes  fine  details  first,
  leaving lower frequencies almost intact for a while.
* after fitting is done, all layers are merged together into  one  bicubic
  spline

IMPORTANT: regularization coefficient used by  this  solver  is  different
           from the one used by  BlockLLS.  Latter  utilizes  nonlinearity
           penalty,  which  is  global  in  nature  (large  regularization
           results in global linear trend being  extracted);  this  solver
           uses another, localized form of penalty, which is suitable  for
           parallel processing.

Notes on memory and performance:
* memory requirements: most memory is consumed  during  modeling   of  the
  higher layers; ~[512*NPoints] bytes is required for a  model  with  full
  hierarchy of grids being generated. However, if you skip a  few  topmost
  layers, you will get nearly constant (wrt. points count and  grid  size)
  memory consumption.
* serial running time: O(K*K)+O(NPoints) for a KxK grid
* parallelism potential: good. You may get  nearly  linear  speed-up  when
  performing fitting with just a few layers. Adding more layers results in
  model becoming more global, which somewhat  reduces  efficiency  of  the
  parallel code.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    NLayers -   number of layers in the model:
                * NLayers>=1 means that up  to  chosen  number  of  bottom
                  layers is fitted
                * NLayers=0 means that maximum number of layers is  chosen
                  (according to current grid size)
                * NLayers<=-1 means that up to |NLayers| topmost layers is
                  skipped
                Recommendations:
                * good "default" value is 2 layers
                * you may need  more  layers,  if  your  dataset  is  very
                  irregular and you want to "patch"  large  holes.  For  a
                  grid step H (equal to AreaWidth/GridSize) you may expect
                  that last layer reproduces variations at distance H (and
                  can patch holes that wide); that higher  layers  operate
                  at distances 2*H, 4*H, 8*H and so on.
                * good value for "bullletproof" mode is  NLayers=0,  which
                  results in complete hierarchy of layers being generated.
    LambdaV -   regularization coefficient, chosen in such a way  that  it
                penalizes bottom layers (fine details) first.
                LambdaV>=0, zero value means that no penalty is applied.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetalgofastddm(spline2dbuilder &state, const ae_int_t nlayers, const double lambdav, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetalgonaivells` function

/*************************************************************************
This  function  allows  you to choose least squares solver used to perform
fitting. This function sets solver algorithm to "NaiveLLS".

IMPORTANT: NaiveLLS is NOT intended to be used in  real  life  code!  This
           algorithm solves problem by generated dense (K^2)x(K^2+NPoints)
           matrix and solves  linear  least  squares  problem  with  dense
           solver.

           It is here just  to  test  BlockLLS  against  reference  solver
           (and maybe for someone trying to compare well optimized  solver
           against straightforward approach to the LLS problem).

More information on naive LLS solver:
* memory requirements: ~[8*K^4+256*NPoints] bytes for KxK grid.
* serial running time: O(K^6+NPoints) for KxK grid
* when compared with BlockLLS,  NaiveLLS  has ~K  larger memory demand and
  ~K^2  larger running time.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    LambdaNS-   nonsmoothness penalty

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetalgonaivells(spline2dbuilder &state, const double lambdans, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetarea` function

/*************************************************************************
This  function  sets  area  where  2D  spline  interpolant  is   built  to
user-defined one: [XA,XB]*[YA,YB]

INPUT PARAMETERS:
    S       -   spline 2D builder object
    XA,XB   -   spatial extent in the first (X) dimension, XA<XB
    YA,YB   -   spatial extent in the second (Y) dimension, YA<YB

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetarea(spline2dbuilder &state, const double xa, const double xb, const double ya, const double yb, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetareaauto` function

/*************************************************************************
This function sets area where 2D spline interpolant is built. "Auto" means
that area extent is determined automatically from dataset extent.

INPUT PARAMETERS:
    S       -   spline 2D builder object

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetareaauto(spline2dbuilder &state, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetconstterm` function

/*************************************************************************
This function sets constant prior term (model is a sum of  bicubic  spline
and global prior, which can be linear, constant, user-defined  constant or
zero).

Constant prior term is determined by least squares fitting.

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetconstterm(spline2dbuilder &state, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetgrid` function

/*************************************************************************
This  function  sets  nodes  count  for  2D spline interpolant. Fitting is
performed on area defined with one of the "setarea"  functions;  this  one
sets number of nodes placed upon the fitting area.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    KX      -   nodes count for the first (X) dimension; fitting  interval
                [XA,XB] is separated into KX-1 subintervals, with KX nodes
                created at the boundaries.
    KY      -   nodes count for the first (Y) dimension; fitting  interval
                [YA,YB] is separated into KY-1 subintervals, with KY nodes
                created at the boundaries.

NOTE: at  least  4  nodes  is  created in each dimension, so KX and KY are
      silently increased if needed.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetgrid(spline2dbuilder &state, const ae_int_t kx, const ae_int_t ky, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetlinterm` function

/*************************************************************************
This function sets linear prior term (model is a sum of bicubic spline and
global  prior,  which  can  be  linear, constant, user-defined constant or
zero).

Linear prior term is determined by least squares fitting.

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetlinterm(spline2dbuilder &state, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetpoints` function

/*************************************************************************
This function adds dataset to the builder object.

This function overrides results of the previous calls, i.e. multiple calls
of this function will result in only the last set being added.

INPUT PARAMETERS:
    S       -   spline 2D builder object
    XY      -   points, array[N,2+D]. One  row  corresponds to  one  point
                in the dataset. First 2  elements  are  coordinates,  next
                D  elements are function values. Array may  be larger than
                specified, in  this  case  only leading [N,NX+NY] elements
                will be used.
    N       -   number of points in the dataset

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetpoints(spline2dbuilder &state, const real_2d_array &xy, const ae_int_t n, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetuserterm` function

/*************************************************************************
This function sets constant prior term (model is a sum of  bicubic  spline
and global prior, which can be linear, constant, user-defined  constant or
zero).

Constant prior term is determined by least squares fitting.

INPUT PARAMETERS:
    S       -   spline builder
    V       -   value for user-defined prior

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetuserterm(spline2dbuilder &state, const double v, const xparams _xparams = alglib::xdefault);

`spline2dbuildersetzeroterm` function

/*************************************************************************
This function sets zero prior term (model is a sum of bicubic  spline  and
global  prior,  which  can  be  linear, constant, user-defined constant or
zero).

INPUT PARAMETERS:
    S       -   spline builder

  -- ALGLIB --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildersetzeroterm(spline2dbuilder &state, const xparams _xparams = alglib::xdefault);

`spline2dbuildhermitev` function

/*************************************************************************
This subroutine builds a Hermite bicubic vector-valued spline.

This function produces merely C1-continuous spline, i.e. the   spline  has
smooth first derivatives.

INPUT PARAMETERS:
    X   -   spline abscissas, array[N]
    N   -   N>=2:
            * if not given, automatically determined as len(X)
            * if given, only leading N elements of X are used
    Y   -   spline ordinates, array[M]
    M   -   M>=2:
            * if not given, automatically determined as len(Y)
            * if given, only leading M elements of Y are used
    F   -   function values, array[M*N*D]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    dFdX-   spline derivatives with respect to X, array[M*N*D]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    dFdY-   spline derivatives with respect to Y, array[M*N*D]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    d2FdXdY-mixed derivatives with respect to X and Y, array[M*N*D]:
            * first D elements store D values at (X[0],Y[0])
            * next D elements store D values at (X[1],Y[0])
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(J*N+I)...D*(J*N+I)+D-1].
    D   -   vector dimension, D>=1:
            * D=1 means scalar-valued bicubic spline
            * D>1 means vector-valued bicubic spline

OUTPUT PARAMETERS:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 2012-2023 by Bochkanov Sergey
*************************************************************************/
void spline2dbuildhermitev(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &f, const real_1d_array &dfdx, const real_1d_array &dfdy, const real_1d_array &d2fdxdy, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);
void spline2dbuildhermitev(const real_1d_array &x, const real_1d_array &y, const real_1d_array &f, const real_1d_array &dfdx, const real_1d_array &dfdy, const real_1d_array &d2fdxdy, const ae_int_t d, spline2dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline2dcalc` function

/*************************************************************************
This subroutine calculates the value of the bilinear or bicubic spline  at
the given point X.

Input parameters:
    C   -   2D spline object.
            Built by spline2dbuildbilinearv or spline2dbuildbicubicv.
    X, Y-   point

Result:
    S(x,y)

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
double spline2dcalc(const spline2dinterpolant &c, const double x, const double y, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`spline2dcalcv` function

/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y-   point

OUTPUT PARAMETERS:
    F   -   array[D] which stores function values.  F is out-parameter and
            it  is  reallocated  after  call to this function. In case you
            want  to    reuse  previously  allocated  F,   you   may   use
            Spline2DCalcVBuf(),  which  reallocates  F only when it is too
            small.

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dcalcv(const spline2dinterpolant &c, const double x, const double y, real_1d_array &f, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dcalcvbuf` function

/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y).

If you need just some specific component of vector-valued spline, you  can
use spline2dcalcvi() function.

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y-   point
    F   -   output buffer, possibly preallocated array. In case array size
            is large enough to store result, it is not reallocated.  Array
            which is too short will be reallocated

OUTPUT PARAMETERS:
    F   -   array[D] (or larger) which stores function values

  -- ALGLIB PROJECT --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dcalcvbuf(const spline2dinterpolant &c, const double x, const double y, real_1d_array &f, const xparams _xparams = alglib::xdefault);

`spline2dcalcvi` function

/*************************************************************************
This subroutine calculates specific component of vector-valued bilinear or
bicubic spline at the given point (X,Y).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y-   point
    I   -   component index, in [0,D). An exception is generated for out
            of range values.

RESULT:
    value of I-th component

  -- ALGLIB PROJECT --
     Copyright 01.02.2018 by Bochkanov Sergey
*************************************************************************/
double spline2dcalcvi(const spline2dinterpolant &c, const double x, const double y, const ae_int_t i, const xparams _xparams = alglib::xdefault);

`spline2dcopy` function

/*************************************************************************
This subroutine makes the copy of the spline model.

Input parameters:
    C   -   spline interpolant

Output parameters:
    CC  -   spline copy

  -- ALGLIB PROJECT --
     Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dcopy(const spline2dinterpolant &c, spline2dinterpolant &cc, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2ddiff` function

/*************************************************************************
This subroutine calculates the value of a bilinear or bicubic spline   and
its derivatives.

Use Spline2DDiff2() if you need second derivatives Sxx and Syy.

Input parameters:
    C   -   spline interpolant.
    X, Y-   point

Output parameters:
    F   -   S(x,y)
    FX  -   dS(x,y)/dX
    FY  -   dS(x,y)/dY

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2ddiff(const spline2dinterpolant &c, const double x, const double y, double &f, double &fx, double &fy, const xparams _xparams = alglib::xdefault);

`spline2ddiff2` function

/*************************************************************************
This subroutine calculates the value of a bilinear or bicubic spline   and
its second derivatives.

Input parameters:
    C   -   spline interpolant.
    X, Y-   point

Output parameters:
    F   -   S(x,y)
    FX  -   dS(x,y)/dX
    FY  -   dS(x,y)/dY
    FXX -   d2S(x,y)/dXdX
    FXY -   d2S(x,y)/dXdY
    FYY -   d2S(x,y)/dYdY

  -- ALGLIB PROJECT --
     Copyright 17.04.2023 by Bochkanov Sergey.

     The second derivatives code was contributed by  Horst  Greiner  under
     public domain terms.
*************************************************************************/
void spline2ddiff2(const spline2dinterpolant &c, const double x, const double y, double &f, double &fx, double &fy, double &fxx, double &fxy, double &fyy, const xparams _xparams = alglib::xdefault);

`spline2ddiff2vi` function

/*************************************************************************
This subroutine calculates the value and the derivatives of I-th component
of a vector-valued bilinear or bicubic spline.

Input parameters:
    C   -   spline interpolant.
    X, Y-   point
    I   -   component index, in [0,D)

Output parameters:
    F   -   S(x,y)
    FX  -   dS(x,y)/dX
    FY  -   dS(x,y)/dY
    FXX -   d2S(x,y)/dXdX
    FXY -   d2S(x,y)/dXdY
    FYY -   d2S(x,y)/dYdY

  -- ALGLIB PROJECT --
     Copyright 17.04.2023 by Bochkanov Sergey.

     The second derivatives code was contributed by  Horst  Greiner  under
     public domain terms.
*************************************************************************/
void spline2ddiff2vi(const spline2dinterpolant &c, const double x, const double y, const ae_int_t i, double &f, double &fx, double &fy, double &fxx, double &fxy, double &fyy, const xparams _xparams = alglib::xdefault);

`spline2ddiffvi` function

/*************************************************************************
This subroutine calculates the value and the derivatives of I-th component
of a  vector-valued bilinear or bicubic spline.

Input parameters:
    C   -   spline interpolant.
    X, Y-   point
    I   -   component index, in [0,D)

Output parameters:
    F   -   S(x,y)
    FX  -   dS(x,y)/dX
    FY  -   dS(x,y)/dY

  -- ALGLIB PROJECT --
     Copyright 05.07.2007 by Bochkanov Sergey
*************************************************************************/
void spline2ddiffvi(const spline2dinterpolant &c, const double x, const double y, const ae_int_t i, double &f, double &fx, double &fy, const xparams _xparams = alglib::xdefault);

`spline2dfit` function

/*************************************************************************
This function fits bicubic spline to current dataset, using current  area/
grid and current LLS solver.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

INPUT PARAMETERS:
    State   -   spline 2D builder object

OUTPUT PARAMETERS:
    S       -   2D spline, fit result
    Rep     -   fitting report, which provides some additional info  about
                errors, R2 coefficient and so on.

  -- ALGLIB --
     Copyright 05.02.2018 by Bochkanov Sergey
*************************************************************************/
void spline2dfit(spline2dbuilder &state, spline2dinterpolant &s, spline2dfitreport &rep, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dlintransf` function

/*************************************************************************
This subroutine performs linear transformation of the spline.

Input parameters:
    C   -   spline interpolant.
    A, B-   transformation coefficients: S2(x,y) = A*S(x,y) + B

Output parameters:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dlintransf(spline2dinterpolant &c, const double a, const double b, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dlintransxy` function

/*************************************************************************
This subroutine performs linear transformation of the spline argument.

Input parameters:
    C       -   spline interpolant
    AX, BX  -   transformation coefficients: x = A*t + B
    AY, BY  -   transformation coefficients: y = A*u + B
Result:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 30.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dlintransxy(spline2dinterpolant &c, const double ax, const double bx, const double ay, const double by, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dresamplebicubic` function

/*************************************************************************
Bicubic spline resampling

Input parameters:
    A           -   function values at the old grid,
                    array[0..OldHeight-1, 0..OldWidth-1]
    OldHeight   -   old grid height, OldHeight>1
    OldWidth    -   old grid width, OldWidth>1
    NewHeight   -   new grid height, NewHeight>1
    NewWidth    -   new grid width, NewWidth>1

Output parameters:
    B           -   function values at the new grid,
                    array[0..NewHeight-1, 0..NewWidth-1]

  -- ALGLIB routine --
     15 May, 2007
     Copyright by Bochkanov Sergey
*************************************************************************/
void spline2dresamplebicubic(const real_2d_array &a, const ae_int_t oldheight, const ae_int_t oldwidth, real_2d_array &b, const ae_int_t newheight, const ae_int_t newwidth, const xparams _xparams = alglib::xdefault);

`spline2dresamplebilinear` function

/*************************************************************************
Bilinear spline resampling

Input parameters:
    A           -   function values at the old grid,
                    array[0..OldHeight-1, 0..OldWidth-1]
    OldHeight   -   old grid height, OldHeight>1
    OldWidth    -   old grid width, OldWidth>1
    NewHeight   -   new grid height, NewHeight>1
    NewWidth    -   new grid width, NewWidth>1

Output parameters:
    B           -   function values at the new grid,
                    array[0..NewHeight-1, 0..NewWidth-1]

  -- ALGLIB routine --
     09.07.2007
     Copyright by Bochkanov Sergey
*************************************************************************/
void spline2dresamplebilinear(const real_2d_array &a, const ae_int_t oldheight, const ae_int_t oldwidth, real_2d_array &b, const ae_int_t newheight, const ae_int_t newwidth, const xparams _xparams = alglib::xdefault);

`spline2dserialize` function

/*************************************************************************
This function serializes data structure to string/stream.

Important properties of s_out:
* it contains alphanumeric characters, dots, underscores, minus signs
* these symbols are grouped into words, which are separated by spaces
  and Windows-style (CR+LF) newlines
* although  serializer  uses  spaces and CR+LF as separators, you can 
  replace any separator character by arbitrary combination of spaces,
  tabs, Windows or Unix newlines. It allows flexible reformatting  of
  the  string in case you want to include it into a text or XML file. 
  But you should not insert separators into the middle of the "words"
  nor should you change the case of letters.
* s_out can be freely moved between 32-bit and 64-bit systems, little
  and big endian machines, and so on. You can serialize structure  on
  32-bit machine and unserialize it on 64-bit one (or vice versa), or
  serialize  it  on  SPARC  and  unserialize  on  x86.  You  can also 
  serialize it in C++ version of ALGLIB and unserialize it in C# one, 
  and vice versa.
*************************************************************************/
void spline2dserialize(const spline2dinterpolant &obj, std::string &s_out);
void spline2dserialize(const spline2dinterpolant &obj, std::ostream &s_out);

`spline2dunpack` function

/*************************************************************************
This subroutine was deprecated in ALGLIB 3.6.0

We recommend you to switch  to  Spline2DUnpackV(),  which is more flexible
and accepts its arguments in more convenient order.

  -- ALGLIB PROJECT --
     Copyright 29.06.2007 by Bochkanov Sergey
*************************************************************************/
void spline2dunpack(const spline2dinterpolant &c, ae_int_t &m, ae_int_t &n, real_2d_array &tbl, const xparams _xparams = alglib::xdefault);

`spline2dunpackv` function

/*************************************************************************
This subroutine unpacks two-dimensional spline into the coefficients table

Input parameters:
    C   -   spline interpolant.

Result:
    M, N-   grid size (x-axis and y-axis)
    D   -   number of components
    Tbl -   coefficients table, unpacked format,
            D - components: [0..(N-1)*(M-1)*D-1, 0..20].
            For T=0..D-1 (component index), I = 0...N-2 (x index),
            J=0..M-2 (y index):
                K :=  T + I*D + J*D*(N-1)

                K-th row stores decomposition for T-th component of the
                vector-valued function

                Tbl[K,0] = X[i]
                Tbl[K,1] = X[i+1]
                Tbl[K,2] = Y[j]
                Tbl[K,3] = Y[j+1]
                Tbl[K,4] = C00
                Tbl[K,5] = C01
                Tbl[K,6] = C02
                Tbl[K,7] = C03
                Tbl[K,8] = C10
                Tbl[K,9] = C11
                ...
                Tbl[K,19] = C33
                Tbl[K,20] = 1 if the cell is present, 0 if the cell is missing.
                            In the latter case Tbl[4..19] are exactly zero.
            On each grid square spline is equals to:
                S(x) = SUM(c[i,j]*(t^i)*(u^j), i=0..3, j=0..3)
                t = x-x[j]
                u = y-y[i]

  -- ALGLIB PROJECT --
     Copyright 16.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline2dunpackv(const spline2dinterpolant &c, ae_int_t &m, ae_int_t &n, ae_int_t &d, real_2d_array &tbl, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline2dunserialize` function

/*************************************************************************
This function unserializes data structure from string/stream.
*************************************************************************/
void spline2dunserialize(const std::string &s_in, spline2dinterpolant &obj);
void spline2dunserialize(const std::istream &s_in, spline2dinterpolant &obj);

spline2d_bicubic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled 
        // at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
        //
        real_1d_array x = "[0.0, 0.5, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array f = "[0.00,0.25,1.00,2.00,2.25,3.00]";
        double vx = 0.25;
        double vy = 0.50;
        double v;
        double dx;
        double dy;
        spline2dinterpolant s;

        // build spline
        spline2dbuildbicubicv(x, 3, y, 2, f, 1, s);

        // calculate S(0.25,0.50)
        v = spline2dcalc(s, vx, vy);
        printf("%.4f\n", double(v)); // EXPECTED: 1.0625

        // calculate derivatives
        spline2ddiff(s, vx, vy, v, dx, dy);
        printf("%.4f\n", double(v)); // EXPECTED: 1.0625
        printf("%.4f\n", double(dx)); // EXPECTED: 0.5000
        printf("%.4f\n", double(dy)); // EXPECTED: 2.0000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline2d_bilinear example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled 
        // at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
        //
        real_1d_array x = "[0.0, 0.5, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array f = "[0.00,0.25,1.00,2.00,2.25,3.00]";
        double vx = 0.25;
        double vy = 0.50;
        double v;
        spline2dinterpolant s;

        // build spline
        spline2dbuildbilinearv(x, 3, y, 2, f, 1, s);

        // calculate S(0.25,0.50)
        v = spline2dcalc(s, vx, vy);
        printf("%.4f\n", double(v)); // EXPECTED: 1.1250
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline2d_copytrans example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We build bilinear spline for f(x,y)=x+2*y for (x,y) in [0,1].
        // Then we apply several transformations to this spline.
        //
        real_1d_array x = "[0.0, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array f = "[0.00,1.00,2.00,3.00]";
        spline2dinterpolant s;
        spline2dinterpolant snew;
        double v;
        spline2dbuildbilinearv(x, 2, y, 2, f, 1, s);

        // copy spline, apply transformation x:=2*xnew, y:=4*ynew
        // evaluate at (xnew,ynew) = (0.25,0.25) - should be same as (x,y)=(0.5,1.0)
        spline2dcopy(s, snew);
        spline2dlintransxy(snew, 2.0, 0.0, 4.0, 0.0);
        v = spline2dcalc(snew, 0.25, 0.25);
        printf("%.4f\n", double(v)); // EXPECTED: 2.500

        // copy spline, apply transformation SNew:=2*S+3
        spline2dcopy(s, snew);
        spline2dlintransf(snew, 2.0, 3.0);
        v = spline2dcalc(snew, 0.5, 1.0);
        printf("%.4f\n", double(v)); // EXPECTED: 8.000

        //
        // Same example, but for vector spline (f0,f1) = {x+2*y, 2*x+y}
        //
        real_1d_array f2 = "[0.00,0.00, 1.00,2.00, 2.00,1.00, 3.00,3.00]";
        real_1d_array vr;
        spline2dbuildbilinearv(x, 2, y, 2, f2, 2, s);

        // copy spline, apply transformation x:=2*xnew, y:=4*ynew
        spline2dcopy(s, snew);
        spline2dlintransxy(snew, 2.0, 0.0, 4.0, 0.0);
        spline2dcalcv(snew, 0.25, 0.25, vr);
        printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [2.500,2.000]

        // copy spline, apply transformation SNew:=2*S+3
        spline2dcopy(s, snew);
        spline2dlintransf(snew, 2.0, 3.0);
        spline2dcalcv(snew, 0.5, 1.0, vr);
        printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [8.000,7.000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline2d_fit_blocklls example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use bicubic spline to reproduce f(x,y)=1/(1+x^2+2*y^2) sampled
        // at irregular points (x,y) from [-1,+1]*[-1,+1]
        //
        // We have 5 such points, located approximately at corners of the area
        // and its center -  but not exactly at the grid. Thus, we have to FIT
        // the spline, i.e. to solve least squares problem
        //
        real_2d_array xy = "[[-0.987,-0.902,0.359],[0.948,-0.992,0.347],[-1.000,1.000,0.333],[1.000,0.973,0.339],[0.017,0.180,0.968]]";

        //
        // First step is to create spline2dbuilder object and set its properties:
        // * d=1 means that we create vector-valued spline with 1 component
        // * we specify dataset xy
        // * we rely on automatic selection of interpolation area
        // * we tell builder that we want to use 5x5 grid for an underlying spline
        // * we choose least squares solver named BlockLLS and configure it by
        //   telling that we want to apply zero nonlinearity penalty.
        //
        // NOTE: you can specify non-zero lambdav if you want to make your spline
        //       more "rigid", i.e. to penalize nonlinearity.
        //
        // NOTE: ALGLIB has two solvers which fit bicubic splines to irregular data,
        //       one of them is BlockLLS and another one is FastDDM. Former is
        //       intended for moderately sized grids (up to 512x512 nodes, although
        //       it may take up to few minutes); it is the most easy to use and
        //       control spline fitting function in the library. Latter, FastDDM,
        //       is intended for efficient solution of large-scale problems
        //       (up to 100.000.000 nodes). Both solvers can be parallelized, but
        //       FastDDM is much more efficient. See comments for more information.
        //
        spline2dbuilder builder;
        ae_int_t d = 1;
        double lambdav = 0.000;
        spline2dbuildercreate(d, builder);
        spline2dbuildersetpoints(builder, xy, 5);
        spline2dbuildersetgrid(builder, 5, 5);
        spline2dbuildersetalgoblocklls(builder, lambdav);

        //
        // Now we are ready to fit and evaluate our results
        //
        spline2dinterpolant s;
        spline2dfitreport rep;
        spline2dfit(builder, s, rep);

        // evaluate results - function value at the grid is reproduced exactly
        double v;
        v = spline2dcalc(s, -1, 1);
        printf("%.2f\n", double(v)); // EXPECTED: 0.333000

        // check maximum error - it must be nearly zero
        printf("%.2f\n", double(rep.maxerror)); // EXPECTED: 0.000
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline2d_unpack example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We build bilinear spline for f(x,y)=x+2*y+3*xy for (x,y) in [0,1].
        // Then we demonstrate how to unpack it.
        //
        real_1d_array x = "[0.0, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array f = "[0.00,1.00,2.00,6.00]";
        real_2d_array c;
        ae_int_t m;
        ae_int_t n;
        ae_int_t d;
        spline2dinterpolant s;

        // build spline
        spline2dbuildbilinearv(x, 2, y, 2, f, 1, s);

        // unpack and test
        spline2dunpackv(s, m, n, d, c);
        printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [[0, 1, 0, 1, 0,2,0,0, 1,3,0,0, 0,0,0,0, 0,0,0,0, 1]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline2d_vector example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We build bilinear vector-valued spline (f0,f1) = {x+2*y, 2*x+y}
        // Spline is built using function values at 2x2 grid: (x,y)=[0,1]*[0,1]
        // Then we perform evaluation at (x,y)=(0.1,0.3)
        //
        real_1d_array x = "[0.0, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array f = "[0.00,0.00, 1.00,2.00, 2.00,1.00, 3.00,3.00]";
        spline2dinterpolant s;
        real_1d_array vr;
        spline2dbuildbilinearv(x, 2, y, 2, f, 2, s);
        spline2dcalcv(s, 0.1, 0.3, vr);
        printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [0.700,0.500]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`spline3d` subpackage

Classes

spline3dinterpolant

Functions

spline3dbuildtrilinearv
spline3dbuildtrilinearvbuf
spline3dcalc
spline3dcalcv
spline3dcalcvbuf
spline3dlintransf
spline3dlintransxyz
spline3dresampletrilinear
spline3dunpackv

Examples

spline3d_trilinear		Trilinear spline interpolation
spline3d_vector		Vector-valued trilinear spline interpolation

`spline3dinterpolant` class

/*************************************************************************
3-dimensional spline inteprolant
*************************************************************************/
class spline3dinterpolant
{
public:
    spline3dinterpolant();
    spline3dinterpolant(const spline3dinterpolant &rhs);
    spline3dinterpolant& operator=(const spline3dinterpolant &rhs);
    virtual ~spline3dinterpolant();
};

`spline3dbuildtrilinearv` function

/*************************************************************************
This subroutine builds trilinear vector-valued spline.

INPUT PARAMETERS:
    X   -   spline abscissas,  array[0..N-1]
    Y   -   spline ordinates,  array[0..M-1]
    Z   -   spline applicates, array[0..L-1]
    F   -   function values, array[0..M*N*L*D-1]:
            * first D elements store D values at (X[0],Y[0],Z[0])
            * next D elements store D values at (X[1],Y[0],Z[0])
            * next D elements store D values at (X[2],Y[0],Z[0])
            * ...
            * next D elements store D values at (X[0],Y[1],Z[0])
            * next D elements store D values at (X[1],Y[1],Z[0])
            * next D elements store D values at (X[2],Y[1],Z[0])
            * ...
            * next D elements store D values at (X[0],Y[0],Z[1])
            * next D elements store D values at (X[1],Y[0],Z[1])
            * next D elements store D values at (X[2],Y[0],Z[1])
            * ...
            * general form - D function values at (X[i],Y[j]) are stored
              at F[D*(N*(M*K+J)+I)...D*(N*(M*K+J)+I)+D-1].
    M,N,
    L   -   grid size, M>=2, N>=2, L>=2
    D   -   vector dimension, D>=1

OUTPUT PARAMETERS:
    C   -   spline interpolant

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dbuildtrilinearv(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &z, const ae_int_t l, const real_1d_array &f, const ae_int_t d, spline3dinterpolant &c, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`spline3dbuildtrilinearvbuf` function

/*************************************************************************
This subroutine builds trilinear vector-valued spline.

Buffered  version   of   Spline3DBuildTrilinearV()  which  reuses   memory
previously allocated in C as much as possible.

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dbuildtrilinearvbuf(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, const real_1d_array &z, const ae_int_t l, const real_1d_array &f, const ae_int_t d, spline3dinterpolant &c, const xparams _xparams = alglib::xdefault);

`spline3dcalc` function

/*************************************************************************
This subroutine calculates the value of the trilinear or tricubic spline at
the given point (X,Y,Z).

INPUT PARAMETERS:
    C   -   coefficients table.
            Built by BuildBilinearSpline or BuildBicubicSpline.
    X, Y,
    Z   -   point

Result:
    S(x,y,z)

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
double spline3dcalc(const spline3dinterpolant &c, const double x, const double y, const double z, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline3dcalcv` function

/*************************************************************************
This subroutine calculates trilinear or tricubic vector-valued spline at the
given point (X,Y,Z).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y,
    Z   -   point

OUTPUT PARAMETERS:
    F   -   array[D] which stores function values.  F is out-parameter and
            it  is  reallocated  after  call to this function. In case you
            want  to    reuse  previously  allocated  F,   you   may   use
            Spline2DCalcVBuf(),  which  reallocates  F only when it is too
            small.

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcalcv(const spline3dinterpolant &c, const double x, const double y, const double z, real_1d_array &f, const xparams _xparams = alglib::xdefault);

Examples: [1]

`spline3dcalcvbuf` function

/*************************************************************************
This subroutine calculates bilinear or bicubic vector-valued spline at the
given point (X,Y,Z).

INPUT PARAMETERS:
    C   -   spline interpolant.
    X, Y,
    Z   -   point
    F   -   output buffer, possibly preallocated array. In case array size
            is large enough to store result, it is not reallocated.  Array
            which is too short will be reallocated

OUTPUT PARAMETERS:
    F   -   array[D] (or larger) which stores function values

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dcalcvbuf(const spline3dinterpolant &c, const double x, const double y, const double z, real_1d_array &f, const xparams _xparams = alglib::xdefault);

`spline3dlintransf` function

/*************************************************************************
This subroutine performs linear transformation of the spline.

INPUT PARAMETERS:
    C   -   spline interpolant.
    A, B-   transformation coefficients: S2(x,y) = A*S(x,y,z) + B

OUTPUT PARAMETERS:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dlintransf(spline3dinterpolant &c, const double a, const double b, const xparams _xparams = alglib::xdefault);

`spline3dlintransxyz` function

/*************************************************************************
This subroutine performs linear transformation of the spline argument.

INPUT PARAMETERS:
    C       -   spline interpolant
    AX, BX  -   transformation coefficients: x = A*u + B
    AY, BY  -   transformation coefficients: y = A*v + B
    AZ, BZ  -   transformation coefficients: z = A*w + B

OUTPUT PARAMETERS:
    C   -   transformed spline

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dlintransxyz(spline3dinterpolant &c, const double ax, const double bx, const double ay, const double by, const double az, const double bz, const xparams _xparams = alglib::xdefault);

`spline3dresampletrilinear` function

/*************************************************************************
Trilinear spline resampling

INPUT PARAMETERS:
    A           -   array[0..OldXCount*OldYCount*OldZCount-1], function
                    values at the old grid, :
                        A[0]        x=0,y=0,z=0
                        A[1]        x=1,y=0,z=0
                        A[..]       ...
                        A[..]       x=oldxcount-1,y=0,z=0
                        A[..]       x=0,y=1,z=0
                        A[..]       ...
                        ...
    OldZCount   -   old Z-count, OldZCount>1
    OldYCount   -   old Y-count, OldYCount>1
    OldXCount   -   old X-count, OldXCount>1
    NewZCount   -   new Z-count, NewZCount>1
    NewYCount   -   new Y-count, NewYCount>1
    NewXCount   -   new X-count, NewXCount>1

OUTPUT PARAMETERS:
    B           -   array[0..NewXCount*NewYCount*NewZCount-1], function
                    values at the new grid:
                        B[0]        x=0,y=0,z=0
                        B[1]        x=1,y=0,z=0
                        B[..]       ...
                        B[..]       x=newxcount-1,y=0,z=0
                        B[..]       x=0,y=1,z=0
                        B[..]       ...
                        ...

  -- ALGLIB routine --
     26.04.2012
     Copyright by Bochkanov Sergey
*************************************************************************/
void spline3dresampletrilinear(const real_1d_array &a, const ae_int_t oldzcount, const ae_int_t oldycount, const ae_int_t oldxcount, const ae_int_t newzcount, const ae_int_t newycount, const ae_int_t newxcount, real_1d_array &b, const xparams _xparams = alglib::xdefault);

`spline3dunpackv` function

/*************************************************************************
This subroutine unpacks tri-dimensional spline into the coefficients table

INPUT PARAMETERS:
    C   -   spline interpolant.

Result:
    N   -   grid size (X)
    M   -   grid size (Y)
    L   -   grid size (Z)
    D   -   number of components
    SType-  spline type. Currently, only one spline type is supported:
            trilinear spline, as indicated by SType=1.
    Tbl -   spline coefficients: [0..(N-1)*(M-1)*(L-1)*D-1, 0..13].
            For T=0..D-1 (component index), I = 0...N-2 (x index),
            J=0..M-2 (y index), K=0..L-2 (z index):
                Q := T + I*D + J*D*(N-1) + K*D*(N-1)*(M-1),

                Q-th row stores decomposition for T-th component of the
                vector-valued function

                Tbl[Q,0] = X[i]
                Tbl[Q,1] = X[i+1]
                Tbl[Q,2] = Y[j]
                Tbl[Q,3] = Y[j+1]
                Tbl[Q,4] = Z[k]
                Tbl[Q,5] = Z[k+1]

                Tbl[Q,6] = C000
                Tbl[Q,7] = C100
                Tbl[Q,8] = C010
                Tbl[Q,9] = C110
                Tbl[Q,10]= C001
                Tbl[Q,11]= C101
                Tbl[Q,12]= C011
                Tbl[Q,13]= C111
            On each grid square spline is equals to:
                S(x) = SUM(c[i,j,k]*(x^i)*(y^j)*(z^k), i=0..1, j=0..1, k=0..1)
                t = x-x[j]
                u = y-y[i]
                v = z-z[k]

            NOTE: format of Tbl is given for SType=1. Future versions of
                  ALGLIB can use different formats for different values of
                  SType.

  -- ALGLIB PROJECT --
     Copyright 26.04.2012 by Bochkanov Sergey
*************************************************************************/
void spline3dunpackv(const spline3dinterpolant &c, ae_int_t &n, ae_int_t &m, ae_int_t &l, ae_int_t &d, ae_int_t &stype, real_2d_array &tbl, const xparams _xparams = alglib::xdefault);

spline3d_trilinear example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use trilinear spline to interpolate f(x,y,z)=x+xy+z sampled 
        // at (x,y,z) from [0.0, 1.0] X [0.0, 1.0] X [0.0, 1.0].
        //
        // We store x, y and z-values at local arrays with same names.
        // Function values are stored in the array F as follows:
        //     f[0]     (x,y,z) = (0,0,0)
        //     f[1]     (x,y,z) = (1,0,0)
        //     f[2]     (x,y,z) = (0,1,0)
        //     f[3]     (x,y,z) = (1,1,0)
        //     f[4]     (x,y,z) = (0,0,1)
        //     f[5]     (x,y,z) = (1,0,1)
        //     f[6]     (x,y,z) = (0,1,1)
        //     f[7]     (x,y,z) = (1,1,1)
        //
        real_1d_array x = "[0.0, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array z = "[0.0, 1.0]";
        real_1d_array f = "[0,1,0,2,1,2,1,3]";
        double vx = 0.50;
        double vy = 0.50;
        double vz = 0.50;
        double v;
        spline3dinterpolant s;

        // build spline
        spline3dbuildtrilinearv(x, 2, y, 2, z, 2, f, 1, s);

        // calculate S(0.5,0.5,0.5)
        v = spline3dcalc(s, vx, vy, vz);
        printf("%.4f\n", double(v)); // EXPECTED: 1.2500
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

spline3d_vector example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // We use trilinear vector-valued spline to interpolate {f0,f1}={x+xy+z,x+xy+yz+z}
        // sampled at (x,y,z) from [0.0, 1.0] X [0.0, 1.0] X [0.0, 1.0].
        //
        // We store x, y and z-values at local arrays with same names.
        // Function values are stored in the array F as follows:
        //     f[0]     f0, (x,y,z) = (0,0,0)
        //     f[1]     f1, (x,y,z) = (0,0,0)
        //     f[2]     f0, (x,y,z) = (1,0,0)
        //     f[3]     f1, (x,y,z) = (1,0,0)
        //     f[4]     f0, (x,y,z) = (0,1,0)
        //     f[5]     f1, (x,y,z) = (0,1,0)
        //     f[6]     f0, (x,y,z) = (1,1,0)
        //     f[7]     f1, (x,y,z) = (1,1,0)
        //     f[8]     f0, (x,y,z) = (0,0,1)
        //     f[9]     f1, (x,y,z) = (0,0,1)
        //     f[10]    f0, (x,y,z) = (1,0,1)
        //     f[11]    f1, (x,y,z) = (1,0,1)
        //     f[12]    f0, (x,y,z) = (0,1,1)
        //     f[13]    f1, (x,y,z) = (0,1,1)
        //     f[14]    f0, (x,y,z) = (1,1,1)
        //     f[15]    f1, (x,y,z) = (1,1,1)
        //
        real_1d_array x = "[0.0, 1.0]";
        real_1d_array y = "[0.0, 1.0]";
        real_1d_array z = "[0.0, 1.0]";
        real_1d_array f = "[0,0, 1,1, 0,0, 2,2, 1,1, 2,2, 1,2, 3,4]";
        double vx = 0.50;
        double vy = 0.50;
        double vz = 0.50;
        spline3dinterpolant s;

        // build spline
        spline3dbuildtrilinearv(x, 2, y, 2, z, 2, f, 2, s);

        // calculate S(0.5,0.5,0.5) - we have vector of values instead of single value
        real_1d_array v;
        spline3dcalcv(s, vx, vy, vz, v);
        printf("%s\n", v.tostring(4).c_str()); // EXPECTED: [1.2500,1.5000]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`ssa` subpackage

Classes

ssamodel

Functions

ssaaddsequence
ssaanalyzelast
ssaanalyzelastwindow
ssaanalyzesequence
ssaappendpointandupdate
ssaappendsequenceandupdate
ssacleardata
ssacreate
ssaforecastavglast
ssaforecastavgsequence
ssaforecastlast
ssaforecastsequence
ssagetbasis
ssagetlrr
ssasetalgoprecomputed
ssasetalgotopkdirect
ssasetalgotopkrealtime
ssasetmemorylimit
ssasetpoweruplength
ssasetseed
ssasetwindow

Examples

ssa_d_basic		Simple SSA analysis demo
ssa_d_forecast		Simple SSA forecasting demo
ssa_d_realtime		Real-time SSA algorithm with fast incremental updates

`ssamodel` class

/*************************************************************************
This object stores state of the SSA model.

You should use ALGLIB functions to work with this object.
*************************************************************************/
class ssamodel
{
public:
    ssamodel();
    ssamodel(const ssamodel &rhs);
    ssamodel& operator=(const ssamodel &rhs);
    virtual ~ssamodel();
};

`ssaaddsequence` function

/*************************************************************************
This function adds data sequence to SSA  model.  Only   single-dimensional
sequences are supported.

What is a sequences? Following definitions/requirements apply:
* a sequence  is  an  array of  values  measured  in  subsequent,  equally
  separated time moments (ticks).
* you may have many sequences  in your  dataset;  say,  one  sequence  may
  correspond to one trading session.
* sequence length should be larger  than current  window  length  (shorter
  sequences will be ignored during analysis).
* analysis is performed within a  sequence; different  sequences  are  NOT
  stacked together to produce one large contiguous stream of data.
* analysis is performed for all  sequences at once, i.e. same set of basis
  vectors is computed for all sequences

INCREMENTAL ANALYSIS

This function is non intended for  incremental updates of previously found
SSA basis. Calling it invalidates  all previous analysis results (basis is
reset and will be recalculated from zero during next analysis).

If  you  want  to  perform   incremental/real-time  SSA,  consider   using
following functions:
* ssaappendpointandupdate() for appending one point
* ssaappendsequenceandupdate() for appending new sequence

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    X               -   array[N], data, can be larger (additional values
                        are ignored)
    N               -   data length, can be automatically determined from
                        the array length. N>=0.

OUTPUT PARAMETERS:
    S               -   SSA model, updated

NOTE: you can clear dataset with ssacleardata()

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaaddsequence(ssamodel &s, const real_1d_array &x, const ae_int_t n, const xparams _xparams = alglib::xdefault);
void ssaaddsequence(ssamodel &s, const real_1d_array &x, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`ssaanalyzelast` function

/*************************************************************************
This function:
* builds SSA basis using internally stored (entire) dataset
* returns reconstruction for the last NTicks of the last sequence

If you want to analyze some other sequence, use ssaanalyzesequence().

Reconstruction phase involves  generation  of  NTicks-WindowWidth  sliding
windows, their decomposition using empirical orthogonal functions found by
SSA, followed by averaging of each data point across  several  overlapping
windows. Thus, every point in the output trend is reconstructed  using  up
to WindowWidth overlapping  windows  (WindowWidth windows exactly  in  the
inner points, just one window at the extremal points).

IMPORTANT: due to averaging this function returns  different  results  for
           different values of NTicks. It is expected and not a bug.

           For example:
           * Trend[NTicks-1] is always same because it is not averaged  in
             any case (same applies to Trend[0]).
           * Trend[NTicks-2] has different values  for  NTicks=WindowWidth
             and NTicks=WindowWidth+1 because former  case  means that  no
             averaging is performed, and latter  case means that averaging
             using two sliding windows  is  performed.  Larger  values  of
             NTicks produce same results as NTicks=WindowWidth+1.
           * ...and so on...

PERFORMANCE: this  function has O((NTicks-WindowWidth)*WindowWidth*NBasis)
             running time. If you work  in  time-constrained  setting  and
             have to analyze just a few last ticks, choosing NTicks  equal
             to WindowWidth+SmoothingLen, with SmoothingLen=1...WindowWidth
             will result in good compromise between noise cancellation and
             analysis speed.

INPUT PARAMETERS:
    S               -   SSA model
    NTicks          -   number of ticks to analyze, Nticks>=1.
                        * special case of NTicks<=WindowWidth  is  handled
                          by analyzing last window and  returning   NTicks
                          last ticks.
                        * special case NTicks>LastSequenceLen  is  handled
                          by prepending result with NTicks-LastSequenceLen
                          zeros.

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], reconstructed trend line
    Noise           -   array[NTicks], the rest of the signal;
                        it holds that ActualData = Trend+Noise.


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.

In  any  case,  only  basis  is  reused. Reconstruction is performed  from
scratch every time you call this function.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the window length (analysis  can  be done,
  but we can not perform reconstruction on the last sequence)

Calling this function in degenerate cases returns following result:
* in any case, NTicks ticks is returned
* trend is assumed to be zero
* noise is initialized by the last sequence; if last sequence  is  shorter
  than the window size, it is moved to  the  end  of  the  array, and  the
  beginning of the noise array is filled by zeros

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaanalyzelast(ssamodel &s, const ae_int_t nticks, real_1d_array &trend, real_1d_array &noise, const xparams _xparams = alglib::xdefault);

`ssaanalyzelastwindow` function

/*************************************************************************
This  function  executes  SSA  on  internally  stored  dataset and returns
analysis  for  the  last  window  of  the  last sequence. Such analysis is
an lightweight alternative for full scale reconstruction (see below).

Typical use case for this function is  real-time  setting,  when  you  are
interested in quick-and-dirty (very quick and very  dirty)  processing  of
just a few last ticks of the trend.

IMPORTANT: full  scale  SSA  involves  analysis  of  the  ENTIRE  dataset,
           with reconstruction being done for  all  positions  of  sliding
           window with subsequent hankelization  (diagonal  averaging)  of
           the resulting matrix.

           Such analysis requires O((DataLen-Window)*Window*NBasis)  FLOPs
           and can be quite costly. However, it has  nice  noise-canceling
           effects due to averaging.

           This function performs REDUCED analysis of the last window.  It
           is much faster - just O(Window*NBasis),  but  its  results  are
           DIFFERENT from that of ssaanalyzelast(). In  particular,  first
           few points of the trend are much more prone to noise.

INPUT PARAMETERS:
    S               -   SSA model

OUTPUT PARAMETERS:
    Trend           -   array[WindowSize], reconstructed trend line
    Noise           -   array[WindowSize], the rest of the signal;
                        it holds that ActualData = Trend+Noise.
    NTicks          -   current WindowSize


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.

In  any  case,  only  basis  is  reused. Reconstruction is performed  from
scratch every time you call this function.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the window length (analysis can  be  done,
  but we can not perform reconstruction on the last sequence)

Calling this function in degenerate cases returns following result:
* in any case, WindowWidth ticks is returned
* trend is assumed to be zero
* noise is initialized by the last sequence; if last sequence  is  shorter
  than the window size, it is moved to  the  end  of  the  array, and  the
  beginning of the noise array is filled by zeros

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaanalyzelastwindow(ssamodel &s, real_1d_array &trend, real_1d_array &noise, ae_int_t &nticks, const xparams _xparams = alglib::xdefault);

`ssaanalyzesequence` function

/*************************************************************************
This function:
* builds SSA basis using internally stored (entire) dataset
* returns reconstruction for the sequence being passed to this function

If  you  want  to  analyze  last  sequence  stored  in   the   model,  use
ssaanalyzelast().

Reconstruction phase involves  generation  of  NTicks-WindowWidth  sliding
windows, their decomposition using empirical orthogonal functions found by
SSA, followed by averaging of each data point across  several  overlapping
windows. Thus, every point in the output trend is reconstructed  using  up
to WindowWidth overlapping  windows  (WindowWidth windows exactly  in  the
inner points, just one window at the extremal points).

PERFORMANCE: this  function has O((NTicks-WindowWidth)*WindowWidth*NBasis)
             running time. If you work  in  time-constrained  setting  and
             have to analyze just a few last ticks, choosing NTicks  equal
             to WindowWidth+SmoothingLen, with SmoothingLen=1...WindowWidth
             will result in good compromise between noise cancellation and
             analysis speed.

INPUT PARAMETERS:
    S               -   SSA model
    Data            -   array[NTicks], can be larger (only NTicks  leading
                        elements will be used)
    NTicks          -   number of ticks to analyze, Nticks>=1.
                        * special case of NTicks<WindowWidth  is   handled
                          by returning zeros as trend, and signal as noise

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], reconstructed trend line
    Noise           -   array[NTicks], the rest of the signal;
                        it holds that ActualData = Trend+Noise.


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.

In  any  case,  only  basis  is  reused. Reconstruction is performed  from
scratch every time you call this function.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* sequence being passed is shorter than the window length

Calling this function in degenerate cases returns following result:
* in any case, NTicks ticks is returned
* trend is assumed to be zero
* noise is initialized by the sequence.

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaanalyzesequence(ssamodel &s, const real_1d_array &data, const ae_int_t nticks, real_1d_array &trend, real_1d_array &noise, const xparams _xparams = alglib::xdefault);
void ssaanalyzesequence(ssamodel &s, const real_1d_array &data, real_1d_array &trend, real_1d_array &noise, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`ssaappendpointandupdate` function

/*************************************************************************
This function appends single point to last data sequence stored in the SSA
model and tries to update model in the  incremental  manner  (if  possible
with current algorithm).

If you want to add more than one point at once:
* if you want to add M points to the same sequence, perform M-1 calls with
  UpdateIts parameter set to 0.0, and last call with non-zero UpdateIts.
* if you want to add new sequence, use ssaappendsequenceandupdate()

Running time of this function does NOT depend on  dataset  size,  only  on
window width and number of singular vectors. Depending on algorithm  being
used, incremental update has complexity:
* for top-K real time   - O(UpdateIts*K*Width^2), with fractional UpdateIts
* for top-K direct      - O(Width^3) for any non-zero UpdateIts
* for precomputed basis - O(1), no update is performed

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    X               -   new point
    UpdateIts       -   >=0,  floating  point (!)  value,  desired  update
                        frequency:
                        * zero value means that point is  stored,  but  no
                          update is performed
                        * integer part of the value means  that  specified
                          number of iterations is always performed
                        * fractional part of  the  value  means  that  one
                          iteration is performed with this probability.

                        Recommended value: 0<UpdateIts<=1.  Values  larger
                        than 1 are VERY seldom  needed.  If  your  dataset
                        changes slowly, you can set it  to  0.1  and  skip
                        90% of updates.

                        In any case, no information is lost even with zero
                        value of UpdateIts! It will be  incorporated  into
                        model, sooner or later.

OUTPUT PARAMETERS:
    S               -   SSA model, updated

NOTE: this function uses internal  RNG  to  handle  fractional  values  of
      UpdateIts. By default it  is  initialized  with  fixed  seed  during
      initial calculation of basis. Thus subsequent calls to this function
      will result in the same sequence of pseudorandom decisions.

      However, if  you  have  several  SSA  models  which  are  calculated
      simultaneously, and if you want to reduce computational  bottlenecks
      by performing random updates at random moments, then fixed  seed  is
      not an option - all updates will fire at same moments.

      You may change it with ssasetseed() function.

NOTE: this function throws an exception if called for empty dataset (there
      is no "last" sequence to modify).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaappendpointandupdate(ssamodel &s, const double x, const double updateits, const xparams _xparams = alglib::xdefault);

`ssaappendsequenceandupdate` function

/*************************************************************************
This function appends new sequence to dataset stored in the SSA  model and
tries to update model in the incremental manner (if possible  with current
algorithm).

Notes:
* if you want to add M sequences at once, perform M-1 calls with UpdateIts
  parameter set to 0.0, and last call with non-zero UpdateIts.
* if you want to add just one point, use ssaappendpointandupdate()

Running time of this function does NOT depend on  dataset  size,  only  on
sequence length, window width and number of singular vectors. Depending on
algorithm being used, incremental update has complexity:
* for top-K real time   - O(UpdateIts*K*Width^2+(NTicks-Width)*Width^2)
* for top-K direct      - O(Width^3+(NTicks-Width)*Width^2)
* for precomputed basis - O(1), no update is performed

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    X               -   new sequence, array[NTicks] or larget
    NTicks          -   >=1, number of ticks in the sequence
    UpdateIts       -   >=0,  floating  point (!)  value,  desired  update
                        frequency:
                        * zero value means that point is  stored,  but  no
                          update is performed
                        * integer part of the value means  that  specified
                          number of iterations is always performed
                        * fractional part of  the  value  means  that  one
                          iteration is performed with this probability.

                        Recommended value: 0<UpdateIts<=1.  Values  larger
                        than 1 are VERY seldom  needed.  If  your  dataset
                        changes slowly, you can set it  to  0.1  and  skip
                        90% of updates.

                        In any case, no information is lost even with zero
                        value of UpdateIts! It will be  incorporated  into
                        model, sooner or later.

OUTPUT PARAMETERS:
    S               -   SSA model, updated

NOTE: this function uses internal  RNG  to  handle  fractional  values  of
      UpdateIts. By default it  is  initialized  with  fixed  seed  during
      initial calculation of basis. Thus subsequent calls to this function
      will result in the same sequence of pseudorandom decisions.

      However, if  you  have  several  SSA  models  which  are  calculated
      simultaneously, and if you want to reduce computational  bottlenecks
      by performing random updates at random moments, then fixed  seed  is
      not an option - all updates will fire at same moments.

      You may change it with ssasetseed() function.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaappendsequenceandupdate(ssamodel &s, const real_1d_array &x, const ae_int_t nticks, const double updateits, const xparams _xparams = alglib::xdefault);
void ssaappendsequenceandupdate(ssamodel &s, const real_1d_array &x, const double updateits, const xparams _xparams = alglib::xdefault);

`ssacleardata` function

/*************************************************************************
This function clears all data stored in the  model  and  invalidates  all
basis components found so far.

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()

OUTPUT PARAMETERS:
    S               -   SSA model, updated

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssacleardata(ssamodel &s, const xparams _xparams = alglib::xdefault);

`ssacreate` function

/*************************************************************************
This function creates SSA model object.  Right after creation model is  in
"dummy" mode - you can add data,  but   analyzing/prediction  will  return
just zeros (it assumes that basis is empty).

HOW TO USE SSA MODEL:

1. create model with ssacreate()
2. add data with one/many ssaaddsequence() calls
3. choose SSA algorithm with one of ssasetalgo...() functions:
   * ssasetalgotopkdirect() for direct one-run analysis
   * ssasetalgotopkrealtime() for algorithm optimized for many  subsequent
     runs with warm-start capabilities
   * ssasetalgoprecomputed() for user-supplied basis
4. set window width with ssasetwindow()
5. perform one of the analysis-related activities:
   a) call ssagetbasis() to get basis
   b) call ssaanalyzelast() ssaanalyzesequence() or ssaanalyzelastwindow()
      to perform analysis (trend/noise separation)
   c) call  one  of   the   forecasting   functions  (ssaforecastlast() or
      ssaforecastsequence()) to perform prediction; alternatively, you can
      extract linear recurrence coefficients with ssagetlrr().
   SSA analysis will be performed during first  call  to  analysis-related
   function. SSA model is smart enough to track all changes in the dataset
   and  model  settings,  to  cache  previously  computed  basis  and   to
   re-evaluate basis only when necessary.

Additionally, if your setting involves constant stream  of  incoming data,
you can perform quick update already calculated  model  with  one  of  the
incremental   append-and-update   functions:  ssaappendpointandupdate() or
ssaappendsequenceandupdate().

NOTE: steps (2), (3), (4) can be performed in arbitrary order.

INPUT PARAMETERS:
    none

OUTPUT PARAMETERS:
    S               -   structure which stores model state

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssacreate(ssamodel &s, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

`ssaforecastavglast` function

/*************************************************************************
This function builds SSA basis and performs forecasting  for  a  specified
number of ticks, returning value of trend.

Forecast is performed as follows:
* SSA  trend  extraction  is  applied to last  M  sliding windows  of  the
  internally stored dataset
* for each of M sliding windows, M predictions are built
* average value of M predictions is returned

This function has following running time:
* O(NBasis*WindowWidth*M) for trend extraction phase (always performed)
* O(WindowWidth*NTicks*M) for forecast phase

NOTE: noise reduction is ALWAYS applied by this algorithm; if you want  to
      apply recurrence relation  to  raw  unprocessed  data,  use  another
      function - ssaforecastsequence() which allows to  turn  on  and  off
      noise reduction phase.

NOTE: combination of several predictions results in lesser sensitivity  to
      noise, but it may produce undesirable discontinuities  between  last
      point of the trend and first point of the prediction. The reason  is
      that  last  point  of  the  trend is usually corrupted by noise, but
      average  value of  several  predictions  is less sensitive to noise,
      thus discontinuity appears. It is not a bug.

INPUT PARAMETERS:
    S               -   SSA model
    M               -   number  of  sliding  windows  to combine, M>=1. If
                        your dataset has less than M sliding windows, this
                        parameter will be silently reduced.
    NTicks          -   number of ticks to forecast, NTicks>=1

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], predicted trend line


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* NTicks  copies  of  the  last  value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=NTicks is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaforecastavglast(ssamodel &s, const ae_int_t m, const ae_int_t nticks, real_1d_array &trend, const xparams _xparams = alglib::xdefault);

`ssaforecastavgsequence` function

/*************************************************************************
This function builds SSA  basis  and  performs  forecasting  for  a  user-
specified sequence, returning value of trend.

Forecasting is done in two stages:
* first,  we  extract  trend  from M last sliding windows of the sequence.
  This stage is optional, you can  turn  it  off  if  you  pass data which
  are already processed with SSA. Of course, you  can  turn  it  off  even
  for raw data, but it is not recommended  -  noise  suppression  is  very
  important for correct prediction.
* then, we apply LRR independently for M sliding windows
* average of M predictions is returned

This function has following running time:
* O(NBasis*WindowWidth*M) for trend extraction phase
* O(WindowWidth*NTicks*M) for forecast phase

NOTE: combination of several predictions results in lesser sensitivity  to
      noise, but it may produce undesirable discontinuities  between  last
      point of the trend and first point of the prediction. The reason  is
      that  last  point  of  the  trend is usually corrupted by noise, but
      average  value of  several  predictions  is less sensitive to noise,
      thus discontinuity appears. It is not a bug.

INPUT PARAMETERS:
    S               -   SSA model
    Data            -   array[NTicks], data to forecast
    DataLen         -   number of ticks in the data, DataLen>=1
    M               -   number  of  sliding  windows  to combine, M>=1. If
                        your dataset has less than M sliding windows, this
                        parameter will be silently reduced.
    ForecastLen     -   number of ticks to predict, ForecastLen>=1
    ApplySmoothing  -   whether to apply smoothing trend extraction or not.
                        if you do not know what to specify, pass true.

OUTPUT PARAMETERS:
    Trend           -   array[ForecastLen], forecasted trend


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* data sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* ForecastLen copies of the last value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=ForecastLen is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaforecastavgsequence(ssamodel &s, const real_1d_array &data, const ae_int_t datalen, const ae_int_t m, const ae_int_t forecastlen, const bool applysmoothing, real_1d_array &trend, const xparams _xparams = alglib::xdefault);
void ssaforecastavgsequence(ssamodel &s, const real_1d_array &data, const ae_int_t m, const ae_int_t forecastlen, real_1d_array &trend, const xparams _xparams = alglib::xdefault);

`ssaforecastlast` function

/*************************************************************************
This function builds SSA basis and performs forecasting  for  a  specified
number of ticks, returning value of trend.

Forecast is performed as follows:
* SSA  trend  extraction  is  applied  to last WindowWidth elements of the
  internally stored dataset; this step is basically a noise reduction.
* linear recurrence relation is applied to extracted trend

This function has following running time:
* O(NBasis*WindowWidth) for trend extraction phase (always performed)
* O(WindowWidth*NTicks) for forecast phase

NOTE: noise reduction is ALWAYS applied by this algorithm; if you want  to
      apply recurrence relation  to  raw  unprocessed  data,  use  another
      function - ssaforecastsequence() which allows to  turn  on  and  off
      noise reduction phase.

NOTE: this algorithm performs prediction using only one - last  -  sliding
      window.  Predictions  produced   by   such   approach   are   smooth
      continuations of the reconstructed  trend  line,  but  they  can  be
      easily corrupted by noise. If you need  noise-resistant  prediction,
      use ssaforecastavglast() function, which averages predictions  built
      using several sliding windows.

INPUT PARAMETERS:
    S               -   SSA model
    NTicks          -   number of ticks to forecast, NTicks>=1

OUTPUT PARAMETERS:
    Trend           -   array[NTicks], predicted trend line


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* last sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* NTicks  copies  of  the  last  value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=NTicks is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaforecastlast(ssamodel &s, const ae_int_t nticks, real_1d_array &trend, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`ssaforecastsequence` function

/*************************************************************************
This function builds SSA  basis  and  performs  forecasting  for  a  user-
specified sequence, returning value of trend.

Forecasting is done in two stages:
* first,  we  extract  trend  from the WindowWidth  last  elements of  the
  sequence. This stage is optional, you  can  turn  it  off  if  you  pass
  data which are already processed with SSA. Of course, you  can  turn  it
  off even for raw data, but it is not recommended - noise suppression  is
  very important for correct prediction.
* then, we apply LRR for last  WindowWidth-1  elements  of  the  extracted
  trend.

This function has following running time:
* O(NBasis*WindowWidth) for trend extraction phase
* O(WindowWidth*NTicks) for forecast phase

NOTE: this algorithm performs prediction using only one - last  -  sliding
      window.  Predictions  produced   by   such   approach   are   smooth
      continuations of the reconstructed  trend  line,  but  they  can  be
      easily corrupted by noise. If you need  noise-resistant  prediction,
      use ssaforecastavgsequence() function,  which  averages  predictions
      built using several sliding windows.

INPUT PARAMETERS:
    S               -   SSA model
    Data            -   array[NTicks], data to forecast
    DataLen         -   number of ticks in the data, DataLen>=1
    ForecastLen     -   number of ticks to predict, ForecastLen>=1
    ApplySmoothing  -   whether to apply smoothing trend extraction or not;
                        if you do not know what to specify, pass True.

OUTPUT PARAMETERS:
    Trend           -   array[ForecastLen], forecasted trend


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Following degenerate cases may happen:
* dataset is empty (no analysis can be done)
* all sequences are shorter than the window length,no analysis can be done
* no algorithm is specified (no analysis can be done)
* data sequence is shorter than the WindowWidth   (analysis  can  be done,
  but we can not perform forecasting on the last sequence)
* window lentgh is 1 (impossible to use for forecasting)
* SSA analysis algorithm is  configured  to  extract  basis  whose size is
  equal to window length (impossible to use for  forecasting;  only  basis
  whose size is less than window length can be used).

Calling this function in degenerate cases returns following result:
* ForecastLen copies of the last value is returned for non-empty task with
  large enough dataset, but with overcomplete  basis  (window  width=1  or
  basis size is equal to window width)
* zero trend with length=ForecastLen is returned for empty task

No analysis is performed in degenerate cases (we immediately return  dummy
values, no basis is ever constructed).

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssaforecastsequence(ssamodel &s, const real_1d_array &data, const ae_int_t datalen, const ae_int_t forecastlen, const bool applysmoothing, real_1d_array &trend, const xparams _xparams = alglib::xdefault);
void ssaforecastsequence(ssamodel &s, const real_1d_array &data, const ae_int_t forecastlen, real_1d_array &trend, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`ssagetbasis` function

/*************************************************************************
This function executes SSA on internally stored dataset and returns  basis
found by current method.

INPUT PARAMETERS:
    S               -   SSA model

OUTPUT PARAMETERS:
    A               -   array[WindowWidth,NBasis],   basis;  vectors  are
                        stored in matrix columns, by descreasing variance
    SV              -   array[NBasis]:
                        * zeros - for model initialized with SSASetAlgoPrecomputed()
                        * singular values - for other algorithms
    WindowWidth     -   current window
    NBasis          -   basis size


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Calling  this  function  in  degenerate  cases  (no  data  or all data are
shorter than window size; no algorithm is specified)  returns  basis  with
just one zero vector.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssagetbasis(ssamodel &s, real_2d_array &a, real_1d_array &sv, ae_int_t &windowwidth, ae_int_t &nbasis, const xparams _xparams = alglib::xdefault);

`ssagetlrr` function

/*************************************************************************
This function returns linear recurrence relation (LRR) coefficients  found
by current SSA algorithm.

INPUT PARAMETERS:
    S               -   SSA model

OUTPUT PARAMETERS:
    A               -   array[WindowWidth-1]. Coefficients  of  the
                        linear recurrence of the form:
                        X[W-1] = X[W-2]*A[W-2] + X[W-3]*A[W-3] + ... + X[0]*A[0].
                        Empty array for WindowWidth=1.
    WindowWidth     -   current window width


CACHING/REUSE OF THE BASIS

Caching/reuse of previous results is performed:
* first call performs full run of SSA; basis is stored in the cache
* subsequent calls reuse previously cached basis
* if you call any function which changes model properties (window  length,
  algorithm, dataset), internal basis will be invalidated.
* the only calls which do NOT invalidate basis are listed below:
  a) ssasetwindow() with same window length
  b) ssaappendpointandupdate()
  c) ssaappendsequenceandupdate()
  d) ssasetalgotopk...() with exactly same K
  Calling these functions will result in reuse of previously found basis.


HANDLING OF DEGENERATE CASES

Calling  this  function  in  degenerate  cases  (no  data  or all data are
shorter than window size; no algorithm is specified) returns zeros.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssagetlrr(ssamodel &s, real_1d_array &a, ae_int_t &windowwidth, const xparams _xparams = alglib::xdefault);

`ssasetalgoprecomputed` function

/*************************************************************************
This  function sets SSA algorithm to "precomputed vectors" algorithm.

This  algorithm  uses  precomputed  set  of  orthonormal  (orthogonal  AND
normalized) basis vectors supplied by user. Thus, basis calculation  phase
is not performed -  we  already  have  our  basis  -  and  only  analysis/
forecasting phase requires actual calculations.

This algorithm may handle "append" requests which add just  one/few  ticks
to the end of the last sequence in O(1) time.

NOTE: this algorithm accepts both basis and window  width,  because  these
      two parameters are naturally aligned.  Calling  this  function  sets
      window width; if you call ssasetwindow() with  other  window  width,
      then during analysis stage algorithm will detect conflict and  reset
      to zero basis.

INPUT PARAMETERS:
    S               -   SSA model
    A               -   array[WindowWidth,NBasis], orthonormalized  basis;
                        this function does NOT control  orthogonality  and
                        does NOT perform any kind of  renormalization.  It
                        is your responsibility to provide it with  correct
                        basis.
    WindowWidth     -   window width, >=1
    NBasis          -   number of basis vectors, 1<=NBasis<=WindowWidth

OUTPUT PARAMETERS:
    S               -   updated model

NOTE: calling this function invalidates basis in all cases.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetalgoprecomputed(ssamodel &s, const real_2d_array &a, const ae_int_t windowwidth, const ae_int_t nbasis, const xparams _xparams = alglib::xdefault);
void ssasetalgoprecomputed(ssamodel &s, const real_2d_array &a, const xparams _xparams = alglib::xdefault);

`ssasetalgotopkdirect` function

/*************************************************************************
This  function sets SSA algorithm to "direct top-K" algorithm.

"Direct top-K" algorithm performs full  SVD  of  the  N*WINDOW  trajectory
matrix (hence its name - direct solver  is  used),  then  extracts  top  K
components. Overall running time is O(N*WINDOW^2), where N is a number  of
ticks in the dataset, WINDOW is window width.

This algorithm may handle "append" requests which add just  one/few  ticks
to the end of the last sequence in O(WINDOW^3) time,  which  is  ~N/WINDOW
times faster than re-computing everything from scratch.

INPUT PARAMETERS:
    S               -   SSA model
    TopK            -   number of components to analyze; TopK>=1.

OUTPUT PARAMETERS:
    S               -   updated model


NOTE: TopK>WindowWidth is silently decreased to WindowWidth during analysis
      phase

NOTE: calling this function invalidates basis, except  for  the  situation
      when this algorithm was already set with same parameters.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetalgotopkdirect(ssamodel &s, const ae_int_t topk, const xparams _xparams = alglib::xdefault);

Examples: [1] [2]

`ssasetalgotopkrealtime` function

/*************************************************************************
This function sets SSA algorithm to "top-K real time algorithm". This algo
extracts K components with largest singular values.

It  is  real-time  version  of  top-K  algorithm  which  is  optimized for
incremental processing and  fast  start-up. Internally  it  uses  subspace
eigensolver for truncated SVD. It results  in  ability  to  perform  quick
updates of the basis when only a few points/sequences is added to dataset.

Performance profile of the algorithm is given below:
* O(K*WindowWidth^2) running time for incremental update  of  the  dataset
  with one of the "append-and-update" functions (ssaappendpointandupdate()
  or ssaappendsequenceandupdate()).
* O(N*WindowWidth^2) running time for initial basis evaluation (N=size  of
  dataset)
* ability  to  split  costly  initialization  across  several  incremental
  updates of the basis (so called "Power-Up" functionality,  activated  by
  ssasetpoweruplength() function)

INPUT PARAMETERS:
    S               -   SSA model
    TopK            -   number of components to analyze; TopK>=1.

OUTPUT PARAMETERS:
    S               -   updated model

NOTE: this  algorithm  is  optimized  for  large-scale  tasks  with  large
      datasets. On toy problems with just  5-10 points it can return basis
      which is slightly different from that returned by  direct  algorithm
      (ssasetalgotopkdirect() function). However, the  difference  becomes
      negligible as dataset grows.

NOTE: TopK>WindowWidth is silently decreased to WindowWidth during analysis
      phase

NOTE: calling this function invalidates basis, except  for  the  situation
      when this algorithm was already set with same parameters.

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetalgotopkrealtime(ssamodel &s, const ae_int_t topk, const xparams _xparams = alglib::xdefault);

Examples: [1]

`ssasetmemorylimit` function

/*************************************************************************
This function sets memory limit of SSA analysis.

Straightforward SSA with sequence length T and window width W needs O(T*W)
memory. It is possible to reduce memory consumption by splitting task into
smaller chunks.

Thus function allows you to specify approximate memory limit (measured  in
double precision numbers used for buffers). Actual memory consumption will
be comparable to the number specified by you.

Default memory limit is 50.000.000 (400Mbytes) in current version.

INPUT PARAMETERS:
    S       -   SSA model
    MemLimit-   memory limit, >=0. Zero value means no limit.

  -- ALGLIB --
     Copyright 20.12.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetmemorylimit(ssamodel &s, const ae_int_t memlimit, const xparams _xparams = alglib::xdefault);

`ssasetpoweruplength` function

/*************************************************************************
This function sets length of power-up cycle for real-time algorithm.

By default, this algorithm performs costly O(N*WindowWidth^2)  init  phase
followed by full run of truncated  EVD.  However,  if  you  are  ready  to
live with a bit lower-quality basis during first few iterations,  you  can
split this O(N*WindowWidth^2) initialization  between  several  subsequent
append-and-update rounds. It results in better latency of the algorithm.

This function invalidates basis/solver, next analysis call will result  in
full recalculation of everything.

INPUT PARAMETERS:
    S       -   SSA model
    PWLen   -   length of the power-up stage:
                * 0 means that no power-up is requested
                * 1 is the same as 0
                * >1 means that delayed power-up is performed

  -- ALGLIB --
     Copyright 03.11.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetpoweruplength(ssamodel &s, const ae_int_t pwlen, const xparams _xparams = alglib::xdefault);

Examples: [1]

`ssasetseed` function

/*************************************************************************
This  function  sets  seed  which  is used to initialize internal RNG when
we make pseudorandom decisions on model updates.

By default, deterministic seed is used - which results in same sequence of
pseudorandom decisions every time you run SSA model. If you  specify  non-
deterministic seed value, then SSA  model  may  return  slightly different
results after each run.

This function can be useful when you have several SSA models updated  with
sseappendpointandupdate() called with 0<UpdateIts<1 (fractional value) and
due to performance limitations want them to perform updates  at  different
moments.

INPUT PARAMETERS:
    S       -   SSA model
    Seed    -   seed:
                * positive values = use deterministic seed for each run of
                  algorithms which depend on random initialization
                * zero or negative values = use non-deterministic seed

  -- ALGLIB --
     Copyright 03.11.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetseed(ssamodel &s, const ae_int_t seed, const xparams _xparams = alglib::xdefault);

`ssasetwindow` function

/*************************************************************************
This function sets window width for SSA model. You should call  it  before
analysis phase. Default window width is 1 (not for real use).

Special notes:
* this function call can be performed at any moment before  first call  to
  analysis-related functions
* changing window width invalidates internally stored basis; if you change
  window width AFTER you call analysis-related  function,  next  analysis
  phase will require re-calculation of  the  basis  according  to  current
  algorithm.
* calling this function with exactly  same window width as current one has
  no effect
* if you specify window width larger  than any data sequence stored in the
  model, analysis will return zero basis.

INPUT PARAMETERS:
    S               -   SSA model created with ssacreate()
    WindowWidth     -   >=1, new window width

OUTPUT PARAMETERS:
    S               -   SSA model, updated

  -- ALGLIB --
     Copyright 30.10.2017 by Bochkanov Sergey
*************************************************************************/
void ssasetwindow(ssamodel &s, const ae_int_t windowwidth, const xparams _xparams = alglib::xdefault);

Examples: [1] [2] [3]

ssa_d_basic example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate SSA trend/noise separation for some toy problem:
        // small monotonically growing series X are analyzed with 3-tick window
        // and "top-K" version of SSA, which selects K largest singular vectors
        // for analysis, with K=1.
        //
        ssamodel s;
        real_1d_array x = "[0,0.5,1,1,1.5,2]";

        //
        // First, we create SSA model, set its properties and add dataset.
        //
        // We use window with width=3 and configure model to use direct SSA
        // algorithm - one which runs exact O(N*W^2) analysis - to extract
        // one top singular vector. Well, it is toy problem :)
        //
        // NOTE: SSA model may store and analyze more than one sequence
        //       (say, different sequences may correspond to data collected
        //       from different devices)
        //
        ssacreate(s);
        ssasetwindow(s, 3);
        ssaaddsequence(s, x);
        ssasetalgotopkdirect(s, 1);

        //
        // Now we begin analysis. Internally SSA model stores everything it needs:
        // data, settings, solvers and so on. Right after first call to analysis-
        // related function it will analyze dataset, build basis and perform analysis.
        //
        // Subsequent calls to analysis functions will reuse previously computed
        // basis, unless you invalidate it by changing model settings (or dataset).
        //
        real_1d_array trend;
        real_1d_array noise;
        ssaanalyzesequence(s, x, trend, noise);
        printf("%s\n", trend.tostring(2).c_str()); // EXPECTED: [0.3815,0.5582,0.7810,1.0794,1.5041,2.0105]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

ssa_d_forecast example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Here we demonstrate SSA forecasting on some toy problem with clearly
        // visible linear trend and small amount of noise.
        //
        ssamodel s;
        real_1d_array x = "[0.05,0.96,2.04,3.11,3.97,5.03,5.98,7.02,8.02]";

        //
        // First, we create SSA model, set its properties and add dataset.
        //
        // We use window with width=3 and configure model to use direct SSA
        // algorithm - one which runs exact O(N*W^2) analysis - to extract
        // two top singular vectors. Well, it is toy problem :)
        //
        // NOTE: SSA model may store and analyze more than one sequence
        //       (say, different sequences may correspond to data collected
        //       from different devices)
        //
        ssacreate(s);
        ssasetwindow(s, 3);
        ssaaddsequence(s, x);
        ssasetalgotopkdirect(s, 2);

        //
        // Now we begin analysis. Internally SSA model stores everything it needs:
        // data, settings, solvers and so on. Right after first call to analysis-
        // related function it will analyze dataset, build basis and perform analysis.
        //
        // Subsequent calls to analysis functions will reuse previously computed
        // basis, unless you invalidate it by changing model settings (or dataset).
        //
        // In this example we show how to use ssaforecastlast() function, which
        // predicts changed in the last sequence of the dataset. If you want to
        // perform prediction for some other sequence, use ssaforecastsequence().
        //
        real_1d_array trend;
        ssaforecastlast(s, 3, trend);

        //
        // Well, we expected it to be [9,10,11]. There exists some difference,
        // which can be explained by the artificial noise in the dataset.
        //
        printf("%s\n", trend.tostring(2).c_str()); // EXPECTED: [9.0005,9.9322,10.8051]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

ssa_d_realtime example

#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"

using namespace alglib;

int main(int argc, char **argv)
{
    try
    {
        //
        // Suppose that you have a constant stream of incoming data, and you want
        // to regularly perform singular spectral analysis of this stream.
        //
        // One full run of direct algorithm costs O(N*Width^2) operations, so
        // the more points you have, the more it costs to rebuild basis from
        // scratch.
        // 
        // Luckily we have incremental SSA algorithm which can perform quick
        // updates of already computed basis in O(K*Width^2) ops, where K
        // is a number of singular vectors extracted. Usually it is orders of
        // magnitude faster than full update of the basis.
        //
        // In this example we start from some initial dataset x0. Then we
        // start appending elements one by one to the end of the last sequence.
        //
        // NOTE: direct algorithm also supports incremental updates, but
        //       with O(Width^3) cost. Typically K<<Width, so specialized
        //       incremental algorithm is still faster.
        //
        ssamodel s1;
        real_2d_array a1;
        real_1d_array sv1;
        ae_int_t w;
        ae_int_t k;
        real_1d_array x0 = "[0.009,0.976,1.999,2.984,3.977,5.002]";
        ssacreate(s1);
        ssasetwindow(s1, 3);
        ssaaddsequence(s1, x0);

        // set algorithm to the real-time version of top-K, K=2
        ssasetalgotopkrealtime(s1, 2);

        // one more interesting feature of the incremental algorithm is "power-up" cycle.
        // even with incremental algorithm initial basis calculation costs O(N*Width^2) ops.
        // if such startup cost is too high for your real-time app, then you may divide
        // initial basis calculation across several model updates. It results in better
        // latency at the price of somewhat lesser precision during first few updates.
        ssasetpoweruplength(s1, 3);

        // now, after we prepared everything, start to add incoming points one by one;
        // in the real life, of course, we will perform some work between subsequent update
        // (analyze something, predict, and so on).
        //
        // After each append we perform one iteration of the real-time solver. Usually
        // one iteration is more than enough to update basis. If you have REALLY tight
        // performance constraints, you may specify fractional amount of iterations,
        // which means that iteration is performed with required probability.
        double updateits = 1.0;
        ssaappendpointandupdate(s1, 5.951, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 7.074, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 7.925, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 8.992, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 9.942, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 11.051, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 11.965, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 13.047, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        ssaappendpointandupdate(s1, 13.970, updateits);
        ssagetbasis(s1, a1, sv1, w, k);

        // Ok, we have our basis in a1[] and singular values at sv1[].
        // But is it good enough? Let's print it.
        printf("%s\n", a1.tostring(3).c_str()); // EXPECTED: [[0.510607,0.753611],[0.575201,0.058445],[0.639081,-0.654717]]

        // Ok, two vectors with 3 components each.
        // But how to understand that is it really good basis?
        // Let's compare it with direct SSA algorithm on the entire sequence.
        ssamodel s2;
        real_2d_array a2;
        real_1d_array sv2;
        real_1d_array x2 = "[0.009,0.976,1.999,2.984,3.977,5.002,5.951,7.074,7.925,8.992,9.942,11.051,11.965,13.047,13.970]";
        ssacreate(s2);
        ssasetwindow(s2, 3);
        ssaaddsequence(s2, x2);
        ssasetalgotopkdirect(s2, 2);
        ssagetbasis(s2, a2, sv2, w, k);

        // it is exactly the same as one calculated with incremental approach!
        printf("%s\n", a2.tostring(3).c_str()); // EXPECTED: [[0.510607,0.753611],[0.575201,0.058445],[0.639081,-0.654717]]
    }
    catch(alglib::ap_error alglib_exception)
    {
        printf("ALGLIB exception with message '%s'\n", alglib_exception.msg.c_str());
        return 1;
    }
    return 0;
}

`stest` subpackage

Functions

onesamplesigntest

Examples

`onesamplesigntest` function

/*************************************************************************
Sign test

This test checks three hypotheses about the median of  the  given  sample.
The following tests are performed:
    * two-tailed test (null hypothesis - the median is equal to the  given
      value)
    * left-tailed test (null hypothesis - the median is  greater  than  or
      equal to the given value)
    * right-tailed test (null hypothesis - the  median  is  less  than  or
      equal to the given value)

Requirements:
    * the scale of measurement should be ordinal, interval or ratio  (i.e.
      the test could not be applied to nominal variables).

The test is non-parametric and doesn't require distribution X to be normal

Input parameters:
    X       -   sample. Array whose index goes from 0 to N-1.
    N       -   size of the sample.
    Median  -   assumed median value.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

While   calculating   p-values   high-precision   binomial    distribution
approximation is used, so significance levels have about 15 exact digits.

  -- ALGLIB --
     Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void onesamplesigntest(const real_1d_array &x, const ae_int_t n, const double median, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`studenttdistr` subpackage

Functions

invstudenttdistribution
studenttdistribution

Examples

`invstudenttdistribution` function

/*************************************************************************
Functional inverse of Student's t distribution

Given probability p, finds the argument t such that stdtr(k,t)
is equal to p.

ACCURACY:

Tested at random 1 <= k <= 100.  The "domain" refers to p:
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE    .001,.999     25000       5.7e-15     8.0e-16
   IEEE    10^-6,.001    25000       2.0e-12     2.9e-14

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double invstudenttdistribution(const ae_int_t k, const double p, const xparams _xparams = alglib::xdefault);

`studenttdistribution` function

/*************************************************************************
Student's t distribution

Computes the integral from minus infinity to t of the Student
t distribution with integer k > 0 degrees of freedom:

                                     t
                                     -
                                    | |
             -                      |         2   -(k+1)/2
            | ( (k+1)/2 )           |  (     x   )
      ----------------------        |  ( 1 + --- )        dx
                    -               |  (      k  )
      sqrt( k pi ) | ( k/2 )        |
                                  | |
                                   -
                                  -inf.

Relation to incomplete beta integral:

       1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z )
where
       z = k/(k + t**2).

For t < -2, this is the method of computation.  For higher t,
a direct method is derived from integration by parts.
Since the function is symmetric about t=0, the area under the
right tail of the density is found by calling the function
with -t instead of t.

ACCURACY:

Tested at random 1 <= k <= 25.  The "domain" refers to t.
                     Relative error:
arithmetic   domain     # trials      peak         rms
   IEEE     -100,-2      50000       5.9e-15     1.4e-15
   IEEE     -2,100      500000       2.7e-15     4.9e-17

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier
*************************************************************************/
double studenttdistribution(const ae_int_t k, const double t, const xparams _xparams = alglib::xdefault);

`studentttests` subpackage

Functions

studentttest1
studentttest2
unequalvariancettest

Examples

`studentttest1` function

/*************************************************************************
One-sample t-test

This test checks three hypotheses about the mean of the given sample.  The
following tests are performed:
    * two-tailed test (null hypothesis - the mean is equal  to  the  given
      value)
    * left-tailed test (null hypothesis - the  mean  is  greater  than  or
      equal to the given value)
    * right-tailed test (null hypothesis - the mean is less than or  equal
      to the given value).

The test is based on the assumption that  a  given  sample  has  a  normal
distribution and  an  unknown  dispersion.  If  the  distribution  sharply
differs from normal, the test will work incorrectly.

INPUT PARAMETERS:
    X       -   sample. Array whose index goes from 0 to N-1.
    N       -   size of sample, N>=0
    Mean    -   assumed value of the mean.

OUTPUT PARAMETERS:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

NOTE: this function correctly handles degenerate cases:
      * when N=0, all p-values are set to 1.0
      * when variance of X[] is exactly zero, p-values are set
        to 1.0 or 0.0, depending on difference between sample mean and
        value of mean being tested.


  -- ALGLIB --
     Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void studentttest1(const real_1d_array &x, const ae_int_t n, const double mean, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`studentttest2` function

/*************************************************************************
Two-sample pooled test

This test checks three hypotheses about the mean of the given samples. The
following tests are performed:
    * two-tailed test (null hypothesis - the means are equal)
    * left-tailed test (null hypothesis - the mean of the first sample  is
      greater than or equal to the mean of the second sample)
    * right-tailed test (null hypothesis - the mean of the first sample is
      less than or equal to the mean of the second sample).

Test is based on the following assumptions:
    * given samples have normal distributions
    * dispersions are equal
    * samples are independent.

Input parameters:
    X       -   sample 1. Array whose index goes from 0 to N-1.
    N       -   size of sample.
    Y       -   sample 2. Array whose index goes from 0 to M-1.
    M       -   size of sample.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

NOTE: this function correctly handles degenerate cases:
      * when N=0 or M=0, all p-values are set to 1.0
      * when both samples has exactly zero variance, p-values are set
        to 1.0 or 0.0, depending on difference between means.

  -- ALGLIB --
     Copyright 18.09.2006 by Bochkanov Sergey
*************************************************************************/
void studentttest2(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`unequalvariancettest` function

/*************************************************************************
Two-sample unpooled test

This test checks three hypotheses about the mean of the given samples. The
following tests are performed:
    * two-tailed test (null hypothesis - the means are equal)
    * left-tailed test (null hypothesis - the mean of the first sample  is
      greater than or equal to the mean of the second sample)
    * right-tailed test (null hypothesis - the mean of the first sample is
      less than or equal to the mean of the second sample).

Test is based on the following assumptions:
    * given samples have normal distributions
    * samples are independent.
Equality of variances is NOT required.

Input parameters:
    X - sample 1. Array whose index goes from 0 to N-1.
    N - size of the sample.
    Y - sample 2. Array whose index goes from 0 to M-1.
    M - size of the sample.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

NOTE: this function correctly handles degenerate cases:
      * when N=0 or M=0, all p-values are set to 1.0
      * when both samples has zero variance, p-values are set
        to 1.0 or 0.0, depending on difference between means.
      * when only one sample has zero variance, test reduces to 1-sample
        version.

  -- ALGLIB --
     Copyright 18.09.2006 by Bochkanov Sergey
*************************************************************************/
void unequalvariancettest(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`svd` subpackage

Functions

rmatrixsvd

Examples

`rmatrixsvd` function

/*************************************************************************
Singular value decomposition of a rectangular matrix.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

The algorithm calculates the singular value decomposition of a matrix of
size MxN: A = U * S * V^T

The algorithm finds the singular values and, optionally, matrices U and V^T.
The algorithm can find both first min(M,N) columns of matrix U and rows of
matrix V^T (singular vectors), and matrices U and V^T wholly (of sizes MxM
and NxN respectively).

Take into account that the subroutine does not return matrix V but V^T.

Input parameters:
    A           -   matrix to be decomposed.
                    Array whose indexes range within [0..M-1, 0..N-1].
    M           -   number of rows in matrix A.
    N           -   number of columns in matrix A.
    UNeeded     -   0, 1 or 2. See the description of the parameter U.
    VTNeeded    -   0, 1 or 2. See the description of the parameter VT.
    AdditionalMemory -
                    If the parameter:
                     * equals 0, the algorithm doesn't use additional
                       memory (lower requirements, lower performance).
                     * equals 1, the algorithm uses additional
                       memory of size min(M,N)*min(M,N) of real numbers.
                       It often speeds up the algorithm.
                     * equals 2, the algorithm uses additional
                       memory of size M*min(M,N) of real numbers.
                       It allows to get a maximum performance.
                    The recommended value of the parameter is 2.

Output parameters:
    W           -   contains singular values in descending order.
    U           -   if UNeeded=0, U isn't changed, the left singular vectors
                    are not calculated.
                    if Uneeded=1, U contains left singular vectors (first
                    min(M,N) columns of matrix U). Array whose indexes range
                    within [0..M-1, 0..Min(M,N)-1].
                    if UNeeded=2, U contains matrix U wholly. Array whose
                    indexes range within [0..M-1, 0..M-1].
    VT          -   if VTNeeded=0, VT isn't changed, the right singular vectors
                    are not calculated.
                    if VTNeeded=1, VT contains right singular vectors (first
                    min(M,N) rows of matrix V^T). Array whose indexes range
                    within [0..min(M,N)-1, 0..N-1].
                    if VTNeeded=2, VT contains matrix V^T wholly. Array whose
                    indexes range within [0..N-1, 0..N-1].

  -- ALGLIB --
     Copyright 2005 by Bochkanov Sergey
*************************************************************************/
bool rmatrixsvd(const real_2d_array &a, const ae_int_t m, const ae_int_t n, const ae_int_t uneeded, const ae_int_t vtneeded, const ae_int_t additionalmemory, real_1d_array &w, real_2d_array &u, real_2d_array &vt, const xparams _xparams = alglib::xdefault);

`trfac` subpackage

Classes

sparsedecompositionanalysis

Functions

cmatrixlu
hpdmatrixcholesky
rmatrixlu
sparsecholesky
sparsecholeskyanalyze
sparsecholeskyfactorize
sparsecholeskyp
sparsecholeskyreload
sparsecholeskyskyline
sparselu
spdmatrixcholesky
spdmatrixcholeskyupdateadd1
spdmatrixcholeskyupdateadd1buf
spdmatrixcholeskyupdatefix
spdmatrixcholeskyupdatefixbuf

Examples

`sparsedecompositionanalysis` class

/*************************************************************************
An analysis of the sparse matrix decomposition, performed prior to  actual
numerical factorization. You should not directly  access  fields  of  this
object - use appropriate ALGLIB functions to work with this object.
*************************************************************************/
class sparsedecompositionanalysis
{
public:
    sparsedecompositionanalysis();
    sparsedecompositionanalysis(const sparsedecompositionanalysis &rhs);
    sparsedecompositionanalysis& operator=(const sparsedecompositionanalysis &rhs);
    virtual ~sparsedecompositionanalysis();
};

`cmatrixlu` function

/*************************************************************************
LU decomposition of a general complex matrix with row pivoting

A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
  Pi - permutation matrix for I and Pivots[I]

INPUT PARAMETERS:
    A       -   array[0..M-1, 0..N-1].
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.


OUTPUT PARAMETERS:
    A       -   matrices L and U in compact form:
                * L is stored under main diagonal
                * U is stored on and above main diagonal
    Pivots  -   permutation matrix in compact form.
                array[0..Min(M-1,N-1)].

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void cmatrixlu(complex_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);
void cmatrixlu(complex_2d_array &a, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);

`hpdmatrixcholesky` function

/*************************************************************************
Cache-oblivious Cholesky decomposition

The algorithm computes Cholesky decomposition  of  a  Hermitian  positive-
definite matrix. The result of an algorithm is a representation  of  A  as
A=U'*U  or A=L*L' (here X' denotes conj(X^T)).

INPUT PARAMETERS:
    A       -   upper or lower triangle of a factorized matrix.
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   if IsUpper=True, then A contains an upper triangle of
                a symmetric matrix, otherwise A contains a lower one.

OUTPUT PARAMETERS:
    A       -   the result of factorization. If IsUpper=True, then
                the upper triangle contains matrix U, so that A = U'*U,
                and the elements below the main diagonal are not modified.
                Similarly, if IsUpper = False.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009-22.01.2018
     Bochkanov Sergey
*************************************************************************/
bool hpdmatrixcholesky(complex_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
bool hpdmatrixcholesky(complex_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);

`rmatrixlu` function

/*************************************************************************
LU decomposition of a general real matrix with row pivoting

A is represented as A = P*L*U, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=min(M,N)-1,
  Pi - permutation matrix for I and Pivots[I]

INPUT PARAMETERS:
    A       -   array[0..M-1, 0..N-1].
    M       -   number of rows in matrix A.
    N       -   number of columns in matrix A.


OUTPUT PARAMETERS:
    A       -   matrices L and U in compact form:
                * L is stored under main diagonal
                * U is stored on and above main diagonal
    Pivots  -   permutation matrix in compact form.
                array[0..Min(M-1,N-1)].

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     10.01.2010
     Bochkanov Sergey
*************************************************************************/
void rmatrixlu(real_2d_array &a, const ae_int_t m, const ae_int_t n, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);
void rmatrixlu(real_2d_array &a, integer_1d_array &pivots, const xparams _xparams = alglib::xdefault);

`sparsecholesky` function

/*************************************************************************
Sparse Cholesky decomposition for a matrix  stored  in  any sparse storage,
without rows/cols permutation.

This function is the most convenient (less parameters to specify), although
the less efficient, version of sparse Cholesky.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern, with the following matrix classes  being  the
           easiest ones to parallelize:
           * large matrices with many nearly-independent sets of rows/cols
           * matrices with large dense blocks on the diagonal
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

Internally it:
* calls SparseCholeskyAnalyze()  function  to  perform  symbolic  analysis
  phase with no permutation being configured.
* calls SparseCholeskyFactorize() function to perform numerical  phase  of
  the factorization

Following alternatives may result in better performance:
* using SparseCholeskyP(), which selects best  pivoting  available,  which
  almost always results in improved sparsity and cache locality
* using  SparseCholeskyAnalyze() and SparseCholeskyFactorize()   functions
  directly,  which  may  improve  performance of repetitive factorizations
  with same sparsity patterns.

The latter also allows one to perform  LDLT  factorization  of  indefinite
matrix (one with strictly diagonal D, which is known  to  be  stable  only
in few special cases, like quasi-definite matrices).

INPUT PARAMETERS:
    A       -   a square NxN sparse matrix, stored in any storage format.
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle.  Another triangle is ignored on  input,  dropped
                on output. Similarly, if IsUpper=False, the lower triangle
                is processed.

OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in CRS format:
                * if IsUpper=True, then the upper triangle contains matrix
                  U such  that  A = U^T*U and the lower triangle is empty.
                * similarly, if IsUpper=False, then lower triangular L  is
                  returned and we have A = L*(L^T).
                Note that THIS function does not  perform  permutation  of
                the rows to reduce fill-in.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False.  Contents  of  A  is  undefined
    in such case.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholesky(sparsematrix &a, const bool isupper, const xparams _xparams = alglib::xdefault);

`sparsecholeskyanalyze` function

/*************************************************************************
Sparse Cholesky/LDLT decomposition: symbolic analysis phase.

This function is a part of the 'expert' sparse Cholesky API:
* SparseCholeskyAnalyze(), that performs symbolic analysis phase and loads
  matrix to be factorized into internal storage
* SparseCholeskySetModType(), that allows to  use  modified  Cholesky/LDLT
  with lower bounds on pivot magnitudes and additional overflow safeguards
* SparseCholeskyFactorize(),  that performs  numeric  factorization  using
  precomputed symbolic analysis and internally stored matrix - and outputs
  result
* SparseCholeskyReload(), that reloads one more matrix with same  sparsity
  pattern into internal storage so  one  may  reuse  previously  allocated
  temporaries and previously performed symbolic analysis

This specific function performs preliminary analysis of the  Cholesky/LDLT
factorization. It allows to choose  different  permutation  types  and  to
choose between classic Cholesky and  indefinite  LDLT  factorization  (the
latter is computed with strictly diagonal D, i.e.  without  Bunch-Kauffman
pivoting).

NOTE: L*D*LT family of factorization may be used to  factorize  indefinite
      matrices. However, numerical stability is guaranteed ONLY for a class
      of quasi-definite matrices.

NOTE: all internal processing is performed with lower triangular  matrices
      stored  in  CRS  format.  Any  other  storage  formats  and/or upper
      triangular storage means  that  one  format  conversion  and/or  one
      transposition will be performed  internally  for  the  analysis  and
      factorization phases. Thus, highest  performance  is  achieved  when
      input is a lower triangular CRS matrix.

INPUT PARAMETERS:
    A           -   sparse square matrix in any sparse storage format.
    IsUpper     -   whether upper or lower  triangle  is  decomposed  (the
                    other one is ignored).
    FactType    -   factorization type:
                    * 0 for traditional Cholesky of SPD matrix
                    * 1 for LDLT decomposition with strictly  diagonal  D,
                        which may have non-positive entries.
    PermType    -   permutation type:
                    *-1 for absence of permutation
                    * 0 for best fill-in reducing  permutation  available,
                        which is 3 in the current version
                    * 1 for supernodal ordering (improves locality and
                      performance, does NOT change fill-in factor)
                    * 2 for original AMD ordering
                    * 3 for  improved  AMD  (approximate  minimum  degree)
                        ordering with better  handling  of  matrices  with
                        dense rows/columns

OUTPUT PARAMETERS:
    Analysis    -   contains:
                    * symbolic analysis of the matrix structure which will
                      be used later to guide numerical factorization.
                    * specific numeric values loaded into internal  memory
                      waiting for the factorization to be performed

This function fails if and only if the matrix A is symbolically degenerate
i.e. has diagonal element which is exactly zero. In  such  case  False  is
returned, contents of Analysis object is undefined.

  -- ALGLIB routine --
     20.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyanalyze(const sparsematrix &a, const bool isupper, const ae_int_t facttype, const ae_int_t permtype, sparsedecompositionanalysis &analysis, const xparams _xparams = alglib::xdefault);

`sparsecholeskyfactorize` function

/*************************************************************************
Sparse Cholesky decomposition: numerical analysis phase.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern, with the following matrix classes  being  the
           easiest ones to parallelize:
           * large matrices with many nearly-independent sets of rows/cols
           * matrices with large dense blocks on the diagonal
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

This function is a part of the 'expert' sparse Cholesky API:
* SparseCholeskyAnalyze(), that performs symbolic analysis phase and loads
  matrix to be factorized into internal storage
* SparseCholeskySetModType(), that allows to  use  modified  Cholesky/LDLT
  with lower bounds on pivot magnitudes and additional overflow safeguards
* SparseCholeskyFactorize(),  that performs  numeric  factorization  using
  precomputed symbolic analysis and internally stored matrix - and outputs
  result
* SparseCholeskyReload(), that reloads one more matrix with same  sparsity
  pattern into internal storage so  one  may  reuse  previously  allocated
  temporaries and previously performed symbolic analysis

Depending on settings specified during SparseCholeskyAnalyze() call it may
produce classic Cholesky or L*D*LT  decomposition  (with strictly diagonal
D), without permutation or with performance-enhancing permutation P.

NOTE: all internal processing is performed with lower triangular  matrices
      stored  in  CRS  format.  Any  other  storage  formats  and/or upper
      triangular storage means  that  one  format  conversion  and/or  one
      transposition will be performed  internally  for  the  analysis  and
      factorization phases. Thus, highest  performance  is  achieved  when
      input is a lower triangular CRS matrix, and lower triangular  output
      is requested.

NOTE: L*D*LT family of factorization may be used to  factorize  indefinite
      matrices. However, numerical stability is guaranteed ONLY for a class
      of quasi-definite matrices.

INPUT PARAMETERS:
    Analysis    -   prior analysis with internally stored matrix which will
                    be factorized
    NeedUpper   -   whether upper triangular or lower triangular output is
                    needed

OUTPUT PARAMETERS:
    A           -   Cholesky decomposition of A stored in lower triangular
                    CRS format, i.e. A=L*L' (or upper triangular CRS, with
                    A=U'*U, depending on NeedUpper parameter).
    D           -   array[N], diagonal factor. If no diagonal  factor  was
                    required during analysis  phase,  still  returned  but
                    filled with 1's
    P           -   array[N], pivots. Permutation matrix P is a product of
                    P(0)*P(1)*...*P(N-1), where P(i) is a  permutation  of
                    row/col I and P[I] (with P[I]>=I).
                    If no permutation was requested during analysis phase,
                    still returned but filled with identity permutation.

The function returns True  when  factorization  resulted  in nondegenerate
matrix. False is returned when factorization fails (Cholesky factorization
of indefinite matrix) or LDLT factorization has exactly zero  elements  at
the diagonal. In the latter case contents of A, D and P is undefined.

The analysis object is not changed during  the  factorization.  Subsequent
calls to SparseCholeskyFactorize() will result in same factorization being
performed one more time.

  -- ALGLIB routine --
     20.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyfactorize(sparsedecompositionanalysis &analysis, const bool needupper, sparsematrix &a, real_1d_array &d, integer_1d_array &p, const xparams _xparams = alglib::xdefault);

`sparsecholeskyp` function

/*************************************************************************
Sparse Cholesky decomposition for a matrix  stored  in  any sparse storage
format, with performance-enhancing permutation of rows/cols.

Present version is configured  to  perform  supernodal  permutation   with
a sparsity reducing ordering.

IMPORTANT: the commercial edition of ALGLIB can parallelize this function.
           Specific speed-up due  to  parallelism  heavily  depends  on  a
           sparsity pattern, with the following matrix classes  being  the
           easiest ones to parallelize:
           * large matrices with many nearly-independent sets of rows/cols
           * matrices with large dense blocks on the diagonal
           See the ALGLIB Reference Manual for more information on how  to
           activate parallelism support.

This function is a wrapper around generic sparse  decomposition  functions
that internally:
* calls SparseCholeskyAnalyze()  function  to  perform  symbolic  analysis
  phase with best available permutation being configured.
* calls SparseCholeskyFactorize() function to perform numerical  phase  of
  the factorization.

NOTE: using  SparseCholeskyAnalyze() and SparseCholeskyFactorize() directly
      may improve  performance  of  repetitive  factorizations  with  same
      sparsity patterns. It also allows one to perform  LDLT factorization
      of  indefinite  matrix  -  a factorization with strictly diagonal D,
      which  is  known to be stable only in few special cases, like quasi-
      definite matrices.

INPUT PARAMETERS:
    A       -   a square NxN sparse matrix, stored in any storage format.
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle.  Another triangle is ignored on  input,  dropped
                on output. Similarly, if IsUpper=False, the lower triangle
                is processed.

OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in CRS format:
                * if IsUpper=True, then the upper triangle contains matrix
                  U such  that  A = U^T*U and the lower triangle is empty.
                * similarly, if IsUpper=False, then lower triangular L  is
                  returned and we have A = L*(L^T).
    P       -   a row/column permutation, a product of P0*P1*...*Pk, k=N-1,
                with Pi being permutation of rows/cols I and P[I]

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False.  Contents  of  A  is  undefined
    in such case.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.09.2020
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyp(sparsematrix &a, const bool isupper, integer_1d_array &p, const xparams _xparams = alglib::xdefault);

`sparsecholeskyreload` function

/*************************************************************************
Sparse  Cholesky  decomposition:  update  internally  stored  matrix  with
another one with exactly same sparsity pattern.

This function is a part of the 'expert' sparse Cholesky API:
* SparseCholeskyAnalyze(), that performs symbolic analysis phase and loads
  matrix to be factorized into internal storage
* SparseCholeskySetModType(), that allows to  use  modified  Cholesky/LDLT
  with lower bounds on pivot magnitudes and additional overflow safeguards
* SparseCholeskyFactorize(),  that performs  numeric  factorization  using
  precomputed symbolic analysis and internally stored matrix - and outputs
  result
* SparseCholeskyReload(), that reloads one more matrix with same  sparsity
  pattern into internal storage so  one  may  reuse  previously  allocated
  temporaries and previously performed symbolic analysis

This specific function replaces internally stored  numerical  values  with
ones from another sparse matrix (but having exactly same sparsity  pattern
as one that was used for initial SparseCholeskyAnalyze() call).

NOTE: all internal processing is performed with lower triangular  matrices
      stored  in  CRS  format.  Any  other  storage  formats  and/or upper
      triangular storage means  that  one  format  conversion  and/or  one
      transposition will be performed  internally  for  the  analysis  and
      factorization phases. Thus, highest  performance  is  achieved  when
      input is a lower triangular CRS matrix.

INPUT PARAMETERS:
    Analysis    -   analysis object
    A           -   sparse square matrix in any sparse storage format.  It
                    MUST have exactly same sparsity pattern as that of the
                    matrix that was passed to SparseCholeskyAnalyze().
                    Any difference (missing elements or additional elements)
                    may result in unpredictable and undefined  behavior  -
                    an algorithm may fail due to memory access violation.
    IsUpper     -   whether upper or lower  triangle  is  decomposed  (the
                    other one is ignored).

OUTPUT PARAMETERS:
    Analysis    -   contains:
                    * symbolic analysis of the matrix structure which will
                      be used later to guide numerical factorization.
                    * specific numeric values loaded into internal  memory
                      waiting for the factorization to be performed

  -- ALGLIB routine --
     20.09.2020
     Bochkanov Sergey
*************************************************************************/
void sparsecholeskyreload(sparsedecompositionanalysis &analysis, const sparsematrix &a, const bool isupper, const xparams _xparams = alglib::xdefault);

`sparsecholeskyskyline` function

/*************************************************************************
Sparse Cholesky decomposition for skyline matrixm using in-place algorithm
without allocating additional storage.

The algorithm computes Cholesky decomposition  of  a  symmetric  positive-
definite sparse matrix. The result of an algorithm is a representation  of
A as A=U^T*U or A=L*L^T

This function allows to perform very efficient decomposition of low-profile
matrices (average bandwidth is ~5-10 elements). For larger matrices it  is
recommended to use supernodal Cholesky decomposition: SparseCholeskyP() or
SparseCholeskyAnalyze()/SparseCholeskyFactorize().

INPUT PARAMETERS:
    A       -   sparse matrix in skyline storage (SKS) format.
    N       -   size of matrix A (can be smaller than actual size of A)
    IsUpper -   if IsUpper=True, then factorization is performed on  upper
                triangle. Another triangle is ignored (it may contant some
                data, but it is not changed).


OUTPUT PARAMETERS:
    A       -   the result of factorization, stored in SKS. If IsUpper=True,
                then the upper  triangle  contains  matrix  U,  such  that
                A = U^T*U. Lower triangle is not changed.
                Similarly, if IsUpper = False. In this case L is returned,
                and we have A = L*(L^T).
                Note that THIS function does not  perform  permutation  of
                rows to reduce bandwidth.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.

NOTE: for  performance  reasons  this  function  does NOT check that input
      matrix  includes  only  finite  values. It is your responsibility to
      make sure that there are no infinite or NAN values in the matrix.

  -- ALGLIB routine --
     16.01.2014
     Bochkanov Sergey
*************************************************************************/
bool sparsecholeskyskyline(sparsematrix &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);

`sparselu` function

/*************************************************************************
Sparse LU decomposition with column pivoting for sparsity and row pivoting
for stability. Input must be square sparse matrix stored in CRS format.

The algorithm  computes  LU  decomposition  of  a  general  square  matrix
(rectangular ones are not supported). The result  of  an  algorithm  is  a
representation of A as A = P*L*U*Q, where:
* L is lower unitriangular matrix
* U is upper triangular matrix
* P = P0*P1*...*PK, K=N-1, Pi - permutation matrix for I and P[I]
* Q = QK*...*Q1*Q0, K=N-1, Qi - permutation matrix for I and Q[I]

This function pivots columns for higher sparsity, and then pivots rows for
stability (larger element at the diagonal).

INPUT PARAMETERS:
    A       -   sparse NxN matrix in CRS format. An exception is generated
                if matrix is non-CRS or non-square.
    PivotType-  pivoting strategy:
                * 0 for best pivoting available (2 in current version)
                * 1 for row-only pivoting (NOT RECOMMENDED)
                * 2 for complete pivoting which produces most sparse outputs

OUTPUT PARAMETERS:
    A       -   the result of factorization, matrices L and U stored in
                compact form using CRS sparse storage format:
                * lower unitriangular L is stored strictly under main diagonal
                * upper triangilar U is stored ON and ABOVE main diagonal
    P       -   row permutation matrix in compact form, array[N]
    Q       -   col permutation matrix in compact form, array[N]

This function always succeeds, i.e. it ALWAYS returns valid factorization,
but for your convenience it also returns  boolean  value  which  helps  to
detect symbolically degenerate matrices:
* function returns TRUE, if the matrix was factorized AND symbolically
  non-degenerate
* function returns FALSE, if the matrix was factorized but U has strictly
  zero elements at the diagonal (the factorization is returned anyway).


  -- ALGLIB routine --
     03.09.2018
     Bochkanov Sergey
*************************************************************************/
bool sparselu(sparsematrix &a, const ae_int_t pivottype, integer_1d_array &p, integer_1d_array &q, const xparams _xparams = alglib::xdefault);

`spdmatrixcholesky` function

/*************************************************************************
Cache-oblivious Cholesky decomposition

The algorithm computes Cholesky decomposition  of  a  symmetric  positive-
definite matrix. The result of an algorithm is a representation  of  A  as
A=U^T*U  or A=L*L^T

INPUT PARAMETERS:
    A       -   upper or lower triangle of a factorized matrix.
                array with elements [0..N-1, 0..N-1].
    N       -   size of matrix A.
    IsUpper -   if IsUpper=True, then A contains an upper triangle of
                a symmetric matrix, otherwise A contains a lower one.

OUTPUT PARAMETERS:
    A       -   the result of factorization. If IsUpper=True, then
                the upper triangle contains matrix U, so that A = U^T*U,
                and the elements below the main diagonal are not modified.
                Similarly, if IsUpper = False.

RESULT:
    If  the  matrix  is  positive-definite,  the  function  returns  True.
    Otherwise, the function returns False. Contents of A is not determined
    in such case.

  ! FREE EDITION OF ALGLIB:
  !
  ! Free Edition of ALGLIB supports following important features for  this
  ! function:
  ! * C++ version: x64 SIMD support using C++ intrinsics
  ! * C#  version: x64 SIMD support using NET5/NetCore hardware intrinsics
  !
  ! We  recommend  you  to  read  'Compiling ALGLIB' section of the ALGLIB
  ! Reference Manual in order  to  find  out  how to activate SIMD support
  ! in ALGLIB.

  ! COMMERCIAL EDITION OF ALGLIB:
  !
  ! Commercial Edition of ALGLIB includes following important improvements
  ! of this function:
  ! * high-performance native backend with same C# interface (C# version)
  ! * multithreading support (C++ and C# versions)
  ! * hardware vendor (Intel) implementations of linear algebra primitives
  !   (C++ and C# versions, x86/x64 platform)
  !
  ! We recommend you to read 'Working with commercial version' section  of
  ! ALGLIB Reference Manual in order to find out how to  use  performance-
  ! related features provided by commercial edition of ALGLIB.

  -- ALGLIB routine --
     15.12.2009
     Bochkanov Sergey
*************************************************************************/
bool spdmatrixcholesky(real_2d_array &a, const ae_int_t n, const bool isupper, const xparams _xparams = alglib::xdefault);
bool spdmatrixcholesky(real_2d_array &a, const bool isupper, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskyupdateadd1` function

/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

This function uses internally allocated buffer which is not saved  between
subsequent  calls.  So,  if  you  perform  a lot  of  subsequent  updates,
we  recommend   you   to   use   "buffered"   version   of  this function:
SPDMatrixCholeskyUpdateAdd1Buf().

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    U       -   array[N], rank-1 update to A: A_mod = A + u*u'
                Exception is thrown if array size is too small.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

NOTE: this function always succeeds, so it does not return completion code

NOTE: this function checks sizes of input arrays, but it does  NOT  checks
      for presence of infinities or NAN's.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdateadd1(real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyupdateadd1(real_2d_array &a, const bool isupper, const real_1d_array &u, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskyupdateadd1buf` function

/*************************************************************************
Update of Cholesky decomposition: rank-1 update to original A.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

See comments for SPDMatrixCholeskyUpdateAdd1() for more information.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    U       -   array[N], rank-1 update to A: A_mod = A + u*u'
                Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdateadd1buf(real_2d_array &a, const ae_int_t n, const bool isupper, const real_1d_array &u, real_1d_array &bufr, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskyupdatefix` function

/*************************************************************************
Update of Cholesky decomposition: "fixing" some variables.

This function uses internally allocated buffer which is not saved  between
subsequent  calls.  So,  if  you  perform  a lot  of  subsequent  updates,
we  recommend   you   to   use   "buffered"   version   of  this function:
SPDMatrixCholeskyUpdateFixBuf().

"FIXING" EXPLAINED:

    Suppose we have N*N positive definite matrix A. "Fixing" some variable
    means filling corresponding row/column of  A  by  zeros,  and  setting
    diagonal element to 1.

    For example, if we fix 2nd variable in 4*4 matrix A, it becomes Af:

        ( A00  A01  A02  A03 )      ( Af00  0   Af02 Af03 )
        ( A10  A11  A12  A13 )      (  0    1    0    0   )
        ( A20  A21  A22  A23 )  =>  ( Af20  0   Af22 Af23 )
        ( A30  A31  A32  A33 )      ( Af30  0   Af32 Af33 )

    If we have Cholesky decomposition of A, it must be recalculated  after
    variables were  fixed.  However,  it  is  possible  to  use  efficient
    algorithm, which needs O(K*N^2)  time  to  "fix"  K  variables,  given
    Cholesky decomposition of original, "unfixed" A.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    Fix     -   array[N], I-th element is True if I-th  variable  must  be
                fixed. Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

NOTE: this function always succeeds, so it does not return completion code

NOTE: this function checks sizes of input arrays, but it does  NOT  checks
      for presence of infinities or NAN's.

NOTE: this  function  is  efficient  only  for  moderate amount of updated
      variables - say, 0.1*N or 0.3*N. For larger amount of  variables  it
      will  still  work,  but  you  may  get   better   performance   with
      straightforward Cholesky.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdatefix(real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, const xparams _xparams = alglib::xdefault);
void spdmatrixcholeskyupdatefix(real_2d_array &a, const bool isupper, const boolean_1d_array &fix, const xparams _xparams = alglib::xdefault);

`spdmatrixcholeskyupdatefixbuf` function

/*************************************************************************
Update of Cholesky  decomposition:  "fixing"  some  variables.  "Buffered"
version which uses preallocated buffer which is saved  between  subsequent
function calls.

See comments for SPDMatrixCholeskyUpdateFix() for more information.

INPUT PARAMETERS:
    A       -   upper or lower Cholesky factor.
                array with elements [0..N-1, 0..N-1].
                Exception is thrown if array size is too small.
    N       -   size of matrix A, N>0
    IsUpper -   if IsUpper=True, then A contains  upper  Cholesky  factor;
                otherwise A contains a lower one.
    Fix     -   array[N], I-th element is True if I-th  variable  must  be
                fixed. Exception is thrown if array size is too small.
    BufR    -   possibly preallocated  buffer;  automatically  resized  if
                needed. It is recommended to  reuse  this  buffer  if  you
                perform a lot of subsequent decompositions.

OUTPUT PARAMETERS:
    A       -   updated factorization.  If  IsUpper=True,  then  the  upper
                triangle contains matrix U, and the elements below the main
                diagonal are not modified. Similarly, if IsUpper = False.

  -- ALGLIB --
     03.02.2014
     Sergey Bochkanov
*************************************************************************/
void spdmatrixcholeskyupdatefixbuf(real_2d_array &a, const ae_int_t n, const bool isupper, const boolean_1d_array &fix, real_1d_array &bufr, const xparams _xparams = alglib::xdefault);

`trigintegrals` subpackage

Functions

hyperbolicsinecosineintegrals
sinecosineintegrals

Examples

`hyperbolicsinecosineintegrals` function

/*************************************************************************
Hyperbolic sine and cosine integrals

Approximates the integrals

                           x
                           -
                          | |   cosh t - 1
  Chi(x) = eul + ln x +   |    -----------  dt,
                        | |          t
                         -
                         0

              x
              -
             | |  sinh t
  Shi(x) =   |    ------  dt
           | |       t
            -
            0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are evaluated by power series for x < 8
and by Chebyshev expansions for x between 8 and 88.
For large x, both functions approach exp(x)/2x.
Arguments greater than 88 in magnitude return MAXNUM.


ACCURACY:

Test interval 0 to 88.
                     Relative error:
arithmetic   function  # trials      peak         rms
   IEEE         Shi      30000       6.9e-16     1.6e-16
       Absolute error, except relative when |Chi| > 1:
   IEEE         Chi      30000       8.4e-16     1.4e-16

Cephes Math Library Release 2.8:  June, 2000
Copyright 1984, 1987, 2000 by Stephen L. Moshier
*************************************************************************/
void hyperbolicsinecosineintegrals(const double x, double &shi, double &chi, const xparams _xparams = alglib::xdefault);

`sinecosineintegrals` function

/*************************************************************************
Sine and cosine integrals

Evaluates the integrals

                         x
                         -
                        |  cos t - 1
  Ci(x) = eul + ln x +  |  --------- dt,
                        |      t
                       -
                        0
            x
            -
           |  sin t
  Si(x) =  |  ----- dt
           |    t
          -
           0

where eul = 0.57721566490153286061 is Euler's constant.
The integrals are approximated by rational functions.
For x > 8 auxiliary functions f(x) and g(x) are employed
such that

Ci(x) = f(x) sin(x) - g(x) cos(x)
Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x)


ACCURACY:
   Test interval = [0,50].
Absolute error, except relative when > 1:
arithmetic   function   # trials      peak         rms
   IEEE        Si        30000       4.4e-16     7.3e-17
   IEEE        Ci        30000       6.9e-16     5.1e-17

Cephes Math Library Release 2.1:  January, 1989
Copyright 1984, 1987, 1989 by Stephen L. Moshier
*************************************************************************/
void sinecosineintegrals(const double x, double &si, double &ci, const xparams _xparams = alglib::xdefault);

`variancetests` subpackage

Functions

ftest
onesamplevariancetest

Examples

`ftest` function

/*************************************************************************
Two-sample F-test

This test checks three hypotheses about dispersions of the given  samples.
The following tests are performed:
    * two-tailed test (null hypothesis - the dispersions are equal)
    * left-tailed test (null hypothesis  -  the  dispersion  of  the first
      sample is greater than or equal to  the  dispersion  of  the  second
      sample).
    * right-tailed test (null hypothesis - the  dispersion  of  the  first
      sample is less than or equal to the dispersion of the second sample)

The test is based on the following assumptions:
    * the given samples have normal distributions
    * the samples are independent.

Input parameters:
    X   -   sample 1. Array whose index goes from 0 to N-1.
    N   -   sample size.
    Y   -   sample 2. Array whose index goes from 0 to M-1.
    M   -   sample size.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 19.09.2006 by Bochkanov Sergey
*************************************************************************/
void ftest(const real_1d_array &x, const ae_int_t n, const real_1d_array &y, const ae_int_t m, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`onesamplevariancetest` function

/*************************************************************************
One-sample chi-square test

This test checks three hypotheses about the dispersion of the given sample
The following tests are performed:
    * two-tailed test (null hypothesis - the dispersion equals  the  given
      number)
    * left-tailed test (null hypothesis - the dispersion is  greater  than
      or equal to the given number)
    * right-tailed test (null hypothesis  -  dispersion is  less  than  or
      equal to the given number).

Test is based on the following assumptions:
    * the given sample has a normal distribution.

Input parameters:
    X           -   sample 1. Array whose index goes from 0 to N-1.
    N           -   size of the sample.
    Variance    -   dispersion value to compare with.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

  -- ALGLIB --
     Copyright 19.09.2006 by Bochkanov Sergey
*************************************************************************/
void onesamplevariancetest(const real_1d_array &x, const ae_int_t n, const double variance, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`wsr` subpackage

Functions

wilcoxonsignedranktest

Examples

`wilcoxonsignedranktest` function

/*************************************************************************
Wilcoxon signed-rank test

This test checks three hypotheses about the median  of  the  given sample.
The following tests are performed:
    * two-tailed test (null hypothesis - the median is equal to the  given
      value)
    * left-tailed test (null hypothesis - the median is  greater  than  or
      equal to the given value)
    * right-tailed test (null hypothesis  -  the  median  is  less than or
      equal to the given value)

Requirements:
    * the scale of measurement should be ordinal, interval or  ratio (i.e.
      the test could not be applied to nominal variables).
    * the distribution should be continuous and symmetric relative to  its
      median.
    * number of distinct values in the X array should be greater than 4

The test is non-parametric and doesn't require distribution X to be normal

Input parameters:
    X       -   sample. Array whose index goes from 0 to N-1.
    N       -   size of the sample.
    Median  -   assumed median value.

Output parameters:
    BothTails   -   p-value for two-tailed test.
                    If BothTails is less than the given significance level
                    the null hypothesis is rejected.
    LeftTail    -   p-value for left-tailed test.
                    If LeftTail is less than the given significance level,
                    the null hypothesis is rejected.
    RightTail   -   p-value for right-tailed test.
                    If RightTail is less than the given significance level
                    the null hypothesis is rejected.

To calculate p-values, special approximation is used. This method lets  us
calculate p-values with two decimal places in interval [0.0001, 1].

"Two decimal places" does not sound very impressive, but in  practice  the
relative error of less than 1% is enough to make a decision.

There is no approximation outside the [0.0001, 1] interval. Therefore,  if
the significance level outlies this interval, the test returns 0.0001.

  -- ALGLIB --
     Copyright 08.09.2006 by Bochkanov Sergey
*************************************************************************/
void wilcoxonsignedranktest(const real_1d_array &x, const ae_int_t n, const double e, double &bothtails, double &lefttail, double &righttail, const xparams _xparams = alglib::xdefault);

`xdebug` subpackage

Classes

xdebugrecord1

Functions

xdebugb1appendcopy
xdebugb1count
xdebugb1not
xdebugb1outeven
xdebugb2count
xdebugb2not
xdebugb2outsin
xdebugb2transpose
xdebugc1appendcopy
xdebugc1neg
xdebugc1outeven
xdebugc1sum
xdebugc2neg
xdebugc2outsincos
xdebugc2sum
xdebugc2transpose
xdebugi1appendcopy
xdebugi1neg
xdebugi1outeven
xdebugi1sum
xdebugi2neg
xdebugi2outsin
xdebugi2sum
xdebugi2transpose
xdebuginitrecord1
xdebugmaskedbiasedproductsum
xdebugr1appendcopy
xdebugr1internalcopyandsum
xdebugr1neg
xdebugr1outeven
xdebugr1sum
xdebugr2internalcopyandsum
xdebugr2neg
xdebugr2outsin
xdebugr2sum
xdebugr2transpose
xdebugupdaterecord1

Examples

`xdebugrecord1` class

/*************************************************************************
This is a debug class intended for testing ALGLIB interface generator.
Never use it in any real life project.

  -- ALGLIB --
     Copyright 20.07.2021 by Bochkanov Sergey
*************************************************************************/
class xdebugrecord1
{
public:
    xdebugrecord1();
    xdebugrecord1(const xdebugrecord1 &rhs);
    xdebugrecord1& operator=(const xdebugrecord1 &rhs);
    virtual ~xdebugrecord1();
    ae_int_t i;
    alglib::complex c;
    real_1d_array a;
};

`xdebugb1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugb1appendcopy(boolean_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb1count` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Counts number of True values in the boolean 1D array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t xdebugb1count(const boolean_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb1not` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by NOT(a[i]).
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugb1not(boolean_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered elements set to True.
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugb1outeven(const ae_int_t n, boolean_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb2count` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Counts number of True values in the boolean 2D array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t xdebugb2count(const boolean_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb2not` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by NOT(a[i]).
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugb2not(boolean_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb2outsin` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sin(3*I+5*J)>0"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugb2outsin(const ae_int_t m, const ae_int_t n, boolean_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugb2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugb2transpose(boolean_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugc1appendcopy(complex_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc1neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -A[I]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugc1neg(complex_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered A[K] set to (x,y) = (K*0.25, K*0.125)
and odd-numbered ones are set to 0.

Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugc1outeven(const ae_int_t n, complex_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc1sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
alglib::complex xdebugc1sum(const complex_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc2neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -a[i,j]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugc2neg(complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc2outsincos` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sin(3*I+5*J),Cos(3*I+5*J)"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugc2outsincos(const ae_int_t m, const ae_int_t n, complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc2sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
alglib::complex xdebugc2sum(const complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugc2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugc2transpose(complex_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugi1appendcopy(integer_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi1neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -A[I]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugi1neg(integer_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered A[I] set to I, and odd-numbered
ones set to 0.

Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugi1outeven(const ae_int_t n, integer_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi1sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t xdebugi1sum(const integer_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi2neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -a[i,j]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugi2neg(integer_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi2outsin` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sign(Sin(3*I+5*J))"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugi2outsin(const ae_int_t m, const ae_int_t n, integer_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi2sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
ae_int_t xdebugi2sum(const integer_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugi2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugi2transpose(integer_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebuginitrecord1` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Creates and returns XDebugRecord1 structure:
* integer and complex fields of Rec1 are set to 1 and 1+i correspondingly
* array field of Rec1 is set to [2,3]

  -- ALGLIB --
     Copyright 27.05.2014 by Bochkanov Sergey
*************************************************************************/
void xdebuginitrecord1(xdebugrecord1 &rec1, const xparams _xparams = alglib::xdefault);

`xdebugmaskedbiasedproductsum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of a[i,j]*(1+b[i,j]) such that c[i,j] is True

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double xdebugmaskedbiasedproductsum(const ae_int_t m, const ae_int_t n, const real_2d_array &a, const real_2d_array &b, const boolean_2d_array &c, const xparams _xparams = alglib::xdefault);

`xdebugr1appendcopy` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Appends copy of array to itself.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugr1appendcopy(real_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr1internalcopyandsum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

Internally it creates a copy of the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double xdebugr1internalcopyandsum(const real_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr1neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -A[I]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugr1neg(real_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr1outeven` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate N-element array with even-numbered A[I] set to I*0.25,
and odd-numbered ones are set to 0.

Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugr1outeven(const ae_int_t n, real_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr1sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double xdebugr1sum(const real_1d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr2internalcopyandsum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

Internally it creates a copy of a.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double xdebugr2internalcopyandsum(const real_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr2neg` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Replace all values in array by -a[i,j]
Array is passed using "shared" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugr2neg(real_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr2outsin` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Generate MxN matrix with elements set to "Sin(3*I+5*J)"
Array is passed using "out" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugr2outsin(const ae_int_t m, const ae_int_t n, real_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr2sum` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Returns sum of elements in the array.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
double xdebugr2sum(const real_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugr2transpose` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Transposes array.
Array is passed using "var" convention.

  -- ALGLIB --
     Copyright 11.10.2013 by Bochkanov Sergey
*************************************************************************/
void xdebugr2transpose(real_2d_array &a, const xparams _xparams = alglib::xdefault);

`xdebugupdaterecord1` function

/*************************************************************************
This is debug function intended for testing ALGLIB interface generator.
Never use it in any real life project.

Creates and returns XDebugRecord1 structure:
* integer and complex fields of Rec1 are set to 1 and 1+i correspondingly
* array field of Rec1 is set to [2,3]

  -- ALGLIB --
     Copyright 27.05.2014 by Bochkanov Sergey
*************************************************************************/
void xdebugupdaterecord1(xdebugrecord1 &rec1, const xparams _xparams = alglib::xdefault);

1 Introduction

1.1 What is ALGLIB

1.2 ALGLIB license

1.3 Documentation license

1.4 Reference Manual and User Guide

1.5 Acknowledgements

2 ALGLIB structure

2.1 Packages

2.2 Subpackages

2.3 Open Source and Commercial versions

3 Compatibility

3.1 CPU

3.2 OS

3.3 Compiler

3.4 Optimization settings

4 Compiling ALGLIB

4.1 Adding to your project

4.2 Configuring for your compiler

4.3 Utilizing SIMD

4.3.1 Overview

4.3.2 Achieving best performance

4.3.3 Achieving best portability

4.3.4 Disabling SIMD instruction sets at compile time

4.4 Utilizing SMP

4.5 Examples (free and commercial editions)

4.5.1 Introduction

4.5.2 Compiling under Windows

4.5.3 Compiling under Linux

5 Using ALGLIB

5.1 Thread-safety

5.2 Global definitions

5.3 Datatypes

5.4 Constants

5.5 Functions

5.6 Working with vectors and matrices

5.7 Using functions: 'expert' and 'friendly' interfaces

5.8 Handling errors

5.9 Working with Level 1 BLAS functions

5.10 Reading data from CSV files

6 Working with commercial version

6.1 Benefits of commercial version

6.2 Working with SIMD support (Intel/AMD users)

6.3 Using multithreading

6.3.1 General information

6.3.2 Compiling ALGLIB in the multithreaded mode

6.3.3 Two kinds of parallelism

6.3.4 Activating parallelism

6.3.5 Controlling cores count

6.3.6 More on parallel callbacks

6.3.7 SMT (CMT/hyper-threading) issues

6.4 Linking with Intel MKL

6.4.1 Using lightweight Intel MKL supplied by ALGLIB Project

6.4.2 Using your own installation of Intel MKL

7 Advanced topics

7.1 Using Red Zones to find memory access violations

7.2 Replacing stdlib rand() as an entropy source with OpenSSL implementation

7.3 Exception-free mode

7.4 Partial compilation

7.5 Testing ALGLIB

8 ALGLIB packages and subpackages

8.1 AlglibMisc package

8.2 DataAnalysis package

8.3 DiffEquations package

8.4 FastTransforms package

8.5 Integration package

8.6 Interpolation package

8.7 LinAlg package

8.8 Optimization package

8.9 Solvers package

8.10 SpecialFunctions package

8.11 Statistics package

ablas subpackage

Functions

Examples

cmatrixcopy function

cmatrixgemm function

cmatrixherk function

cmatrixlefttrsm function

cmatrixmv function

cmatrixrank1 function

8.1 `AlglibMisc` package

8.2 `DataAnalysis` package

8.3 `DiffEquations` package

8.4 `FastTransforms` package

8.5 `Integration` package

8.6 `Interpolation` package

8.7 `LinAlg` package

8.8 `Optimization` package

8.9 `Solvers` package

8.10 `SpecialFunctions` package

8.11 `Statistics` package

`ablas` subpackage

`cmatrixcopy` function

`cmatrixgemm` function

`cmatrixherk` function

`cmatrixlefttrsm` function

`cmatrixmv` function

`cmatrixrank1` function

`cmatrixrighttrsm` function

`cmatrixsyrk` function

`cmatrixtranspose` function

`rmatrixcopy` function

`rmatrixenforcesymmetricity` function

`rmatrixgemm` function

`rmatrixgemv` function

`rmatrixgencopy` function

`rmatrixger` function

`rmatrixlefttrsm` function

`rmatrixmv` function

`rmatrixrank1` function

`rmatrixrighttrsm` function

`rmatrixsymv` function

`rmatrixsyrk` function

`rmatrixsyvmv` function

`rmatrixtranspose` function

`rmatrixtrsv` function

`rvectorcopy` function

`airyf` subpackage

`airy` function

`autogk` subpackage

`autogkreport` class

`autogkstate` class

`autogkintegrate` function

`autogkiteration` function

`autogkresults` function

`autogksingular` function

`autogksmooth` function

`autogksmoothw` function

`basestat` subpackage

`cov2` function

`covm` function

`covm2` function

`pearsoncorr2` function

`pearsoncorrelation` function

`pearsoncorrm` function

`pearsoncorrm2` function

`rankdata` function

`rankdatacentered` function

`sampleadev` function

`samplekurtosis` function

`samplemean` function

`samplemedian` function

`samplemoments` function

`samplepercentile` function

`sampleskewness` function

`samplevariance` function

`spearmancorr2` function

`spearmancorrm` function

`spearmancorrm2` function

`spearmanrankcorrelation` function

`bdss` subpackage

`dsoptimalsplit2` function