1 Introduction

1.1 What is ALGLIB

ALGLIB is a cross-platform numerical analysis 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)
Continuous and mixed-integer optimization
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 precompiled performance backends (based on Intel MKL) that accelerate low-level tasks like basic dense and sparse linear algebra. 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 respects:

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 and SIMD-optimized. Open source ALGLIB is single-threaded and can not efficiently use modern multicore CPU's; it also lacks commercial SIMD kernels.
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 the 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)

All mainstream CPU's (in particular, x86, x86_64, ARM, RISC-V) 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, AE_OS=AE_LINUX or AE_OS=AE_POSIX at compile time, depending on OS being used. Former corresponds to any OS from the Windows family (from most recent Windows versions and downb to 32/64-bit Windows XP), while latter two means almost any LINUX edition or generic POSIX-compatible OS. Specifying AE_OS=AE_LINUX allows ALGLIB to use Linux-specific performance-related OS extensions, while AE_OS=AE_POSIX allows ALGLIB to run on 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 mainstream compilers (MSVC, GCC, Clang, ICC) satisfy these requirements. Most specialized compilers for DSPs also satisfy them.

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).
ae_uint64_t is defined as unsigned _int64 (MSVC) or as unsigned long long (GCC).
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 4.00+, Commercial Edition provides a better set of SIMD kernels than the Free one.

The Commercial Edition also includes so-called Performance Backends - optional precompiled shared libraries that include vendor-optimized implementations of linear algebra and other functions, e.g. Intel MKL backend. These are different from SIMD kernels - the latter were developed by us and are deeply integrated into ALGLIB, while the former are optional external third-party components. ALGLIB can be compiled without a Performance Backend linked, but it can not be compiled without SIMD kernel units (you can compile them in the generic C mode, though).

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 functions, 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 and optimization settings of your choice

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 architecture settings
it is recommended to use the best optimization settings for the entire library; it will not compromise its portability

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 the compile time, depending on the OS being used. The former corresponds to any modern OS from the Windows family (the oldest OS we support is 32/64-bit Windows XP), while the latter means almost any POSIX-compatible OS (or any OS from the Linux family). ALGLIB will utilize OS capabilities for thread management: 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

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 (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.

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

> 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.

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 alglib???_??mkl.so shared library, which contains special lightweight MKL distribution shipped with ALGLIB.

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 the 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 a 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 the 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.
Performance backends. Commercial version of ALGLIB includes a standardized interface to various optimized libraries provided by hardware vendors (Intel). It also comes with precompiled versions of these libraries which can be linked with ALGLIB without installing vendor software development tools. ALGLIB uses these libraries (when available) to accelerate its own functions, relying on its own optimized implementations if you decide not to link it with third party components.

6.2 Working with SIMD support (Intel/AMD users)

Commercial ALGLIB for C++ can utilize SIMD instructions supported by Intel and AMD processors more extensively than the Free Edition. 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.

In order to turn on x64-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.

Configuration of SIMD support is discussed in a more detailed manner in the separate section of the Reference Manual.

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 its own synchronization framework. Under POSIX-compatible OS (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 OS modern from the Windows family (and down to an antique 32/64-bit Windows XP), 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::setglobalflags 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.

There are also additional values that are used to control interaction with performance backends. These values are discussed in the corresponding manual section.

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.

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.

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 the 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/hyper-threading issues

Simultaneous multithreading (SMT), also known as Hyper-threading, is a CPU design where several (usually two) logical cores share resources of one physical core. Say, on a 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 us 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 precompiled performance backends

6.4.1 Introduction

ALGLIB is a self-contained numerical toolkit that comes with its own implementation of all of its functions. ALGLIB also has its own multithreading and SIMD kernels, so it provides a good performance without linking with any external library.

Nevertheless, it is possible to further accelerate ALGLIB by linking with optimized software libraries provided by hardware vendors. These libraries can be used to accelerate low-to-mid-level operations like BLAS or sparse direct solvers (depending on a specific backend chosen). The acceleration can be due to more efficient SIMD usage (most typically), or due to different algorithms used. ALGLIB can export accelerated functionality as its 'primary' function - e.g., a request to solve a linear system can be forwarded directly to a performance backend. It can also use backend to accelerate higher-level functions that depend on this functionality (e.g., SQP or interior point optimizers spend a lot of time doing basic linear algebra).

ALGLIB comes with several precompiled backends (shared libraries with standardized APIs) that can be linked without downloading entire SDK from a hardware vendor website. For exampe, in order to use Intel MKL you do not have to install entire oneAPI toolkit - using a single shared library included into commercial ALGLIB is enough. You can use backend libraries as a part of ALGLIB under a combination of ALGLIB License Agreement and corresponding license terms for the backend library (included into ALGLIB archive). Doing so is allowed by license agreements with corresponding vendors.

It is also possible to link ALGLIB with your own installation of a vendor's library. For example, it makes sense if you already use the same vendor library in your application. Below we will discuss both options.

6.4.2 Using precompiled performance backends

A precompiled performance backend is a shared library (DLL and import LIB under Windows, SO under Linux/POSIX) that sits in the performance-backends/backend-name directory of your commercial ALGLIB distribution. You can treat it as a black box that provides accelerated functions.

All precompiled backends export their functions via single unified API used by ALGLIB. Thus, they can be linked interchangeably without making any changes to the way ALGLIB itself is compiled. If a function (e.g., direct sparse solver) is missing in this specific backend, it simply returns 'unsupported' and ALGLIB falls back to its own implementation. No matter which specific backend is linked, you can always use full set of ALGLIB functions.

The core dispatcher that handles requests for a potentially accelerated functionality is located in the ap.cpp file. Depending on specific preprocessor symbols #defined during compilation and on problem sizes, it can forward a call to a generic C implementation, to one of the SIMD-optimized versions located in kernels_avx2/fma/etc.cpp, or to a performance backend currently linked:

if no CPU or backend-related symbols is #defined, a generic C implementation is used
if AE_CPU is #defined, a call can be forwarded to an ALGLIB own SIMD kernel
if ALGLIB_LINK_PERFORMANCE_BACKEND (or one of backend-specific symbols) is #defined, a call can be forwarded to a backend
if both CPU and backend symbols are #defined, ALGLIB dynamically decides what to do depending on problem metrics and additional flags passed by you at run time. BLAS/LAPACK-related calls are typically forwarded to a vendor backend, because it is assumed to be better optimized. For higher-level functions (e.g. direct sparse solvers) the choice is typically done at user side by specifying additional flags.

ALGLIB provides the flexibility to define performance backend settings at both global and function-specific levels. Global settings can be configured using the alglib::setglobalflags 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 settings 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 backend-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 can have the following backend-related values:

alglib::backend_linalg and alglib::no_backend_linalg - BLAS/LAPACK usage.
alglib::backend_dss and alglib::no_backend_dss - direct sparse solver usage.

There are also additional values that are used to control parallelism. These values are discussed in the corresponding manual section.

You can combine values pertaining to different subsets of backend functionality (basic linear algebra, direct sparse solvers) using the OR operator, such as alglib::backend_linalg|alglib::no_backend_dss. However, attempting to combine conflicting values like alglib::backend_linalg|alglib::no_backend_linalg will result in an undefined behavior.

If a value for a specific type of backend functionality is not provided at the function-local level (e.g., only alglib::backend_linalg is passed which defines settings for BLAS but omits DSS), then this specific setting defaults to the global settings as determined by alglib::setglobalflags. In cases where no global settings are specified, ALGLIB chooses a default option on its own.

IMPORTANT: Activating backend does not automatically mean that computations will always be passed to the backend. ALGLIB assesses whether the problem at hand is sufficiently large so that the additional overhead from calling backend is negligible.

6.4.3 Performance backend: mklmini

This performance backend is a lightweight subset of Intel MKL that includes only BLAS and LAPACK kernels. The backend includes only single-threaded versions of BLAS/LAPACK, with parallelism done at ALGLIB level.

Compiling and linking with the backend. In order to activate ALGLIB-backend connection, ALGLIB itself must be compiled as follows:

ALGLIB_LINK_PERFORMANCE_BACKEND must be #defined at the project level in order to activate ALGLIB-backend connection.
additionally, AE_OS=AE_WINDOWS (AE_OS=AE_POSIX, AE_OS=AE_LINUX) must be #defined to activate multithreading capabilities
additionally, AE_CPU=AE_INTEL must be #defined to use SIMD instructions provided by x86/x64 CPU's for the rest of ALGLIB

After that, an appropriate shared library from performance-backends/mklmini has to be linked with your program and ALGLIB:

x64/x86 Windows. An import library (.lib file) for the backend has to be included in your project, and a dynamic library (.dll file) has to be present during run-time.
x64 Linux. A shared library (.so file) has to be specified during compile-time, and has to be present during run-time.

Each library in the performance-backends/mklmini directory is a highly portable standalone binary with no external dependencies, except for standard system libraries. You do not have to install Intel MKL yourself in order to use this backend.

Compatibility. The backend is highly compatible with all user configurations: serial applications, parallel applications using OS threads, parallel applications using various OpenMP implementations.

Activating the backend. BLAS/LAPACK kernels are active by default, unless explicitly turned off with the alglib::no_backend_linalg flag.

Performance. Linking with this backend results in the following performance-related changes:

BLAS/LAPACK. Merely linking with the backend accelerates all parts of ALGLIB that depend on dense BLAS/LAPACK functionality, because BLAS/LAPACK calls are always forwarded to the vendor library. Some limited acceleration is provided for algorithms dependent on sparse BLAS; however, direct sparse solvers are not accelerated by this flag. We have never observed performance deterioration from switching to MKL dense BLAS/LAPACK kernels, typical results range from several-fold improvement to zero (for problems that do not depend on BLAS/LAPACK).
Parallelism. This backend includes only single-threaded versions of BLAS/LAPACK kernels that are parallelized at ALGLIB level. Full compatibility with ALGLIB parallelism flags.

6.4.4 Performance backend: mklfull

This performance backend includes a larger subset of Intel MKL than mklmini: BLAS/LAPACK kernels plus PARDISO (a direct sparse solver by Intel) for x86/x64 architectures. This backend includes only single-threaded versions of BLAS/LAPACK kernels and PARDISO. PARDISO is always executed serially, BLAS/LAPACK are parallelized at ALGLIB level.

Compiling and linking with the backend. In order to activate ALGLIB-backend connection, ALGLIB itself must be compiled as follows:

ALGLIB_LINK_PERFORMANCE_BACKEND must be #defined at the project level in order to activate ALGLIB-backend connection.
additionally, AE_OS=AE_WINDOWS (AE_OS=AE_POSIX, AE_OS=AE_LINUX) must be #defined to activate multithreading capabilities
additionally, AE_CPU=AE_INTEL must be #defined to use SIMD instructions provided by x86/x64 CPU's for the rest of ALGLIB

After that, an appropriate shared library from performance-backends/mklfull has to be linked with your program and ALGLIB:

x64/x86 Windows. An import library (.lib file) for the backend has to be included in your project, and a dynamic library (.dll file) has to be present during run-time.
x64 Linux. A shared library (.so file) has to be specified during compile-time, and has to be present during run-time.

Each library in the performance-backends/mklfull directory is a highly portable standalone binary with no external dependencies, except for standard system libraries. You do not have to install Intel MKL yourself in order to use this backend.

Compatibility. The backend is highly compatible with all user configurations: serial applications, parallel applications using OS threads, parallel applications using various OpenMP implementations.

Activating the backend. The backend includes two functionally distinct parts: BLAS/LAPACK kernels and DSS (Direct Sparse Solver) functionality. BLAS/LAPACK kernels are active by default, unless explicitly turned off with the alglib::no_backend_linalg flag. DSS functionality is turned off by default (because PARDISO is not always superior to ALGLIB), but can be activated by passing alglib::backend_dss flag to ALGLIB.

Performance. Linking with this backend results in the following performance-related changes:

BLAS/LAPACK. Merely linking with the backend accelerates all parts of ALGLIB that depend on dense BLAS/LAPACK functionality, because BLAS/LAPACK calls are always forwarded to the vendor library. Some limited acceleration is provided for algorithms dependent on sparse BLAS; however, direct sparse solvers are not accelerated by this flag. We have never observed performance deterioration from switching to MKL dense BLAS/LAPACK kernels, typical results range from several-fold improvement to zero (for problems that do not depend on BLAS/LAPACK).
DSS. PARDISO direct sparse solver (DSS) is disabled by default, but can be enabled with alglib::backend_dss flag. For large sparse matrices with big amounts of fill-in PARDISO often provides several-fold improvement over ALGLIB DSS. However, for highly sparse problems ALGLIB DSS is typically much faster than PARDISO - from several times to an order of magnitude. Performance changes are most noticeable for sparse direct solvers, continuous and mixed-integer optimizers. The short summary is that ALGLIB DSS provides uniform performance for all problem classes, while PARDISO shines only for some specific problem types.
Parallelism. This backend includes only single-threaded versions of BLAS/LAPACK kernels and PARDISO. BLAS/LAPACK are parallelized at ALGLIB level, with the full compatibility with ALGLIB parallelism flags. PARDISO is always executed serially, although due to its thread-safety it can be used from multiple threads (either ALGLIB worker threads or user threads).

6.5 Linking with performance backends installed on your system

6.5.1 Using your own installation of Intel MKL

If you want to use your own installation of MKL, then you should compile ALGLIB as follows:

1. Add to your project alglib2pbl.c file from the /performance-backends directory and compile it (as C file) along with the rest of ALGLIB. This C file implements interface between ALGLIB and performance backends. Having this file in your project and defining appropriate preprocessor definition (see below) 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.

2. ALGLIB and alglib2pbl.c should be compiled with all the necessary CPU and OS-specific preprocessor definitions (AE_OS=AE_WINDOWS or AE_OS=AE_LINUX, and AE_CPU=AE_INTEL). Additionally, you should #define AE_MKL at the global level. This definition tells ALGLIB to use all functions of Intel MKL (BLAS, LAPACK, direct sparse solvers, etc). Additionally, you can #define AE_LIGHTWEIGHT_PBL if you want ALGLIB to use only BLAS/LAPACK kernels.

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.

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)

Functions

airy

Examples

Integrating f=exp(x) by adaptive integrator

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

Functions

dsoptimalsplit2
dsoptimalsplit2fast

Examples

Functions

rmatrixbdsvd

Examples

Functions

besseli0
besseli1
besselj0
besselj1
besseljn
besselk0
besselk1
besselkn
bessely0
bessely1
besselyn

Examples

Functions

beta

Examples

Functions

binomialcdistribution
binomialdistribution
invbinomialdistribution

Examples

Functions

chebyshevcalculate
chebyshevcoefficients
chebyshevsum
fromchebyshev

Examples

Functions

chisquarecdistribution
chisquaredistribution
invchisquaredistribution

Examples

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

Functions

convc1d
convc1dbuf
convc1dcircular
convc1dcircularbuf
convc1dcircularinv
convc1dcircularinvbuf
convc1dinv
convc1dinvbuf
convr1d
convr1dbuf
convr1dcircular
convr1dcircularbuf
convr1dcircularinv
convr1dcircularinvbuf
convr1dinv
convr1dinvbuf

Examples

Functions

corrc1d
corrc1dbuf
corrc1dcircular
corrc1dcircularbuf
corrr1d
corrr1dbuf
corrr1dcircular
corrr1dcircularbuf

Examples

Functions

pearsoncorrelationsignificance
spearmanrankcorrelationsignificance

Examples

Functions

kmeansgenerate

Examples

Functions

dawsonintegral

Examples

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

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

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

Functions

ellipticintegrale
ellipticintegralk
ellipticintegralkhighprecision
incompleteellipticintegrale
incompleteellipticintegralk

Examples

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

Functions

exponentialintegralei
exponentialintegralen

Examples

Functions

fcdistribution
fdistribution
invfdistribution

Examples

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

Functions

fhtr1d
fhtr1dinv

Examples

Functions

filterema
filterlrma
filtersma

Examples

filters_d_ema		EMA(alpha) filter
filters_d_lrma		LRMA(k) filter
filters_d_sma		SMA(k) filter

Functions

fitspherels
fitspheremc
fitspheremi
fitspheremz
fitspherex

Examples

Functions

fresnelintegral

Examples

Functions

gammafunction
lngamma

Examples

Functions

gkqgenerategaussjacobi
gkqgenerategausslegendre
gkqgeneraterec
gkqlegendrecalc
gkqlegendretbl

Examples

Functions

gqgenerategausshermite
gqgenerategaussjacobi
gqgenerategausslaguerre
gqgenerategausslegendre
gqgenerategausslobattorec
gqgenerategaussradaurec
gqgeneraterec

Examples

Functions

hermitecalculate
hermitecoefficients
hermitesum

Examples

Classes

hqrndstate

Functions

hqrndcontinuous
hqrnddiscrete
hqrndexponential
hqrndnormal
hqrndnormal2
hqrndnormalm
hqrndnormalv
hqrndrandomize
hqrndseed
hqrnduniformi
hqrnduniformr
hqrndunit2

Examples

Functions

incompletebeta
invincompletebeta

Examples

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

Functions

incompletegamma
incompletegammac
invincompletegammac

Examples

Functions

nsfitspheremcc
nsfitspheremic
nsfitspheremzc
nsfitspherex
spline1dfitcubic
spline1dfithermite
spline1dfitpenalized
spline1dfitpenalizedw

Examples

Functions

rmatrixinvupdatecolumn
rmatrixinvupdaterow
rmatrixinvupdatesimple
rmatrixinvupdateuv

Examples

Classes

sparsesolverreport
sparsesolverstate

Functions

sparsesolvegmres
sparsesolvercreate
sparsesolverooccontinue
sparsesolveroocgetrequestdata
sparsesolveroocgetrequestdata1
sparsesolveroocgetrequestinfo
sparsesolveroocsendresult
sparsesolveroocstart
sparsesolveroocstop
sparsesolverrequesttermination
sparsesolverresults
sparsesolversetalgogmres
sparsesolversetcond
sparsesolversetstartingpoint
sparsesolversetxrep
sparsesolversolve
sparsesolversolvesymmetric
sparsesolvesymmetricgmres

Examples

Functions

jacobianellipticfunctions

Examples

Functions

jarqueberatest

Examples

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

Functions

laguerrecalculate
laguerrecoefficients
laguerresum

Examples

Functions

fisherlda
fisherldan

Examples

Functions

legendrecalculate
legendrecoefficients
legendresum

Examples

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

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

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

Classes

logitmodel
mnlreport

Functions

mnlavgce
mnlavgerror
mnlavgrelerror
mnlclserror
mnlpack
mnlprocess
mnlprocessi
mnlrelclserror
mnlrmserror
mnltrainh
mnlunpack

Examples

Classes

minlpreport
minlpstate

Functions

minlpaddlc2
minlpaddlc2dense
minlpcreate
minlpoptimize
minlpresults
minlpresultsbuf
minlpsetalgodss
minlpsetalgoipm
minlpsetbc
minlpsetbcall
minlpsetbci
minlpsetcost
minlpsetlc
minlpsetlc2
minlpsetlc2dense
minlpsetscale

Examples

lpsolvers_basic

Basic linear programming example

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

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 `MINLP` package
minlpsolvers		Mixed-integer nonlinear programming solver (structured and black-box problems)

8.9 `Optimization` package
lpsolvers		Linear programming suite
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
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 (core types)

8.10 `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

8.11 `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

8.12 `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
mcmc		Markov Chain Monte-Carlo
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

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/hyper-threading issues

6.4 Linking with precompiled performance backends

6.4.1 Introduction

6.4.2 Using precompiled performance backends

6.4.3 Performance backend: mklmini

6.4.4 Performance backend: mklfull

6.5 Linking with performance backends installed on your system

6.5.1 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 MINLP package

8.9 Optimization package

8.10 Solvers package

8.11 SpecialFunctions package

8.12 Statistics package

ablas subpackage

Functions

Examples

cmatrixcopy 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 `MINLP` package

8.9 `Optimization` package

8.10 `Solvers` package

8.11 `SpecialFunctions` package

8.12 `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