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 Improving performance (CPU-specific and OS-specific optimizations) 5 Working with commercial version 5.1 Benefits of commercial version 5.2 Working with SSE support (Intel/AMD users) 5.3 Using multithreading 5.3.1 SMT (CMT/hyper-threading) issues 5.4 Linking with Intel MKL 5.4.1 Using lightweight Intel MKL supplied by ALGLIB Project 5.4.2 Using your own installation of Intel MKL 5.5 Examples - compiling commercial edition of ALGLIB 5.5.1 Introduction 5.5.2 Compiling under Windows 6 Using ALGLIB 6.1 Thread-safety 6.2 Global definitions 6.3 Datatypes 6.4 Constants 6.5 Functions 6.6 Working with vectors and matrices 6.7 Using functions: 'expert' and 'friendly' interfaces 6.8 Handling errors 6.9 Working with Level 1 BLAS functions 6.10 Reading data from CSV files 7 Advanced topics 7.1 Testing ALGLIB 8 ALGLIB packages and subpackages 8.1AlglibMiscpackage 8.2DataAnalysispackage 8.3DiffEquationspackage 8.4FastTransformspackage 8.5Integrationpackage 8.6Interpolationpackage 8.7LinAlgpackage 8.8Optimizationpackage 8.9Solverspackage 8.10SpecialFunctionspackage 8.11Statisticspackage
ALGLIB is a cross-platform numerical analysis and data mining library. It supports several programming languages (C++, C#, Pascal, VBA) and several operating systems (Windows, *nix family).
ALGLIB features include:
ALGLIB Project (the company behind ALGLIB) delivers to you several editions of ALGLIB:
Free Edition is a serial version without multithreading support or extensive low-level optimizations (generic C or C# code). Commercial Edition is a heavily optimized version of ALGLIB. It supports multithreading, it was extensively optimized, and (on Intel platforms) - our commercial users may enjoy precompiled version of ALGLIB which internally calls Intel MKL to accelerate low-level tasks. We obtained license from Intel corp., which allows us to integrate Intel MKL into ALGLIB, so you don't have to buy separate license from Intel.
ALGLIB Free Edition is 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 Commercial Edition is distributed under license which is friendly to commericial users. A copy of the commercial license can be found at http://www.alglib.net/commercial.php.
This reference manual is licensed under BSD-like documentation license:
Copyright 1994-2017 Sergey Bochkanov, ALGLIB Project. All rights reserved.
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.
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 Reference Manual is a place to go.
Free software needs free documentation - that's why 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 it is? 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 ALGLIB User Guide is distributed under less-permissive personal-use-only license.
ALGLIB was not possible without contribution of the next open source projects:
We also want to thank developers of the Intel's local development center (Nizhny Novgorod branch) for their help during MKL integration.
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. Pre-3.0 versions of ALGLIB included more than 100 units, but it was difficult to work with such large number of files. Since ALGLIB 3.0 all units are merged into 11 packages and two support units:
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.
But many packages require only these two to work, and many other packages need significantly less than 13 packages.
For example, statistics.cpp requires two packages mentioned above and only one additional package - specialfunctions.cpp.
There is one more concept to learn - subpackages.
Every package was created from several source files.
For example (as of ALGLIB 3.0.0), linalg.cpp was created by merging together 14 .cpp files (C++ interface) and 14 .c files (computational core).
These files provide different functionality: one of them calculates triangular factorizations, another generates random matrices, and so on.
We've merged source code, but what to do with their documentation?
Of course, we can merge their documentation (as we've merged units) in one big list of functions and data structures, but such list will be hard to read. Instead, we have decided to merge source code, but leave documentation separate.
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.
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 open source one in following aspects:
This documentation applies to both versions of ALGLIB. Detailed description of commercial version can be found below.
ALGLIB is compatible with any CPU which:
Most mainstream CPU's (in particular, x86, x86_64, ARM and SPARC) satisfy these requirements.
As for Intel architecture, ALGLIB works with both FPU-based and SSE-based implementations of floating point math. 80-bit internal representation used by Intel FPU is not a problem for ALGLIB.
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 commercial version of ALGLIB,
you should compile it in OS-specific mode by #defining either AE_OS=AE_WINDOWS
or AE_OS=AE_POSIX at compile time, depending on OS being used.
Former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family,
while latter means almost any POSIX-compatible OS.
It applies only to commercial version of ALGLIB.
Open source version is always OS-agnostic, even in the presence of OS-specific definitions.
ALGLIB is compatible with any C++ compiler which:
x/0 will return INF.
But at least we must be able to compare double precision value with infinity or NAN without raising exception.All modern compilers (in particular, reasonably new versions of MSVC, GCC and Sun Studio) satisfy these requirements.
However, some very old compilers (ten years old version of Borland C++ Builder, for example) may emit code which does not correctly work with IEEE special values. If you use one of these old compilers, we recommend you to run ALGLIB test suite to ensure that library works.
ALGLIB is compatible with any kind of optimizing compiler as long as:
Generally, all kinds of optimization that were marked by compiler vendor as "safe" are possible. For example, ALGLIB can be compiled:
From the other side, following "unsafe" optimizations will break ALGLIB:
Adding ALGLIB to your project is easy - just pick packages you need and... add them to your project! Under most used compilers (GCC, MSVC, Sun Studio) 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:
In any case, compiling ALGLIB is so simple that even without project file you can do it in several minutes.
If you use modern versions of MSVC, GCC or Sun Studio, 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 tries to autodetect your compiler and to define these types in compiler-specific manner:
ae_int32_t is defined as int, because this type is 32 bits wide in all modern compilers.ae_int64_t is defined as _int64 (MSVC) or as signed long long (GCC, Sun Studio).ae_int_t is defined as ptrdiff_t.In most cases, it is enough. But if anything goes wrong, you have several options:
stdint.h, you can define AE_HAVE_STDINT conditional symbolAE_INT32_T and/or AE_INT64_T and/or AE_INT_T symbols.
Just assign datatype name to them, and ALGLIB will automatically use your definition. You can define only one or two types (those which are not defined automatically).You can improve performance of ALGLIB in a several ways:
ALGLIB has two-layered structure: some set of basic performance-critical primitives is implemented using optimized code, 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 everywhere from Intel to SPARC. However, you can tell ALGLIB that it will work under particular architecture by defining appropriate macro at the global level:
AE_CPU=AE_INTEL - to tell ALGLIB that it will work under Intel
When AE_CPU macro is defined and equals to the AE_INTEL, it enables SSE2 support.
ALGLIB will use cpuid instruction to determine SSE2 presence at run-time and - in case we have SSE2 - to use SSE2-capable code.
ALGLIB uses SSE2 intrinsics which are portable across different compilers and efficient enough for most practical purposes.
As of ALGLIB 3.4, SSE2 support is available for MSVC, GCC and Sun Studio users only.
Commercial version of ALGLIB for C++ features four important improvements over open source one:
ALGLIB for C++ can utilize SSE2 set of instructions supported by all modern Intel and AMD x86 processors. This feature is optional and must be explicitly turned on during compile-time. If you do not activate it, ALGLIB will use generic C code, without any processor-specific assembly/intrinsics.
Thus, if you turn on this feature, your code will run faster on x86_32 and x86_64 processors, but will be unportable to non-x86 platforms (and Intel MIC platform, which is not exactly x86 and does not support SSE!). From the other side, if you do not activate this feature, your code will be portable to almost any modern CPU (SPARC, ARM, ...).
In order to turn on x86-specific optimizations,
you should define AE_CPU=AE_INTEL preprocessor definition at global level.
It will tell ALGLIB to use SSE intrinsics supported by GCC, MSVC and Intel compilers.
Additionally you should tell compiler to generate SSE-capable code.
It can be done in the project settings of your IDE or in the command line:
GCC example: > g++ -msse2 -I. -DAE_CPU=AE_INTEL *.cpp -lm MSVC example: > cl /I. /EHsc /DAE_CPU=AE_INTEL *.cpp
Commercial version of ALGLIB includes out-of-the-box support for multithreading. Many (not all) computationally intensive problems can be solved in multithreaded mode. You should read comments on specific ALGLIB functions to determine what can be multithreaded and what can not.
ALGLIB does not depend on vendor/compiler support for technologies like OpenMP/MPI/... Under Windows ALGLIB uses OS threads and custom synchronization framework. Under POSIX-compatible OS (Solaris, Linux, FreeBSD, NetBSD, OpenBSD, ...) ALGLIB uses POSIX Threads (standard *nix library which is shipped with any POSIX system) with its threading and synchronization primitives. It gives ALGLIB unprecedented portability across operating systems and compilers. ALGLIB does not depend on presence of any custom multithreading library or compiler support for any multithreading technology.
If you want to use multithreaded capabilities of ALGLIB, you should:
Let explain it in more details...
1.
You should compile ALGLIB in OS-specific mode by #defining either
AE_OS=AE_WINDOWS or AE_OS=AE_POSIX at compile time, depending on OS being used.
Former corresponds to any modern OS (32/64-bit Windows XP and later) from Windows family,
while latter means almost any POSIX-compatible OS.
When compiling on POSIX, do not forget to link ALGLIB with libpthread library.
2.
ALGLIB automatically determines number of cores on application startup.
On Windows it is done using GetSystemInfo() call.
On POSIX systems ALGLIB performs sysconf(_SC_NPROCESSORS_ONLN) system call.
This system call is supported by all modern POSIX-compatible systems: Solaris, Linux, FreeBSD, NetBSD, OpenBSD.
By default, ALGLIB uses all available cores except for one.
Say, on 4-core system it will use three cores - unless being told to use more or less.
It will keep your system responsive during lengthy computations.
Such behavior may be changed with setnworkers() call:
alglib::setnworkers(0) = use all coresalglib::setnworkers(-1) = leave one core unusedalglib::setnworkers(-2) = leave two cores unusedalglib::setnworkers(+2) = use 2 cores (even if you have more)
You may want to specify maximum number of worker threads during compile time
by means of preprocessor definition AE_NWORKERS=N.
You can add this definition to compiler command line or change corresponding project settings in your IDE.
Here N can be any positive number.
ALGLIB will use exactly N worker threads, unless being told to use less by setnworkers() call.
Some old POSIX-compatible operating systems do not support sysconf(_SC_NPROCESSORS_ONLN) system call
which is required in order to automatically determine number of active cores.
On these systems you should specify number of cores manually at compile time.
Without it ALGLIB will run in single-threaded mode.
3. When you use commercial edition of ALGLIB, you may choose between serial and multithreaded versions of SMP-capable functions:
smp_ prefix in its name) creates (or wakes up) worker threads,
inserts task in the worker queue, and waits for completion of the task.
All processing is done in context of worker thread(s).You should carefully decide what version of function to use. Starting/stopping worker thread costs tens of thousands of CPU cycles. Thus you won't get multithreading speedup on small computational problems.
Simultaneous multithreading (SMT) also known as Hyper-threading (Intel) and Cluster-based Multithreading (AMD) is a CPU design where several (usually two) logical cores share resources of one physical core. Say, on dual-core system with 2x HT scale factor you will see 4 logical cores. Each pair of these 4 cores, however, share same hardware resources. Thus, you may get only marginal speedup when running highly optimized software which fully utilizes CPU resources.
Say, if one thread occupies floating-point unit, another thread on the same physical core may work with integer numbers at the same time without any performance penalties. In this case you may get some speedup due to having additional cores. But if both threads keep FPU unit 100% busy, they won't get any multithreaded speedup.
So, if 2 math-intensive threads are dispatched by OS scheduler to different physical cores, you will get 2x speedup due to use of multithreading. But if these threads are dispatched to different logical cores - but same physical core - you won't get any speedup at all! One physical core will be 100% busy, and another one will be 100% idle. From the other side, if you start four threads instead of two, your system will be 100% utilized independently of thread scheduling details.
Let we stress it one more time - multithreading speedup on SMT systems is highly dependent on number of threads you are running and decisions made by OS scheduler. It is not 100% deterministic! With "true SMP" when you run 2 threads, you get 2x speedup (or 1.95, or 1.80 - it depends on algorithm, but this factor is always same). With SMT when you run 2 threads you may get your 2x speedup - or no speedup at all. Modern OS schedulers do a good job on single-socket hardware, but even in this "simple" case they give no guarantees of fair distribution of hardware resources. And things become a bit tricky when you work with multi-socket hardware. On SMT systems the only guaranteed way to 100% utilize your CPU is to create as many worker threads as there are logical cores. In this case OS scheduler has no chance to make its work in a wrong way.
Commercial edition of ALGLIB includes MKL extensions - special lightweight distribution of Intel MKL, highly optimized numerical library from Intel - and precompiled ALGLIB-MKL interface libraries. Linking your programs with MKL extensions allows you to run ALGLIB with maximum performance.
Current version of ALGLIB features Windows-only MKL extensions, but in future ALGLIB releases we will introduce MKL extensions for Linux systems.
Unlike the rest of the library, MKL extensions are distributed in binary-only form. ALGLIB itself is still distributed in source code form, but Intel MKL and ALGLIB-MKL interface are distributed as precompiled dynamic/static libraries. We can not distribute them in source because of license restrictions associated with Intel MKL. Also due to license restrictions we can not give you direct access to MKL functionality. You may use MKL to accelerate ALGLIB - without paying for MKL license - but you may not call its functions directly. It is technically possible, but strictly prohibited by both MKL's EULA and ALGLIB License Agreement. If you want to work with MKL, you should buy separate license from Intel.
MKL extensions are located in the /cpp/mkl-windows subdirectory of the ALGLIB distribution.
This directory includes:
In order to activate MKL extensions you should:
AE_OS=AE_WINDOWS (to activate multithreading capabilities)AE_CPU=AE_INTEL (to use SSE instructions provided by x86/x64 CPU's)AE_MKL (to use Intel MKL functions in ALGLIB)"/DAE_OS=AE_WINDOWS /DAE_CPU=AE_INTEL /DAE_MKL"
mkl4alglib_32/64.lib - and link it with your application.
mkl4alglib_32/64.dll into directory where it can be found during application startup
(usually - application dir)
Examples on linking from command line can be found in the next section.
If you bought separate license for Intel MKL, and want to use your own
installation of MKL - and not our lightweight distribution - then you
should compile ALGLIB as it was told in the previous section, with all necessary preprocessor definitions.
But instead of linking with mkl4alglib dynamic library,
you should add to your project mkl4alglib.c file from mkl-interface directory
and compile it (as C file) along with the rest of ALGLIB.
This C file implements interface between MKL and ALGLIB.
Having this file in your project and defining AE_MKL preprocessor definition
results in ALGLIB using MKL functions.
However, this C file is just interface! It is your responsibility to make sure that C/C++ compiler can find MKL headers, and appropriate MKL static/dynamic libraries are linked to your application.
If you link ALGLIB with your own installation if Intel MKL, you may do so on any OS where MKL works - Windows or Linux.
In this section we'll consider different compilation scenarios for commercial version of ALGLIB - from simple platform-agnostic compilation to 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 (cpp folder with ALGLIB distribution).
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
#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 number of worker threads: "4" means "use 4 cores".
// This line is ignored if AE_OS is UNDEFINED.
alglib::setnworkers(4);
// Perform matrix-matrix product.
// We call function with "smp_" prefix, which means that ALGLIB
// will try to execute it in parallel manner whenever it is possible.
flops = 2*pow((double)n, (double)3);
timeneeded = counter();
alglib::smp_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 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. 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 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
> copy cpp\src\*.* .
> 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.
OS-agnostic mode, ALGLIB knows it is x86/x64
> copy cpp\src\*.* .
> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL *.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.
ALGLIB knows it is Windows on x86/x64 CPU
> copy cpp\src\*.* .
> cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /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) mkl4alglib static library,
which is import library for special lightweight MKL distribution, shipped with ALGLIB for no additional price.
We also should copy to current directory appropriate mkl4alglib DLL file which contains Intel MKL.
Linking with MKL extensions
> copy cpp\src\*.* . > copy cpp\mkl-windows\mkl4alglib_64.lib . > copy cpp\mkl-windows\mkl4alglib_64.dll . > cl /I. /EHsc /Fedemo.exe /Ox /DAE_CPU=AE_INTEL /DAE_OS=AE_WINDOWS /DAE_MKL demo.cpp mkl4alglib_64.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!
Both open source and commercial versions of ALGLIB are 100% thread-safe as long as different user threads work with different instances of objects/arrays. Thread-safety is guaranteed by having no global shared variables.
However, any kind of sharing ALGLIB objects/arrays between different threads is potentially hazardous. Even when this object is seemingly used in read-only mode!
Say, you use ALGLIB neural network NET to process two input vectors X0 and X1,
and get two output vectors Y0 and Y1.
You may decide that neural network is used in read-only mode which does not change state of NET,
because output is written to distinct arrays Y.
Thus, you may want to process these vectors from parallel threads.
But it is not read-only operation, even if it looks like this!
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!
ALGLIB defines several conditional symbols (all start with "AE_" which means "ALGLIB environment") and two namespaces:
alglib_impl (contains computational core) and alglib (contains C++ interface).
Although this manual mentions both alglib_impl and alglib namespaces, only alglib namespace should be used by you.
It contains user-friendly C++ interface with automatic memory management, exception handling and all other nice features.
alglib_impl is less user-friendly, is less documented, and it is too easy to crash your system or cause memory leak if you use it directly.
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 libraryalglib::complex - double precision complex datatype, safer replacement for std::complexalglib::ap_error - exception which is thrown by libraryboolean_1d_array - 1-dimensional boolean arrayinteger_1d_array - 1-dimensional integer arrayreal_1d_array - 1-dimensional real (double precision) arraycomplex_1d_array - 1-dimensional complex arrayboolean_2d_array - 2-dimensional boolean arrayinteger_2d_array - 2-dimensional integer arrayreal_2d_array - 2-dimensional real (double precision) arraycomplex_2d_array - 2-dimensional complex arrayPackage-specific datatypes are classes which can be divided into two distinct groups:
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 largeralglib::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 largeralglib::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 infinityalglib::fp_neginf - negative infinity
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_MAXalglib::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 NANalglib::fp_isposinf - checks whether number is +INFalglib::fp_isneginf - checks whether number is -INFalglib::fp_isinf - checks whether number is +INF or -INFalglib::fp_isfinite - checks whether number is finite value (possibly subnormalized)
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);
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());
Most ALGLIB functions provide two interfaces: 'expert' and 'friendly'. What is the difference between two? When you use 'friendly' interface, ALGLIB:
When you use 'expert' interface, ALGLIB:
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
'Friendly' interface - matrix semantics:
#include "linalg.h" ... alglib::real_2d_array a; alglib::matinvreport rep; alglib::ae_int_t info; // // 'Friendly' interface: spdmatrixinverse() accepts and returns symmetric matrix // // symmetric positive definite matrix a = "[[2,1],[1,2]]"; // after this line A will contain [[0.66,-0.33],[-0.33,0.66]] // which is symmetric too alglib::spdmatrixinverse(a, info, rep); // you may try to pass nonsymmetric matrix a = "[[2,1],[0,2]]"; // but exception will be thrown in such case alglib::spdmatrixinverse(a, info, rep);
Same function but with 'expert' interface:
#include "linalg.h" ... alglib::real_2d_array a; alglib::matinvreport rep; alglib::ae_int_t info; // // 'Expert' interface, spdmatrixinverse() // // only upper triangle is used; a[1][0] is initialized by NAN, // but it can be arbitrary number a = "[[2,1],[NAN,2]]"; // after this line A will contain [[0.66,-0.33],[NAN,0.66]] // only upper triangle is modified alglib::spdmatrixinverse(a, 2 /* N */, true /* upper triangle is used */, info, rep);
ALGLIB uses two error handling strategies:
What is actually done depends on function being used and error being reported:
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.
ALGLIB (ap.h header) includes following Level 1 BLAS functions:
alglib::vdotproduct() family, which allows to calculate dot product of two real or complex vectorsalglib::vmove() family, which allows to copy real/complex vector to another location with optimal multiplication by real/complex valuealglib::vmoveneg() family, which allows to copy real/complex vector to another location with multiplication by -1alglib::vadd() and alglib::vsub() families, which allows to add or subtract two real/complex vectors with optimal multiplication by real/complex valuealglib::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);
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:
See comments on alglib::read_csv() function for more information about its functionality.
There are two test suites in ALGLIB: computational tests and interface 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).
Running test suite is easy - just
test_c.cpp or test_i.cpp) along with the rest of the library
If you want to be sure that ALGLIB will work with some sophisticated optimization settings, set corresponding flags during compile time.
If your compiler/system are not in the list of supported ones, we recommend you to run both test suites. But if you are running out of time, run at least test_i.cpp.
8.1
| ||
| hqrnd | High quality random numbers generator | |
| nearestneighbor | Nearest neighbor search: approximate and exact | |
| xdebug | Debug functions to test ALGLIB interface generator | |
8.2
| ||
| 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 | |
| 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 | |
8.3
| ||
| odesolver | Ordinary differential equation solver | |
8.4
| ||
| conv | Fast real/complex convolution | |
| corr | Fast real/complex cross-correlation | |
| fft | Real/complex FFT | |
| fht | Real Fast Hartley Transform | |
8.5
| ||
| autogk | Adaptive 1-dimensional integration | |
| gkq | Gauss-Kronrod quadrature generator | |
| gq | Gaussian quadrature generator | |
8.6
| ||
| idwint | Inverse distance weighting: interpolation/fitting | |
| lsfit | Fitting with least squates target function (linear and nonlinear least-squares) | |
| nsfit | Fitting with non-least-squares target functions (ones involving min/max operations, etc) | |
| 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 | |
| spline3d | 3D spline interpolation | |
8.7
| ||
| 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 estimate | |
| schur | Schur decomposition | |
| sparse | Sparse matrices | |
| spdgevd | Generalized symmetric eigensolver | |
| svd | Singular value decomposition | |
| trfac | LU and Cholesky decompositions (dense and sparse) | |
8.8
| ||
| 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 | |
| minlbfgs | Limited memory BFGS optimizer | |
| minlm | Improved Levenberg-Marquardt optimizer | |
| minnlc | Nonlinearly constrained optimizer | |
| minns | Nonsmooth constrained optimizer | |
| minqp | Quadratic programming with bound and linear equality/inequality constraints | |
8.9
| ||
| densesolver | Dense linear system solver | |
| lincg | Sparse linear CG solver | |
| linlsqr | Sparse linear LSQR solver | |
| nleq | Solvers for nonlinear equations | |
| polynomialsolver | Polynomial solver | |
8.10
| ||
| 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 | Normal distribution | |
| poissondistr | Poisson distribution | |
| psif | Psi function | |
| studenttdistr | Student's t-distribution | |
| trigintegrals | Trigonometric integrals | |
8.11
| ||
| basestat | Mean, variance, covariance, correlation, etc. | |
| correlationtests | Hypothesis testing: correlation tests | |
| jarquebera | Hypothesis testing: Jarque-Bera test | |
| mannwhitneyu | Hypothesis testing: Mann-Whitney-U test | |
| stest | Hypothesis testing: sign test | |
| studentttests | Hypothesis testing: Student's t-test | |
| variancetests | Hypothesis testing: F-test and one-sample variance test | |
| wsr | Hypothesis testing: Wilcoxon signed rank test | |
ablas subpackage| ablas_d_gemm | Matrix multiplication (single-threaded) | |
| ablas_d_syrk | Symmetric rank-K update (single-threaded) | |
| ablas_smp_gemm | Matrix multiplication (multithreaded) | |
| ablas_smp_syrk | Symmetric rank-K update (multithreaded) |
cmatrixcopy function/************************************************************************* Copy Input parameters: M - number of rows N - number of columns A - source matrix, MxN submatrix is copied and transposed IA - submatrix offset (row index) JA - submatrix offset (column index) B - destination matrix, must be large enough to store result IB - submatrix offset (row index) JB - submatrix offset (column index) *************************************************************************/void alglib::cmatrixcopy( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t ia, ae_int_t ja, complex_2d_array& b, ae_int_t ib, ae_int_t jb);
cmatrixgemm function/************************************************************************* This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where: * C is MxN general matrix * op1(A) is MxK matrix * op2(B) is KxN matrix * "op" may be identity transformation, transposition, conjugate transposition Additional info: * cache-oblivious algorithm is used. * multiplication result replaces C. If Beta=0, C elements are not used in calculations (not multiplied by zero - just not referenced) * if Alpha=0, A is not used (not multiplied by zero - just not referenced) * if both Beta and Alpha are zero, C is filled by zeros. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. IMPORTANT: This function does NOT preallocate output matrix C, it MUST be preallocated by caller prior to calling this function. In case C does not have enough space to store result, exception will be generated. INPUT PARAMETERS M - matrix size, M>0 N - matrix size, N>0 K - matrix size, K>0 Alpha - coefficient A - matrix IA - submatrix offset JA - submatrix offset OpTypeA - transformation type: * 0 - no transformation * 1 - transposition * 2 - conjugate transposition B - matrix IB - submatrix offset JB - submatrix offset OpTypeB - transformation type: * 0 - no transformation * 1 - transposition * 2 - conjugate transposition Beta - coefficient C - matrix (PREALLOCATED, large enough to store result) IC - submatrix offset JC - submatrix offset -- ALGLIB routine -- 16.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixgemm( ae_int_t m, ae_int_t n, ae_int_t k, alglib::complex alpha, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, complex_2d_array b, ae_int_t ib, ae_int_t jb, ae_int_t optypeb, alglib::complex beta, complex_2d_array& c, ae_int_t ic, ae_int_t jc); void alglib::smp_cmatrixgemm( ae_int_t m, ae_int_t n, ae_int_t k, alglib::smp_complex alpha, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, complex_2d_array b, ae_int_t ib, ae_int_t jb, ae_int_t optypeb, alglib::smp_complex beta, complex_2d_array& c, ae_int_t ic, ae_int_t jc);
cmatrixherk function/************************************************************************* This subroutine calculates C=alpha*A*A^H+beta*C or C=alpha*A^H*A+beta*C where: * C is NxN Hermitian matrix given by its upper/lower triangle * A is NxK matrix when A*A^H is calculated, KxN matrix otherwise Additional info: * cache-oblivious algorithm is used. * multiplication result replaces C. If Beta=0, C elements are not used in calculations (not multiplied by zero - just not referenced) * if Alpha=0, A is not used (not multiplied by zero - just not referenced) * if both Beta and Alpha are zero, C is filled by zeros. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS N - matrix size, N>=0 K - matrix size, K>=0 Alpha - coefficient A - matrix IA - submatrix offset (row index) JA - submatrix offset (column index) OpTypeA - multiplication type: * 0 - A*A^H is calculated * 2 - A^H*A is calculated Beta - coefficient C - preallocated input/output matrix IC - submatrix offset (row index) JC - submatrix offset (column index) IsUpper - whether upper or lower triangle of C is updated; this function updates only one half of C, leaving other half unchanged (not referenced at all). -- ALGLIB routine -- 16.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixherk( ae_int_t n, ae_int_t k, double alpha, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, double beta, complex_2d_array& c, ae_int_t ic, ae_int_t jc, bool isupper); void alglib::smp_cmatrixherk( ae_int_t n, ae_int_t k, double alpha, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, double beta, complex_2d_array& c, ae_int_t ic, ae_int_t jc, bool isupper);
cmatrixlefttrsm function/************************************************************************* This subroutine calculates op(A^-1)*X where: * X is MxN general matrix * A is MxM upper/lower triangular/unitriangular matrix * "op" may be identity transformation, transposition, conjugate transposition Multiplication result replaces X. Cache-oblivious algorithm is used. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS N - matrix size, N>=0 M - matrix size, N>=0 A - matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1] I1 - submatrix offset J1 - submatrix offset IsUpper - whether matrix is upper triangular IsUnit - whether matrix is unitriangular OpType - transformation type: * 0 - no transformation * 1 - transposition * 2 - conjugate transposition X - matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1] I2 - submatrix offset J2 - submatrix offset -- ALGLIB routine -- 15.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixlefttrsm( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, complex_2d_array& x, ae_int_t i2, ae_int_t j2); void alglib::smp_cmatrixlefttrsm( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, complex_2d_array& x, ae_int_t i2, ae_int_t j2);
cmatrixmv function/************************************************************************* Matrix-vector product: y := op(A)*x INPUT PARAMETERS: M - number of rows of op(A) M>=0 N - number of columns of op(A) N>=0 A - target matrix IA - submatrix offset (row index) JA - submatrix offset (column index) OpA - operation type: * OpA=0 => op(A) = A * OpA=1 => op(A) = A^T * OpA=2 => op(A) = A^H X - input vector IX - subvector offset IY - subvector offset Y - preallocated matrix, must be large enough to store result OUTPUT PARAMETERS: Y - vector which stores result if M=0, then subroutine does nothing. if N=0, Y is filled by zeros. -- ALGLIB routine -- 28.01.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixmv( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t opa, complex_1d_array x, ae_int_t ix, complex_1d_array& y, ae_int_t iy);
cmatrixrank1 function/************************************************************************* Rank-1 correction: A := A + u*v' INPUT PARAMETERS: M - number of rows N - number of columns A - target matrix, MxN submatrix is updated IA - submatrix offset (row index) JA - submatrix offset (column index) U - vector #1 IU - subvector offset V - vector #2 IV - subvector offset *************************************************************************/void alglib::cmatrixrank1( ae_int_t m, ae_int_t n, complex_2d_array& a, ae_int_t ia, ae_int_t ja, complex_1d_array& u, ae_int_t iu, complex_1d_array& v, ae_int_t iv);
cmatrixrighttrsm function/************************************************************************* This subroutine calculates X*op(A^-1) where: * X is MxN general matrix * A is NxN upper/lower triangular/unitriangular matrix * "op" may be identity transformation, transposition, conjugate transposition Multiplication result replaces X. Cache-oblivious algorithm is used. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS N - matrix size, N>=0 M - matrix size, N>=0 A - matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1] I1 - submatrix offset J1 - submatrix offset IsUpper - whether matrix is upper triangular IsUnit - whether matrix is unitriangular OpType - transformation type: * 0 - no transformation * 1 - transposition * 2 - conjugate transposition X - matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1] I2 - submatrix offset J2 - submatrix offset -- ALGLIB routine -- 15.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixrighttrsm( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, complex_2d_array& x, ae_int_t i2, ae_int_t j2); void alglib::smp_cmatrixrighttrsm( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, complex_2d_array& x, ae_int_t i2, ae_int_t j2);
cmatrixsyrk function/************************************************************************* This subroutine is an older version of CMatrixHERK(), one with wrong name (it is HErmitian update, not SYmmetric). It is left here for backward compatibility. -- ALGLIB routine -- 16.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixsyrk( ae_int_t n, ae_int_t k, double alpha, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, double beta, complex_2d_array& c, ae_int_t ic, ae_int_t jc, bool isupper); void alglib::smp_cmatrixsyrk( ae_int_t n, ae_int_t k, double alpha, complex_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, double beta, complex_2d_array& c, ae_int_t ic, ae_int_t jc, bool isupper);
cmatrixtranspose function/************************************************************************* Cache-oblivous complex "copy-and-transpose" Input parameters: M - number of rows N - number of columns A - source matrix, MxN submatrix is copied and transposed IA - submatrix offset (row index) JA - submatrix offset (column index) B - destination matrix, must be large enough to store result IB - submatrix offset (row index) JB - submatrix offset (column index) *************************************************************************/void alglib::cmatrixtranspose( ae_int_t m, ae_int_t n, complex_2d_array a, ae_int_t ia, ae_int_t ja, complex_2d_array& b, ae_int_t ib, ae_int_t jb);
rmatrixcopy function/************************************************************************* Copy Input parameters: M - number of rows N - number of columns A - source matrix, MxN submatrix is copied and transposed IA - submatrix offset (row index) JA - submatrix offset (column index) B - destination matrix, must be large enough to store result IB - submatrix offset (row index) JB - submatrix offset (column index) *************************************************************************/void alglib::rmatrixcopy( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t ia, ae_int_t ja, real_2d_array& b, ae_int_t ib, ae_int_t jb);
rmatrixenforcesymmetricity function/************************************************************************* This code enforces symmetricy of the matrix by copying Upper part to lower one (or vice versa). INPUT PARAMETERS: A - matrix N - number of rows/columns IsUpper - whether we want to copy upper triangle to lower one (True) or vice versa (False). *************************************************************************/void alglib::rmatrixenforcesymmetricity( real_2d_array& a, ae_int_t n, bool isupper);
rmatrixgemm function/************************************************************************* This subroutine calculates C = alpha*op1(A)*op2(B) +beta*C where: * C is MxN general matrix * op1(A) is MxK matrix * op2(B) is KxN matrix * "op" may be identity transformation, transposition Additional info: * cache-oblivious algorithm is used. * multiplication result replaces C. If Beta=0, C elements are not used in calculations (not multiplied by zero - just not referenced) * if Alpha=0, A is not used (not multiplied by zero - just not referenced) * if both Beta and Alpha are zero, C is filled by zeros. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. IMPORTANT: This function does NOT preallocate output matrix C, it MUST be preallocated by caller prior to calling this function. In case C does not have enough space to store result, exception will be generated. INPUT PARAMETERS M - matrix size, M>0 N - matrix size, N>0 K - matrix size, K>0 Alpha - coefficient A - matrix IA - submatrix offset JA - submatrix offset OpTypeA - transformation type: * 0 - no transformation * 1 - transposition B - matrix IB - submatrix offset JB - submatrix offset OpTypeB - transformation type: * 0 - no transformation * 1 - transposition Beta - coefficient C - PREALLOCATED output matrix, large enough to store result IC - submatrix offset JC - submatrix offset -- ALGLIB routine -- 2009-2013 Bochkanov Sergey *************************************************************************/void alglib::rmatrixgemm( ae_int_t m, ae_int_t n, ae_int_t k, double alpha, real_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, real_2d_array b, ae_int_t ib, ae_int_t jb, ae_int_t optypeb, double beta, real_2d_array& c, ae_int_t ic, ae_int_t jc); void alglib::smp_rmatrixgemm( ae_int_t m, ae_int_t n, ae_int_t k, double alpha, real_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, real_2d_array b, ae_int_t ib, ae_int_t jb, ae_int_t optypeb, double beta, real_2d_array& c, ae_int_t ic, ae_int_t jc);
rmatrixlefttrsm function/************************************************************************* This subroutine calculates op(A^-1)*X where: * X is MxN general matrix * A is MxM upper/lower triangular/unitriangular matrix * "op" may be identity transformation, transposition Multiplication result replaces X. Cache-oblivious algorithm is used. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS N - matrix size, N>=0 M - matrix size, N>=0 A - matrix, actial matrix is stored in A[I1:I1+M-1,J1:J1+M-1] I1 - submatrix offset J1 - submatrix offset IsUpper - whether matrix is upper triangular IsUnit - whether matrix is unitriangular OpType - transformation type: * 0 - no transformation * 1 - transposition X - matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1] I2 - submatrix offset J2 - submatrix offset -- ALGLIB routine -- 15.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixlefttrsm( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, real_2d_array& x, ae_int_t i2, ae_int_t j2); void alglib::smp_rmatrixlefttrsm( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, real_2d_array& x, ae_int_t i2, ae_int_t j2);
rmatrixmv function/************************************************************************* Matrix-vector product: y := op(A)*x INPUT PARAMETERS: M - number of rows of op(A) N - number of columns of op(A) A - target matrix IA - submatrix offset (row index) JA - submatrix offset (column index) OpA - operation type: * OpA=0 => op(A) = A * OpA=1 => op(A) = A^T X - input vector IX - subvector offset IY - subvector offset Y - preallocated matrix, must be large enough to store result OUTPUT PARAMETERS: Y - vector which stores result if M=0, then subroutine does nothing. if N=0, Y is filled by zeros. -- ALGLIB routine -- 28.01.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixmv( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t opa, real_1d_array x, ae_int_t ix, real_1d_array& y, ae_int_t iy);
rmatrixrank1 function/************************************************************************* Rank-1 correction: A := A + u*v' INPUT PARAMETERS: M - number of rows N - number of columns A - target matrix, MxN submatrix is updated IA - submatrix offset (row index) JA - submatrix offset (column index) U - vector #1 IU - subvector offset V - vector #2 IV - subvector offset *************************************************************************/void alglib::rmatrixrank1( ae_int_t m, ae_int_t n, real_2d_array& a, ae_int_t ia, ae_int_t ja, real_1d_array& u, ae_int_t iu, real_1d_array& v, ae_int_t iv);
rmatrixrighttrsm function/************************************************************************* This subroutine calculates X*op(A^-1) where: * X is MxN general matrix * A is NxN upper/lower triangular/unitriangular matrix * "op" may be identity transformation, transposition Multiplication result replaces X. Cache-oblivious algorithm is used. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS N - matrix size, N>=0 M - matrix size, N>=0 A - matrix, actial matrix is stored in A[I1:I1+N-1,J1:J1+N-1] I1 - submatrix offset J1 - submatrix offset IsUpper - whether matrix is upper triangular IsUnit - whether matrix is unitriangular OpType - transformation type: * 0 - no transformation * 1 - transposition X - matrix, actial matrix is stored in X[I2:I2+M-1,J2:J2+N-1] I2 - submatrix offset J2 - submatrix offset -- ALGLIB routine -- 15.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixrighttrsm( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, real_2d_array& x, ae_int_t i2, ae_int_t j2); void alglib::smp_rmatrixrighttrsm( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t i1, ae_int_t j1, bool isupper, bool isunit, ae_int_t optype, real_2d_array& x, ae_int_t i2, ae_int_t j2);
rmatrixsyrk function/************************************************************************* This subroutine calculates C=alpha*A*A^T+beta*C or C=alpha*A^T*A+beta*C where: * C is NxN symmetric matrix given by its upper/lower triangle * A is NxK matrix when A*A^T is calculated, KxN matrix otherwise Additional info: * cache-oblivious algorithm is used. * multiplication result replaces C. If Beta=0, C elements are not used in calculations (not multiplied by zero - just not referenced) * if Alpha=0, A is not used (not multiplied by zero - just not referenced) * if both Beta and Alpha are zero, C is filled by zeros. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Because starting/stopping worker thread always ! involves some overhead, parallelism starts to be profitable for N's ! larger than 128. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS N - matrix size, N>=0 K - matrix size, K>=0 Alpha - coefficient A - matrix IA - submatrix offset (row index) JA - submatrix offset (column index) OpTypeA - multiplication type: * 0 - A*A^T is calculated * 2 - A^T*A is calculated Beta - coefficient C - preallocated input/output matrix IC - submatrix offset (row index) JC - submatrix offset (column index) IsUpper - whether C is upper triangular or lower triangular -- ALGLIB routine -- 16.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixsyrk( ae_int_t n, ae_int_t k, double alpha, real_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, double beta, real_2d_array& c, ae_int_t ic, ae_int_t jc, bool isupper); void alglib::smp_rmatrixsyrk( ae_int_t n, ae_int_t k, double alpha, real_2d_array a, ae_int_t ia, ae_int_t ja, ae_int_t optypea, double beta, real_2d_array& c, ae_int_t ic, ae_int_t jc, bool isupper);
rmatrixtranspose function/************************************************************************* Cache-oblivous real "copy-and-transpose" Input parameters: M - number of rows N - number of columns A - source matrix, MxN submatrix is copied and transposed IA - submatrix offset (row index) JA - submatrix offset (column index) B - destination matrix, must be large enough to store result IB - submatrix offset (row index) JB - submatrix offset (column index) *************************************************************************/void alglib::rmatrixtranspose( ae_int_t m, ae_int_t n, real_2d_array a, ae_int_t ia, ae_int_t ja, real_2d_array& b, ae_int_t ib, ae_int_t jb);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array a = "[[2,1],[1,3]]";
real_2d_array b = "[[2,1],[0,1]]";
real_2d_array c = "[[0,0],[0,0]]";
//
// rmatrixgemm() function allows us to calculate matrix product C:=A*B or
// to perform more general operation, C:=alpha*op1(A)*op2(B)+beta*C,
// where A, B, C are rectangular matrices, op(X) can be X or X^T,
// alpha and beta are scalars.
//
// This function:
// * can apply transposition and/or multiplication by scalar to operands
// * can use arbitrary part of matrices A/B (given by submatrix offset)
// * can store result into arbitrary part of C
// * for performance reasons requires C to be preallocated
//
// Parameters of this function are:
// * M, N, K - sizes of op1(A) (which is MxK), op2(B) (which
// is KxN) and C (which is MxN)
// * Alpha - coefficient before A*B
// * A, IA, JA - matrix A and offset of the submatrix
// * OpTypeA - transformation type:
// 0 - no transformation
// 1 - transposition
// * B, IB, JB - matrix B and offset of the submatrix
// * OpTypeB - transformation type:
// 0 - no transformation
// 1 - transposition
// * Beta - coefficient before C
// * C, IC, JC - preallocated matrix C and offset of the submatrix
//
// Below we perform simple product C:=A*B (alpha=1, beta=0)
//
// IMPORTANT: this function works with preallocated C, which must be large
// enough to store multiplication result.
//
ae_int_t m = 2;
ae_int_t n = 2;
ae_int_t k = 2;
double alpha = 1.0;
ae_int_t ia = 0;
ae_int_t ja = 0;
ae_int_t optypea = 0;
ae_int_t ib = 0;
ae_int_t jb = 0;
ae_int_t optypeb = 0;
double beta = 0.0;
ae_int_t ic = 0;
ae_int_t jc = 0;
rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[4,3],[2,4]]
//
// Now we try to apply some simple transformation to operands: C:=A*B^T
//
optypeb = 1;
rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[5,1],[5,3]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// rmatrixsyrk() function allows us to calculate symmetric rank-K update
// C := beta*C + alpha*A'*A, where C is square N*N matrix, A is square K*N
// matrix, alpha and beta are scalars. It is also possible to update by
// adding A*A' instead of A'*A.
//
// Parameters of this function are:
// * N, K - matrix size
// * Alpha - coefficient before A
// * A, IA, JA - matrix and submatrix offsets
// * OpTypeA - multiplication type:
// * 0 - A*A^T is calculated
// * 2 - A^T*A is calculated
// * Beta - coefficient before C
// * C, IC, JC - preallocated input/output matrix and submatrix offsets
// * IsUpper - whether upper or lower triangle of C is updated;
// this function updates only one half of C, leaving
// other half unchanged (not referenced at all).
//
// Below we will show how to calculate simple product C:=A'*A
//
// NOTE: beta=0 and we do not use previous value of C, but still it
// MUST be preallocated.
//
ae_int_t n = 2;
ae_int_t k = 1;
double alpha = 1.0;
ae_int_t ia = 0;
ae_int_t ja = 0;
ae_int_t optypea = 2;
double beta = 0.0;
ae_int_t ic = 0;
ae_int_t jc = 0;
bool isupper = true;
real_2d_array a = "[[1,2]]";
// preallocate space to store result
real_2d_array c = "[[0,0],[0,0]]";
// calculate product, store result into upper part of c
rmatrixsyrk(n, k, alpha, a, ia, ja, optypea, beta, c, ic, jc, isupper);
// output result.
// IMPORTANT: lower triangle of C was NOT updated!
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[1,2],[0,4]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// In this example we assume that you already know how to work with
// rmatrixgemm() function. Below we concentrate on its multithreading
// capabilities.
//
// SMP edition of ALGLIB includes smp_rmatrixgemm() - multithreaded
// version of rmatrixgemm() function. In the basic edition of ALGLIB
// (GPL edition or commercial version without SMP support) this function
// just calls single-threaded stub. So, you may call this function from
// ANY edition of ALGLIB, but only in SMP edition it will work in really
// multithreaded mode.
//
// In order to use multithreading, you have to:
// 1) Install SMP edition of ALGLIB.
// 2) This step is specific for C++ users: you should activate OS-specific
// capabilities of ALGLIB by defining AE_OS=AE_POSIX (for *nix systems)
// or AE_OS=AE_WINDOWS (for Windows systems).
// C# users do not have to perform this step because C# programs are
// portable across different systems without OS-specific tuning.
// 3) Allow ALGLIB to know about number of worker threads to use:
// a) autodetection (C++, C#):
// ALGLIB will automatically determine number of CPU cores and
// (by default) will use all cores except for one. Say, on 4-core
// system it will use three cores - unless you manually told it
// to use more or less. It will keep your system responsive during
// lengthy computations.
// 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)
// b) manual specification (C++, C#):
// You may want to specify maximum number of worker threads during
// compile time by means of preprocessor definition AE_NWORKERS.
// For C++ it will be "AE_NWORKERS=X" where X can be any positive number.
// For C# it is "AE_NWORKERSX", where X should be replaced by number of
// workers (AE_NWORKERS2, AE_NWORKERS3, AE_NWORKERS4, ...).
// You can add this definition to compiler command line or change
// corresponding project settings in your IDE.
//
// After you installed and configured SMP edition of ALGLIB, you may choose
// between serial and multithreaded versions of SMP-capable functions:
// * serial version works as usual, in the context of the calling thread
// * multithreaded version (with "smp_" prefix) creates (or wakes up) worker
// threads, inserts task in the worker queue, and waits for completion of
// the task. All processing is done in context of worker thread(s).
//
// NOTE: because starting/stopping worker threads costs thousands of CPU cycles,
// you should not use multithreading for lightweight computational problems.
//
// NOTE: some old POSIX-compatible operating systems do not support
// sysconf(_SC_NPROCESSORS_ONLN) system call which is required in order
// to automatically determine number of active cores. On these systems
// you should specify number of cores manually at compile time.
// Without it ALGLIB will run in single-threaded mode.
//
// Now, back to our example. In this example we will show you:
// * how to call SMP version of rmatrixgemm(). Because we work with tiny 2x2
// matrices, we won't expect to see ANY speedup from using multithreading.
// The only purpose of this demo is to show how to call SMP functions.
// * how to modify number of worker threads used by ALGLIB
//
real_2d_array a = "[[2,1],[1,3]]";
real_2d_array b = "[[2,1],[0,1]]";
real_2d_array c = "[[0,0],[0,0]]";
ae_int_t m = 2;
ae_int_t n = 2;
ae_int_t k = 2;
double alpha = 1.0;
ae_int_t ia = 0;
ae_int_t ja = 0;
ae_int_t optypea = 0;
ae_int_t ib = 0;
ae_int_t jb = 0;
ae_int_t optypeb = 0;
double beta = 0.0;
ae_int_t ic = 0;
ae_int_t jc = 0;
// serial code
c = "[[0,0],[0,0]]";
rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
// SMP code with default number of worker threads
c = "[[0,0],[0,0]]";
smp_rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[4,3],[2,4]]
// override number of worker threads - use two cores
alglib::setnworkers(+2);
c = "[[0,0],[0,0]]";
smp_rmatrixgemm(m, n, k, alpha, a, ia, ja, optypea, b, ib, jb, optypeb, beta, c, ic, jc);
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[4,3],[2,4]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// In this example we assume that you already know how to work with
// rmatrixsyrk() function. Below we concentrate on its multithreading
// capabilities.
//
// SMP edition of ALGLIB includes smp_rmatrixsyrk() - multithreaded
// version of rmatrixsyrk() function. In the basic edition of ALGLIB
// (GPL edition or commercial version without SMP support) this function
// just calls single-threaded stub. So, you may call this function from
// ANY edition of ALGLIB, but only in SMP edition it will work in really
// multithreaded mode.
//
// In order to use multithreading, you have to:
// 1) Install SMP edition of ALGLIB.
// 2) This step is specific for C++ users: you should activate OS-specific
// capabilities of ALGLIB by defining AE_OS=AE_POSIX (for *nix systems)
// or AE_OS=AE_WINDOWS (for Windows systems).
// C# users do not have to perform this step because C# programs are
// portable across different systems without OS-specific tuning.
// 3) Allow ALGLIB to know about number of worker threads to use:
// a) autodetection (C++, C#):
// ALGLIB will automatically determine number of CPU cores and
// (by default) will use all cores except for one. Say, on 4-core
// system it will use three cores - unless you manually told it
// to use more or less. It will keep your system responsive during
// lengthy computations.
// 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)
// b) manual specification (C++, C#):
// You may want to specify maximum number of worker threads during
// compile time by means of preprocessor definition AE_NWORKERS.
// For C++ it will be "AE_NWORKERS=X" where X can be any positive number.
// For C# it is "AE_NWORKERSX", where X should be replaced by number of
// workers (AE_NWORKERS2, AE_NWORKERS3, AE_NWORKERS4, ...).
// You can add this definition to compiler command line or change
// corresponding project settings in your IDE.
//
// After you installed and configured SMP edition of ALGLIB, you may choose
// between serial and multithreaded versions of SMP-capable functions:
// * serial version works as usual, in the context of the calling thread
// * multithreaded version (with "smp_" prefix) creates (or wakes up) worker
// threads, inserts task in the worker queue, and waits for completion of
// the task. All processing is done in context of worker thread(s).
//
// NOTE: because starting/stopping worker threads costs thousands of CPU cycles,
// you should not use multithreading for lightweight computational problems.
//
// NOTE: some old POSIX-compatible operating systems do not support
// sysconf(_SC_NPROCESSORS_ONLN) system call which is required in order
// to automatically determine number of active cores. On these systems
// you should specify number of cores manually at compile time.
// Without it ALGLIB will run in single-threaded mode.
//
// Now, back to our example. In this example we will show you:
// * how to call SMP version of rmatrixsyrk(). Because we work with tiny 2x2
// matrices, we won't expect to see ANY speedup from using multithreading.
// The only purpose of this demo is to show how to call SMP functions.
// * how to modify number of worker threads used by ALGLIB
//
ae_int_t n = 2;
ae_int_t k = 1;
double alpha = 1.0;
ae_int_t ia = 0;
ae_int_t ja = 0;
ae_int_t optypea = 2;
double beta = 0.0;
ae_int_t ic = 0;
ae_int_t jc = 0;
bool isupper = true;
real_2d_array a = "[[1,2]]";
real_2d_array c = "[[]]";
//
// Default number of worker threads.
// Preallocate space to store result, call multithreaded version, test.
//
// NOTE: this function updates only one triangular part of C. In our
// example we choose to update upper triangle.
//
c = "[[0,0],[0,0]]";
smp_rmatrixsyrk(n, k, alpha, a, ia, ja, optypea, beta, c, ic, jc, isupper);
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[1,2],[0,4]]
//
// Override default number of worker threads (set to 2).
// Preallocate space to store result, call multithreaded version, test.
//
// NOTE: this function updates only one triangular part of C. In our
// example we choose to update upper triangle.
//
alglib::setnworkers(+2);
c = "[[0,0],[0,0]]";
smp_rmatrixsyrk(n, k, alpha, a, ia, ja, optypea, beta, c, ic, jc, isupper);
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [[1,2],[0,4]]
return 0;
}
airyf subpackageairy function/************************************************************************* Airy function Solution of the differential equation y"(x) = xy. The function returns the two independent solutions Ai, Bi and their first derivatives Ai'(x), Bi'(x). Evaluation is by power series summation for small x, by rational minimax approximations for large x. ACCURACY: Error criterion is absolute when function <= 1, relative when function > 1, except * denotes relative error criterion. For large negative x, the absolute error increases as x^1.5. For large positive x, the relative error increases as x^1.5. Arithmetic domain function # trials peak rms IEEE -10, 0 Ai 10000 1.6e-15 2.7e-16 IEEE 0, 10 Ai 10000 2.3e-14* 1.8e-15* IEEE -10, 0 Ai' 10000 4.6e-15 7.6e-16 IEEE 0, 10 Ai' 10000 1.8e-14* 1.5e-15* IEEE -10, 10 Bi 30000 4.2e-15 5.3e-16 IEEE -10, 10 Bi' 30000 4.9e-15 7.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/void alglib::airy( double x, double& ai, double& aip, double& bi, double& bip);
autogk subpackage| autogk_d1 | Integrating f=exp(x) by adaptive integrator |
autogkreport class/************************************************************************* Integration report: * TerminationType = completetion code: * -5 non-convergence of Gauss-Kronrod nodes calculation subroutine. * -1 incorrect parameters were specified * 1 OK * Rep.NFEV countains number of function calculations * Rep.NIntervals contains number of intervals [a,b] was partitioned into. *************************************************************************/class autogkreport { ae_int_t terminationtype; ae_int_t nfev; ae_int_t nintervals; };
autogkstate class/************************************************************************* This structure stores state of the integration algorithm. Although this class has public fields, they are not intended for external use. You should use ALGLIB functions to work with this class: * autogksmooth()/AutoGKSmoothW()/... to create objects * autogkintegrate() to begin integration * autogkresults() to get results *************************************************************************/class autogkstate { };
autogkintegrate function/************************************************************************* This function is used to launcn iterations of ODE solver It accepts following parameters: diff - callback which calculates dy/dx for given y and x obj - optional object which is passed to diff; can be NULL -- ALGLIB -- Copyright 07.05.2009 by Bochkanov Sergey *************************************************************************/void autogkintegrate(autogkstate &state, void (*func)(double x, double xminusa, double bminusx, double &y, void *ptr), void *ptr = NULL);
Examples: [1]
autogkresults function/************************************************************************* Adaptive integration results Called after AutoGKIteration returned False. Input parameters: State - algorithm state (used by AutoGKIteration). Output parameters: V - integral(f(x)dx,a,b) Rep - optimization report (see AutoGKReport description) -- ALGLIB -- Copyright 14.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::autogkresults( autogkstate state, double& v, autogkreport& rep);
Examples: [1]
autogksingular function/************************************************************************* Integration on a finite interval [A,B]. Integrand have integrable singularities at A/B. F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B, with known alpha/beta (alpha>-1, beta>-1). If alpha/beta are not known, estimates from below can be used (but these estimates should be greater than -1 too). One of alpha/beta variables (or even both alpha/beta) may be equal to 0, which means than function F(x) is non-singular at A/B. Anyway (singular at bounds or not), function F(x) is supposed to be continuous on (A,B). Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result is calculated with accuracy close to the machine precision. INPUT PARAMETERS: A, B - interval boundaries (A<B, A=B or A>B) Alpha - power-law coefficient of the F(x) at A, Alpha>-1 Beta - power-law coefficient of the F(x) at B, Beta>-1 OUTPUT PARAMETERS State - structure which stores algorithm state SEE ALSO AutoGKSmooth, AutoGKSmoothW, AutoGKResults. -- ALGLIB -- Copyright 06.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::autogksingular( double a, double b, double alpha, double beta, autogkstate& state);
autogksmooth function/************************************************************************* Integration of a smooth function F(x) on a finite interval [a,b]. Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result is calculated with accuracy close to the machine precision. Algorithm works well only with smooth integrands. It may be used with continuous non-smooth integrands, but with less performance. It should never be used with integrands which have integrable singularities at lower or upper limits - algorithm may crash. Use AutoGKSingular in such cases. INPUT PARAMETERS: A, B - interval boundaries (A<B, A=B or A>B) OUTPUT PARAMETERS State - structure which stores algorithm state SEE ALSO AutoGKSmoothW, AutoGKSingular, AutoGKResults. -- ALGLIB -- Copyright 06.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::autogksmooth(double a, double b, autogkstate& state);
Examples: [1]
autogksmoothw function/************************************************************************* Integration of a smooth function F(x) on a finite interval [a,b]. This subroutine is same as AutoGKSmooth(), but it guarantees that interval [a,b] is partitioned into subintervals which have width at most XWidth. Subroutine can be used when integrating nearly-constant function with narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth subroutine can overlook them. INPUT PARAMETERS: A, B - interval boundaries (A<B, A=B or A>B) OUTPUT PARAMETERS State - structure which stores algorithm state SEE ALSO AutoGKSmooth, AutoGKSingular, AutoGKResults. -- ALGLIB -- Copyright 06.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::autogksmoothw( double a, double b, double xwidth, autogkstate& state);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "integration.h"
using namespace alglib;
void int_function_1_func(double x, double xminusa, double bminusx, double &y, void *ptr)
{
// this callback calculates f(x)=exp(x)
y = exp(x);
}
int main(int argc, char **argv)
{
//
// This example demonstrates integration of f=exp(x) on [0,1]:
// * first, autogkstate is initialized
// * then we call integration function
// * and finally we obtain results with autogkresults() call
//
double a = 0;
double b = 1;
autogkstate s;
double v;
autogkreport rep;
autogksmooth(a, b, s);
alglib::autogkintegrate(s, int_function_1_func);
autogkresults(s, v, rep);
printf("%.2f\n", double(v)); // EXPECTED: 1.7182
return 0;
}
basestat subpackage| basestat_d_base | Basic functionality (moments, adev, median, percentile) | |
| basestat_d_c2 | Correlation (covariance) between two random variables | |
| basestat_d_cm | Correlation (covariance) between components of random vector | |
| basestat_d_cm2 | Correlation (covariance) between two random vectors |
cov2 function/************************************************************************* 2-sample covariance Input parameters: X - sample 1 (array indexes: [0..N-1]) Y - sample 2 (array indexes: [0..N-1]) N - N>=0, sample size: * if given, only N leading elements of X/Y are processed * if not given, automatically determined from input sizes Result: covariance (zero for N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/double alglib::cov2(real_1d_array x, real_1d_array y); double alglib::cov2(real_1d_array x, real_1d_array y, ae_int_t n);
Examples: [1]
covm function/************************************************************************* Covariance matrix SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! with covariance matrices smaller than 128*128. INPUT PARAMETERS: X - array[N,M], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation N - N>=0, number of observations: * if given, only leading N rows of X are used * if not given, automatically determined from input size M - M>0, number of variables: * if given, only leading M columns of X are used * if not given, automatically determined from input size OUTPUT PARAMETERS: C - array[M,M], covariance matrix (zero if N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::covm(real_2d_array x, real_2d_array& c); void alglib::covm( real_2d_array x, ae_int_t n, ae_int_t m, real_2d_array& c); void alglib::smp_covm(real_2d_array x, real_2d_array& c); void alglib::smp_covm( real_2d_array x, ae_int_t n, ae_int_t m, real_2d_array& c);
Examples: [1]
covm2 function/************************************************************************* Cross-covariance matrix SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! with covariance matrices smaller than 128*128. INPUT PARAMETERS: X - array[N,M1], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation Y - array[N,M2], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation N - N>=0, number of observations: * if given, only leading N rows of X/Y are used * if not given, automatically determined from input sizes M1 - M1>0, number of variables in X: * if given, only leading M1 columns of X are used * if not given, automatically determined from input size M2 - M2>0, number of variables in Y: * if given, only leading M1 columns of X are used * if not given, automatically determined from input size OUTPUT PARAMETERS: C - array[M1,M2], cross-covariance matrix (zero if N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::covm2(real_2d_array x, real_2d_array y, real_2d_array& c); void alglib::covm2( real_2d_array x, real_2d_array y, ae_int_t n, ae_int_t m1, ae_int_t m2, real_2d_array& c); void alglib::smp_covm2(real_2d_array x, real_2d_array y, real_2d_array& c); void alglib::smp_covm2( real_2d_array x, real_2d_array y, ae_int_t n, ae_int_t m1, ae_int_t m2, real_2d_array& c);
Examples: [1]
pearsoncorr2 function/************************************************************************* Pearson product-moment correlation coefficient Input parameters: X - sample 1 (array indexes: [0..N-1]) Y - sample 2 (array indexes: [0..N-1]) N - N>=0, sample size: * if given, only N leading elements of X/Y are processed * if not given, automatically determined from input sizes Result: Pearson product-moment correlation coefficient (zero for N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/double alglib::pearsoncorr2(real_1d_array x, real_1d_array y); double alglib::pearsoncorr2(real_1d_array x, real_1d_array y, ae_int_t n);
Examples: [1]
pearsoncorrelation function/************************************************************************* Obsolete function, we recommend to use PearsonCorr2(). -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/double alglib::pearsoncorrelation( real_1d_array x, real_1d_array y, ae_int_t n);
pearsoncorrm function/************************************************************************* Pearson product-moment correlation matrix SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! with correlation matrices smaller than 128*128. INPUT PARAMETERS: X - array[N,M], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation N - N>=0, number of observations: * if given, only leading N rows of X are used * if not given, automatically determined from input size M - M>0, number of variables: * if given, only leading M columns of X are used * if not given, automatically determined from input size OUTPUT PARAMETERS: C - array[M,M], correlation matrix (zero if N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::pearsoncorrm(real_2d_array x, real_2d_array& c); void alglib::pearsoncorrm( real_2d_array x, ae_int_t n, ae_int_t m, real_2d_array& c); void alglib::smp_pearsoncorrm(real_2d_array x, real_2d_array& c); void alglib::smp_pearsoncorrm( real_2d_array x, ae_int_t n, ae_int_t m, real_2d_array& c);
Examples: [1]
pearsoncorrm2 function/************************************************************************* Pearson product-moment cross-correlation matrix SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! with correlation matrices smaller than 128*128. INPUT PARAMETERS: X - array[N,M1], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation Y - array[N,M2], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation N - N>=0, number of observations: * if given, only leading N rows of X/Y are used * if not given, automatically determined from input sizes M1 - M1>0, number of variables in X: * if given, only leading M1 columns of X are used * if not given, automatically determined from input size M2 - M2>0, number of variables in Y: * if given, only leading M1 columns of X are used * if not given, automatically determined from input size OUTPUT PARAMETERS: C - array[M1,M2], cross-correlation matrix (zero if N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::pearsoncorrm2( real_2d_array x, real_2d_array y, real_2d_array& c); void alglib::pearsoncorrm2( real_2d_array x, real_2d_array y, ae_int_t n, ae_int_t m1, ae_int_t m2, real_2d_array& c); void alglib::smp_pearsoncorrm2( real_2d_array x, real_2d_array y, real_2d_array& c); void alglib::smp_pearsoncorrm2( real_2d_array x, real_2d_array y, ae_int_t n, ae_int_t m1, ae_int_t m2, real_2d_array& c);
Examples: [1]
rankdata function/************************************************************************* This function replaces data in XY by their ranks: * XY is processed row-by-row * rows are processed separately * tied data are correctly handled (tied ranks are calculated) * ranking starts from 0, ends at NFeatures-1 * sum of within-row values is equal to (NFeatures-1)*NFeatures/2 SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! ones where expected operations count is less than 100.000 INPUT PARAMETERS: XY - array[NPoints,NFeatures], dataset NPoints - number of points NFeatures- number of features OUTPUT PARAMETERS: XY - data are replaced by their within-row ranks; ranking starts from 0, ends at NFeatures-1 -- ALGLIB -- Copyright 18.04.2013 by Bochkanov Sergey *************************************************************************/void alglib::rankdata(real_2d_array& xy); void alglib::rankdata( real_2d_array& xy, ae_int_t npoints, ae_int_t nfeatures); void alglib::smp_rankdata(real_2d_array& xy); void alglib::smp_rankdata( real_2d_array& xy, ae_int_t npoints, ae_int_t nfeatures);
rankdatacentered function/************************************************************************* This function replaces data in XY by their CENTERED ranks: * XY is processed row-by-row * rows are processed separately * tied data are correctly handled (tied ranks are calculated) * centered ranks are just usual ranks, but centered in such way that sum of within-row values is equal to 0.0. * centering is performed by subtracting mean from each row, i.e it changes mean value, but does NOT change higher moments SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! ones where expected operations count is less than 100.000 INPUT PARAMETERS: XY - array[NPoints,NFeatures], dataset NPoints - number of points NFeatures- number of features OUTPUT PARAMETERS: XY - data are replaced by their within-row ranks; ranking starts from 0, ends at NFeatures-1 -- ALGLIB -- Copyright 18.04.2013 by Bochkanov Sergey *************************************************************************/void alglib::rankdatacentered(real_2d_array& xy); void alglib::rankdatacentered( real_2d_array& xy, ae_int_t npoints, ae_int_t nfeatures); void alglib::smp_rankdatacentered(real_2d_array& xy); void alglib::smp_rankdatacentered( real_2d_array& xy, ae_int_t npoints, ae_int_t nfeatures);
sampleadev function/************************************************************************* ADev Input parameters: X - sample N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X Output parameters: ADev- ADev -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::sampleadev(real_1d_array x, double& adev); void alglib::sampleadev(real_1d_array x, ae_int_t n, double& adev);
Examples: [1]
samplekurtosis function/************************************************************************* Calculation of the kurtosis. INPUT PARAMETERS: X - sample N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X NOTE: This function return result which calculated by 'SampleMoments' function and stored at 'Kurtosis' variable. -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/double alglib::samplekurtosis(real_1d_array x); double alglib::samplekurtosis(real_1d_array x, ae_int_t n);
samplemean function/************************************************************************* Calculation of the mean. INPUT PARAMETERS: X - sample N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X NOTE: This function return result which calculated by 'SampleMoments' function and stored at 'Mean' variable. -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/double alglib::samplemean(real_1d_array x); double alglib::samplemean(real_1d_array x, ae_int_t n);
samplemedian function/************************************************************************* Median calculation. Input parameters: X - sample (array indexes: [0..N-1]) N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X Output parameters: Median -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::samplemedian(real_1d_array x, double& median); void alglib::samplemedian(real_1d_array x, ae_int_t n, double& median);
Examples: [1]
samplemoments function/************************************************************************* Calculation of the distribution moments: mean, variance, skewness, kurtosis. INPUT PARAMETERS: X - sample N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X OUTPUT PARAMETERS Mean - mean. Variance- variance. Skewness- skewness (if variance<>0; zero otherwise). Kurtosis- kurtosis (if variance<>0; zero otherwise). NOTE: variance is calculated by dividing sum of squares by N-1, not N. -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::samplemoments( real_1d_array x, double& mean, double& variance, double& skewness, double& kurtosis); void alglib::samplemoments( real_1d_array x, ae_int_t n, double& mean, double& variance, double& skewness, double& kurtosis);
Examples: [1]
samplepercentile function/************************************************************************* Percentile calculation. Input parameters: X - sample (array indexes: [0..N-1]) N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X P - percentile (0<=P<=1) Output parameters: V - percentile -- ALGLIB -- Copyright 01.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::samplepercentile(real_1d_array x, double p, double& v); void alglib::samplepercentile( real_1d_array x, ae_int_t n, double p, double& v);
Examples: [1]
sampleskewness function/************************************************************************* Calculation of the skewness. INPUT PARAMETERS: X - sample N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X NOTE: This function return result which calculated by 'SampleMoments' function and stored at 'Skewness' variable. -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/double alglib::sampleskewness(real_1d_array x); double alglib::sampleskewness(real_1d_array x, ae_int_t n);
samplevariance function/************************************************************************* Calculation of the variance. INPUT PARAMETERS: X - sample N - N>=0, sample size: * if given, only leading N elements of X are processed * if not given, automatically determined from size of X NOTE: This function return result which calculated by 'SampleMoments' function and stored at 'Variance' variable. -- ALGLIB -- Copyright 06.09.2006 by Bochkanov Sergey *************************************************************************/double alglib::samplevariance(real_1d_array x); double alglib::samplevariance(real_1d_array x, ae_int_t n);
spearmancorr2 function/************************************************************************* Spearman's rank correlation coefficient Input parameters: X - sample 1 (array indexes: [0..N-1]) Y - sample 2 (array indexes: [0..N-1]) N - N>=0, sample size: * if given, only N leading elements of X/Y are processed * if not given, automatically determined from input sizes Result: Spearman's rank correlation coefficient (zero for N=0 or N=1) -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/double alglib::spearmancorr2(real_1d_array x, real_1d_array y); double alglib::spearmancorr2( real_1d_array x, real_1d_array y, ae_int_t n);
Examples: [1]
spearmancorrm function/************************************************************************* Spearman's rank correlation matrix SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! with correlation matrices smaller than 128*128. INPUT PARAMETERS: X - array[N,M], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation N - N>=0, number of observations: * if given, only leading N rows of X are used * if not given, automatically determined from input size M - M>0, number of variables: * if given, only leading M columns of X are used * if not given, automatically determined from input size OUTPUT PARAMETERS: C - array[M,M], correlation matrix (zero if N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::spearmancorrm(real_2d_array x, real_2d_array& c); void alglib::spearmancorrm( real_2d_array x, ae_int_t n, ae_int_t m, real_2d_array& c); void alglib::smp_spearmancorrm(real_2d_array x, real_2d_array& c); void alglib::smp_spearmancorrm( real_2d_array x, ae_int_t n, ae_int_t m, real_2d_array& c);
Examples: [1]
spearmancorrm2 function/************************************************************************* Spearman's rank cross-correlation matrix SMP EDITION OF ALGLIB: ! This function can utilize multicore capabilities of your system. In ! order to do this you have to call version with "smp_" prefix, which ! indicates that multicore code will be used. ! ! This note is given for users of SMP edition; if you use GPL edition, ! or commercial edition of ALGLIB without SMP support, you still will ! be able to call smp-version of this function, but all computations ! will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. ! ! You should remember that starting/stopping worker thread always have ! non-zero cost. Although multicore version is pretty efficient on ! large problems, we do not recommend you to use it on small problems - ! with correlation matrices smaller than 128*128. INPUT PARAMETERS: X - array[N,M1], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation Y - array[N,M2], sample matrix: * J-th column corresponds to J-th variable * I-th row corresponds to I-th observation N - N>=0, number of observations: * if given, only leading N rows of X/Y are used * if not given, automatically determined from input sizes M1 - M1>0, number of variables in X: * if given, only leading M1 columns of X are used * if not given, automatically determined from input size M2 - M2>0, number of variables in Y: * if given, only leading M1 columns of X are used * if not given, automatically determined from input size OUTPUT PARAMETERS: C - array[M1,M2], cross-correlation matrix (zero if N=0 or N=1) -- ALGLIB -- Copyright 28.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::spearmancorrm2( real_2d_array x, real_2d_array y, real_2d_array& c); void alglib::spearmancorrm2( real_2d_array x, real_2d_array y, ae_int_t n, ae_int_t m1, ae_int_t m2, real_2d_array& c); void alglib::smp_spearmancorrm2( real_2d_array x, real_2d_array y, real_2d_array& c); void alglib::smp_spearmancorrm2( real_2d_array x, real_2d_array y, ae_int_t n, ae_int_t m1, ae_int_t m2, real_2d_array& c);
Examples: [1]
spearmanrankcorrelation function/************************************************************************* Obsolete function, we recommend to use SpearmanCorr2(). -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/double alglib::spearmanrankcorrelation( real_1d_array x, real_1d_array y, ae_int_t n);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_1d_array x = "[0,1,4,9,16,25,36,49,64,81]";
double mean;
double variance;
double skewness;
double kurtosis;
double adev;
double p;
double v;
//
// Here we demonstrate calculation of sample moments
// (mean, variance, skewness, kurtosis)
//
samplemoments(x, mean, variance, skewness, kurtosis);
printf("%.1f\n", double(mean)); // EXPECTED: 28.5
printf("%.1f\n", double(variance)); // EXPECTED: 801.1667
printf("%.1f\n", double(skewness)); // EXPECTED: 0.5751
printf("%.1f\n", double(kurtosis)); // EXPECTED: -1.2666
//
// Average deviation
//
sampleadev(x, adev);
printf("%.1f\n", double(adev)); // EXPECTED: 23.2
//
// Median and percentile
//
samplemedian(x, v);
printf("%.1f\n", double(v)); // EXPECTED: 20.5
p = 0.5;
samplepercentile(x, p, v);
printf("%.1f\n", double(v)); // EXPECTED: 20.5
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We have two samples - x and y, and want to measure dependency between them
//
real_1d_array x = "[0,1,4,9,16,25,36,49,64,81]";
real_1d_array y = "[0,1,2,3,4,5,6,7,8,9]";
double v;
//
// Three dependency measures are calculated:
// * covariation
// * Pearson correlation
// * Spearman rank correlation
//
v = cov2(x, y);
printf("%.2f\n", double(v)); // EXPECTED: 82.5
v = pearsoncorr2(x, y);
printf("%.2f\n", double(v)); // EXPECTED: 0.9627
v = spearmancorr2(x, y);
printf("%.2f\n", double(v)); // EXPECTED: 1.000
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// X is a sample matrix:
// * I-th row corresponds to I-th observation
// * J-th column corresponds to J-th variable
//
real_2d_array x = "[[1,0,1],[1,1,0],[-1,1,0],[-2,-1,1],[-1,0,9]]";
real_2d_array c;
//
// Three dependency measures are calculated:
// * covariation
// * Pearson correlation
// * Spearman rank correlation
//
// Result is stored into C, with C[i,j] equal to correlation
// (covariance) between I-th and J-th variables of X.
//
covm(x, c);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.80,0.60,-1.40],[0.60,0.70,-0.80],[-1.40,-0.80,14.70]]
pearsoncorrm(x, c);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.000,0.535,-0.272],[0.535,1.000,-0.249],[-0.272,-0.249,1.000]]
spearmancorrm(x, c);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[1.000,0.556,-0.306],[0.556,1.000,-0.750],[-0.306,-0.750,1.000]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "statistics.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// X and Y are sample matrices:
// * I-th row corresponds to I-th observation
// * J-th column corresponds to J-th variable
//
real_2d_array x = "[[1,0,1],[1,1,0],[-1,1,0],[-2,-1,1],[-1,0,9]]";
real_2d_array y = "[[2,3],[2,1],[-1,6],[-9,9],[7,1]]";
real_2d_array c;
//
// Three dependency measures are calculated:
// * covariation
// * Pearson correlation
// * Spearman rank correlation
//
// Result is stored into C, with C[i,j] equal to correlation
// (covariance) between I-th variable of X and J-th variable of Y.
//
covm2(x, y, c);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[4.100,-3.250],[2.450,-1.500],[13.450,-5.750]]
pearsoncorrm2(x, y, c);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[0.519,-0.699],[0.497,-0.518],[0.596,-0.433]]
spearmancorrm2(x, y, c);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [[0.541,-0.649],[0.216,-0.433],[0.433,-0.135]]
return 0;
}
bdss subpackagedsoptimalsplit2 function/************************************************************************* Optimal binary classification Algorithms finds optimal (=with minimal cross-entropy) binary partition. Internal subroutine. INPUT PARAMETERS: A - array[0..N-1], variable C - array[0..N-1], class numbers (0 or 1). N - array size OUTPUT PARAMETERS: Info - completetion code: * -3, all values of A[] are same (partition is impossible) * -2, one of C[] is incorrect (<0, >1) * -1, incorrect pararemets were passed (N<=0). * 1, OK Threshold- partiton boundary. Left part contains values which are strictly less than Threshold. Right part contains values which are greater than or equal to Threshold. PAL, PBL- probabilities P(0|v<Threshold) and P(1|v<Threshold) PAR, PBR- probabilities P(0|v>=Threshold) and P(1|v>=Threshold) CVE - cross-validation estimate of cross-entropy -- ALGLIB -- Copyright 22.05.2008 by Bochkanov Sergey *************************************************************************/void alglib::dsoptimalsplit2( real_1d_array a, integer_1d_array c, ae_int_t n, ae_int_t& info, double& threshold, double& pal, double& pbl, double& par, double& pbr, double& cve);
dsoptimalsplit2fast function/************************************************************************* Optimal partition, internal subroutine. Fast version. Accepts: A array[0..N-1] array of attributes array[0..N-1] C array[0..N-1] array of class labels TiesBuf array[0..N] temporaries (ties) CntBuf array[0..2*NC-1] temporaries (counts) Alpha centering factor (0<=alpha<=1, recommended value - 0.05) BufR array[0..N-1] temporaries BufI array[0..N-1] temporaries Output: Info error code (">0"=OK, "<0"=bad) RMS training set RMS error CVRMS leave-one-out RMS error Note: content of all arrays is changed by subroutine; it doesn't allocate temporaries. -- ALGLIB -- Copyright 11.12.2008 by Bochkanov Sergey *************************************************************************/void alglib::dsoptimalsplit2fast( real_1d_array& a, integer_1d_array& c, integer_1d_array& tiesbuf, integer_1d_array& cntbuf, real_1d_array& bufr, integer_1d_array& bufi, ae_int_t n, ae_int_t nc, double alpha, ae_int_t& info, double& threshold, double& rms, double& cvrms);
bdsvd subpackagermatrixbdsvd function/************************************************************************* Singular value decomposition of a bidiagonal matrix (extended algorithm) COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. The algorithm performs the singular value decomposition of a bidiagonal matrix B (upper or lower) representing it as B = Q*S*P^T, where Q and P - orthogonal matrices, S - diagonal matrix with non-negative elements on the main diagonal, in descending order. The algorithm finds singular values. In addition, the algorithm can calculate matrices Q and P (more precisely, not the matrices, but their product with given matrices U and VT - U*Q and (P^T)*VT)). Of course, matrices U and VT can be of any type, including identity. Furthermore, the algorithm can calculate Q'*C (this product is calculated more effectively than U*Q, because this calculation operates with rows instead of matrix columns). The feature of the algorithm is its ability to find all singular values including those which are arbitrarily close to 0 with relative accuracy close to machine precision. If the parameter IsFractionalAccuracyRequired is set to True, all singular values will have high relative accuracy close to machine precision. If the parameter is set to False, only the biggest singular value will have relative accuracy close to machine precision. The absolute error of other singular values is equal to the absolute error of the biggest singular value. Input parameters: D - main diagonal of matrix B. Array whose index ranges within [0..N-1]. E - superdiagonal (or subdiagonal) of matrix B. Array whose index ranges within [0..N-2]. N - size of matrix B. IsUpper - True, if the matrix is upper bidiagonal. IsFractionalAccuracyRequired - THIS PARAMETER IS IGNORED SINCE ALGLIB 3.5.0 SINGULAR VALUES ARE ALWAYS SEARCHED WITH HIGH ACCURACY. U - matrix to be multiplied by Q. Array whose indexes range within [0..NRU-1, 0..N-1]. The matrix can be bigger, in that case only the submatrix [0..NRU-1, 0..N-1] will be multiplied by Q. NRU - number of rows in matrix U. C - matrix to be multiplied by Q'. Array whose indexes range within [0..N-1, 0..NCC-1]. The matrix can be bigger, in that case only the submatrix [0..N-1, 0..NCC-1] will be multiplied by Q'. NCC - number of columns in matrix C. VT - matrix to be multiplied by P^T. Array whose indexes range within [0..N-1, 0..NCVT-1]. The matrix can be bigger, in that case only the submatrix [0..N-1, 0..NCVT-1] will be multiplied by P^T. NCVT - number of columns in matrix VT. Output parameters: D - singular values of matrix B in descending order. U - if NRU>0, contains matrix U*Q. VT - if NCVT>0, contains matrix (P^T)*VT. C - if NCC>0, contains matrix Q'*C. Result: True, if the algorithm has converged. False, if the algorithm hasn't converged (rare case). NOTE: multiplication U*Q is performed by means of transposition to internal buffer, multiplication and backward transposition. It helps to avoid costly columnwise operations and speed-up algorithm. Additional information: The type of convergence is controlled by the internal parameter TOL. If the parameter is greater than 0, the singular values will have relative accuracy TOL. If TOL<0, the singular values will have absolute accuracy ABS(TOL)*norm(B). By default, |TOL| falls within the range of 10*Epsilon and 100*Epsilon, where Epsilon is the machine precision. It is not recommended to use TOL less than 10*Epsilon since this will considerably slow down the algorithm and may not lead to error decreasing. History: * 31 March, 2007. changed MAXITR from 6 to 12. -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1999. *************************************************************************/bool alglib::rmatrixbdsvd( real_1d_array& d, real_1d_array e, ae_int_t n, bool isupper, bool isfractionalaccuracyrequired, real_2d_array& u, ae_int_t nru, real_2d_array& c, ae_int_t ncc, real_2d_array& vt, ae_int_t ncvt);
bessel subpackagebesseli0 function/************************************************************************* Modified Bessel function of order zero Returns modified Bessel function of order zero of the argument. The function is defined as i0(x) = j0( ix ). The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 5.8e-16 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besseli0(double x);
besseli1 function/************************************************************************* Modified Bessel function of order one Returns modified Bessel function of order one of the argument. The function is defined as i1(x) = -i j1( ix ). The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.9e-15 2.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besseli1(double x);
besselj0 function/************************************************************************* Bessel function of order zero Returns Bessel function of order zero of the argument. The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval the following rational approximation is used: 2 2 (w - r ) (w - r ) P (w) / Q (w) 1 2 3 8 2 where w = x and the two r's are zeros of the function. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 60000 4.2e-16 1.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besselj0(double x);
besselj1 function/************************************************************************* Bessel function of order one Returns Bessel function of order one of the argument. The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 24 term Chebyshev expansion is used. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 30000 2.6e-16 1.1e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besselj1(double x);
besseljn function/************************************************************************* Bessel function of integer order Returns Bessel function of order n, where n is a (possibly negative) integer. The ratio of jn(x) to j0(x) is computed by backward recurrence. First the ratio jn/jn-1 is found by a continued fraction expansion. Then the recurrence relating successive orders is applied until j0 or j1 is reached. If n = 0 or 1 the routine for j0 or j1 is called directly. ACCURACY: Absolute error: arithmetic range # trials peak rms IEEE 0, 30 5000 4.4e-16 7.9e-17 Not suitable for large n or x. Use jv() (fractional order) instead. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besseljn(ae_int_t n, double x);
besselk0 function/************************************************************************* Modified Bessel function, second kind, order zero Returns modified Bessel function of the second kind of order zero of the argument. The range is partitioned into the two intervals [0,8] and (8, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Tested at 2000 random points between 0 and 8. Peak absolute error (relative when K0 > 1) was 1.46e-14; rms, 4.26e-15. Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.2e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besselk0(double x);
besselk1 function/************************************************************************* Modified Bessel function, second kind, order one Computes the modified Bessel function of the second kind of order one of the argument. The range is partitioned into the two intervals [0,2] and (2, infinity). Chebyshev polynomial expansions are employed in each interval. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.2e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besselk1(double x);
besselkn function/************************************************************************* Modified Bessel function, second kind, integer order Returns modified Bessel function of the second kind of order n of the argument. The range is partitioned into the two intervals [0,9.55] and (9.55, infinity). An ascending power series is used in the low range, and an asymptotic expansion in the high range. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 90000 1.8e-8 3.0e-10 Error is high only near the crossover point x = 9.55 between the two expansions used. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besselkn(ae_int_t nn, double x);
bessely0 function/************************************************************************* Bessel function of the second kind, order zero Returns Bessel function of the second kind, of order zero, of the argument. The domain is divided into the intervals [0, 5] and (5, infinity). In the first interval a rational approximation R(x) is employed to compute y0(x) = R(x) + 2 * log(x) * j0(x) / PI. Thus a call to j0() is required. In the second interval, the Hankel asymptotic expansion is employed with two rational functions of degree 6/6 and 7/7. ACCURACY: Absolute error, when y0(x) < 1; else relative error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.3e-15 1.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/double alglib::bessely0(double x);
bessely1 function/************************************************************************* Bessel function of second kind of order one Returns Bessel function of the second kind of order one of the argument. The domain is divided into the intervals [0, 8] and (8, infinity). In the first interval a 25 term Chebyshev expansion is used, and a call to j1() is required. In the second, the asymptotic trigonometric representation is employed using two rational functions of degree 5/5. ACCURACY: Absolute error: arithmetic domain # trials peak rms IEEE 0, 30 30000 1.0e-15 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/double alglib::bessely1(double x);
besselyn function/************************************************************************* Bessel function of second kind of integer order Returns Bessel function of order n, where n is a (possibly negative) integer. The function is evaluated by forward recurrence on n, starting with values computed by the routines y0() and y1(). If n = 0 or 1 the routine for y0 or y1 is called directly. ACCURACY: Absolute error, except relative when y > 1: arithmetic domain # trials peak rms IEEE 0, 30 30000 3.4e-15 4.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::besselyn(ae_int_t n, double x);
betaf subpackagebeta function/************************************************************************* Beta function - - | (a) | (b) beta( a, b ) = -----------. - | (a+b) For large arguments the logarithm of the function is evaluated using lgam(), then exponentiated. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 30000 8.1e-14 1.1e-14 Cephes Math Library Release 2.0: April, 1987 Copyright 1984, 1987 by Stephen L. Moshier *************************************************************************/double alglib::beta(double a, double b);
binomialdistr subpackagebinomialcdistribution function/************************************************************************* Complemented binomial distribution Returns the sum of the terms k+1 through n of the Binomial probability density: n -- ( n ) j n-j > ( ) p (1-p) -- ( j ) j=k+1 The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula y = bdtrc( k, n, p ) = incbet( k+1, n-k, p ). The arguments must be positive, with p ranging from 0 to 1. ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 6.7e-15 8.2e-16 For p between 0 and .001: IEEE 0,100 100000 1.5e-13 2.7e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::binomialcdistribution(ae_int_t k, ae_int_t n, double p);
binomialdistribution function/************************************************************************* Binomial distribution Returns the sum of the terms 0 through k of the Binomial probability density: k -- ( n ) j n-j > ( ) p (1-p) -- ( j ) j=0 The terms are not summed directly; instead the incomplete beta integral is employed, according to the formula y = bdtr( k, n, p ) = incbet( n-k, k+1, 1-p ). The arguments must be positive, with p ranging from 0 to 1. ACCURACY: Tested at random points (a,b,p), with p between 0 and 1. a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 4.3e-15 2.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::binomialdistribution(ae_int_t k, ae_int_t n, double p);
invbinomialdistribution function/************************************************************************* Inverse binomial distribution Finds the event probability p such that the sum of the terms 0 through k of the Binomial probability density is equal to the given cumulative probability y. This is accomplished using the inverse beta integral function and the relation 1 - p = incbi( n-k, k+1, y ). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between 0.001 and 1: IEEE 0,100 100000 2.3e-14 6.4e-16 IEEE 0,10000 100000 6.6e-12 1.2e-13 For p between 10^-6 and 0.001: IEEE 0,100 100000 2.0e-12 1.3e-14 IEEE 0,10000 100000 1.5e-12 3.2e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invbinomialdistribution(ae_int_t k, ae_int_t n, double y);
chebyshev subpackagechebyshevcalculate function/************************************************************************* Calculation of the value of the Chebyshev polynomials of the first and second kinds. Parameters: r - polynomial kind, either 1 or 2. n - degree, n>=0 x - argument, -1 <= x <= 1 Result: the value of the Chebyshev polynomial at x *************************************************************************/double alglib::chebyshevcalculate(ae_int_t r, ae_int_t n, double x);
chebyshevcoefficients function/************************************************************************* Representation of Tn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/void alglib::chebyshevcoefficients(ae_int_t n, real_1d_array& c);
chebyshevsum function/************************************************************************* Summation of Chebyshev polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*T0(x) + c[1]*T1(x) + ... + c[N]*TN(x) or c[0]*U0(x) + c[1]*U1(x) + ... + c[N]*UN(x) depending on the R. Parameters: r - polynomial kind, either 1 or 2. n - degree, n>=0 x - argument Result: the value of the Chebyshev polynomial at x *************************************************************************/double alglib::chebyshevsum( real_1d_array c, ae_int_t r, ae_int_t n, double x);
fromchebyshev function/************************************************************************* Conversion of a series of Chebyshev polynomials to a power series. Represents A[0]*T0(x) + A[1]*T1(x) + ... + A[N]*Tn(x) as B[0] + B[1]*X + ... + B[N]*X^N. Input parameters: A - Chebyshev series coefficients N - degree, N>=0 Output parameters B - power series coefficients *************************************************************************/void alglib::fromchebyshev(real_1d_array a, ae_int_t n, real_1d_array& b);
chisquaredistr subpackagechisquarecdistribution function/************************************************************************* Complemented Chi-square distribution Returns the area under the right hand tail (from x to infinity) of the Chi square probability density function with v degrees of freedom: inf. - 1 | | v/2-1 -t/2 P( x | v ) = ----------- | t e dt v/2 - | | 2 | (v/2) - x where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chdtr( v, x ) = igamc( v/2.0, x/2.0 ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::chisquarecdistribution(double v, double x);
chisquaredistribution function/************************************************************************* Chi-square distribution Returns the area under the left hand tail (from 0 to x) of the Chi square probability density function with v degrees of freedom. x - 1 | | v/2-1 -t/2 P( x | v ) = ----------- | t e dt v/2 - | | 2 | (v/2) - 0 where x is the Chi-square variable. The incomplete gamma integral is used, according to the formula y = chdtr( v, x ) = igam( v/2.0, x/2.0 ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::chisquaredistribution(double v, double x);
invchisquaredistribution function/************************************************************************* Inverse of complemented Chi-square distribution Finds the Chi-square argument x such that the integral from x to infinity of the Chi-square density is equal to the given cumulative probability y. This is accomplished using the inverse gamma integral function and the relation x/2 = igami( df/2, y ); ACCURACY: See inverse incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invchisquaredistribution(double v, double y);
clustering subpackage| clst_ahc | Simple hierarchical clusterization with Euclidean distance function | |
| clst_distance | Clusterization with different metric types | |
| clst_kclusters | Obtaining K top clusters from clusterization tree | |
| clst_kmeans | Simple k-means clusterization | |
| clst_linkage | Clusterization with different linkage types |
ahcreport class/************************************************************************* This structure is used to store results of the agglomerative hierarchical clustering (AHC). Following information is returned: * TerminationType - completion code: * 1 for successful completion of algorithm * -5 inappropriate combination of clustering algorithm and distance function was used. As for now, it is possible only when Ward's method is called for dataset with non-Euclidean distance function. In case negative completion code is returned, other fields of report structure are invalid and should not be used. * NPoints contains number of points in the original dataset * Z contains information about merges performed (see below). Z contains indexes from the original (unsorted) dataset and it can be used when you need to know what points were merged. However, it is not convenient when you want to build a dendrograd (see below). * if you want to build dendrogram, you can use Z, but it is not good option, because Z contains indexes from unsorted dataset. Dendrogram built from such dataset is likely to have intersections. So, you have to reorder you points before building dendrogram. Permutation which reorders point is returned in P. Another representation of merges, which is more convenient for dendorgram construction, is returned in PM. * more information on format of Z, P and PM can be found below and in the examples from ALGLIB Reference Manual. FORMAL DESCRIPTION OF FIELDS: NPoints number of points Z array[NPoints-1,2], contains indexes of clusters linked in pairs to form clustering tree. I-th row corresponds to I-th merge: * Z[I,0] - index of the first cluster to merge * Z[I,1] - index of the second cluster to merge * Z[I,0]<Z[I,1] * clusters are numbered from 0 to 2*NPoints-2, with indexes from 0 to NPoints-1 corresponding to points of the original dataset, and indexes from NPoints to 2*NPoints-2 correspond to clusters generated by subsequent merges (I-th row of Z creates cluster with index NPoints+I). IMPORTANT: indexes in Z[] are indexes in the ORIGINAL, unsorted dataset. In addition to Z algorithm outputs permutation which rearranges points in such way that subsequent merges are performed on adjacent points (such order is needed if you want to build dendrogram). However, indexes in Z are related to original, unrearranged sequence of points. P array[NPoints], permutation which reorders points for dendrogram construction. P[i] contains index of the position where we should move I-th point of the original dataset in order to apply merges PZ/PM. PZ same as Z, but for permutation of points given by P. The only thing which changed are indexes of the original points; indexes of clusters remained same. MergeDist array[NPoints-1], contains distances between clusters being merged (MergeDist[i] correspond to merge stored in Z[i,...]): * CLINK, SLINK and average linkage algorithms report "raw", unmodified distance metric. * Ward's method reports weighted intra-cluster variance, which is equal to ||Ca-Cb||^2 * Sa*Sb/(Sa+Sb). Here A and B are clusters being merged, Ca is a center of A, Cb is a center of B, Sa is a size of A, Sb is a size of B. PM array[NPoints-1,6], another representation of merges, which is suited for dendrogram construction. It deals with rearranged points (permutation P is applied) and represents merges in a form which different from one used by Z. For each I from 0 to NPoints-2, I-th row of PM represents merge performed on two clusters C0 and C1. Here: * C0 contains points with indexes PM[I,0]...PM[I,1] * C1 contains points with indexes PM[I,2]...PM[I,3] * indexes stored in PM are given for dataset sorted according to permutation P * PM[I,1]=PM[I,2]-1 (only adjacent clusters are merged) * PM[I,0]<=PM[I,1], PM[I,2]<=PM[I,3], i.e. both clusters contain at least one point * heights of "subdendrograms" corresponding to C0/C1 are stored in PM[I,4] and PM[I,5]. Subdendrograms corresponding to single-point clusters have height=0. Dendrogram of the merge result has height H=max(H0,H1)+1. NOTE: there is one-to-one correspondence between merges described by Z and PM. I-th row of Z describes same merge of clusters as I-th row of PM, with "left" cluster from Z corresponding to the "left" one from PM. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/class ahcreport { ae_int_t terminationtype; ae_int_t npoints; integer_1d_array p; integer_2d_array z; integer_2d_array pz; integer_2d_array pm; real_1d_array mergedist; };
clusterizerstate class/************************************************************************* This structure is a clusterization engine. You should not try to access its fields directly. Use ALGLIB functions in order to work with this object. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/class clusterizerstate { };
kmeansreport class/************************************************************************* This structure is used to store results of the k-means clustering algorithm. Following information is always returned: * NPoints contains number of points in the original dataset * TerminationType contains completion code, negative on failure, positive on success * K contains number of clusters For positive TerminationType we return: * NFeatures contains number of variables in the original dataset * C, which contains centers found by algorithm * CIdx, which maps points of the original dataset to clusters FORMAL DESCRIPTION OF FIELDS: NPoints number of points, >=0 NFeatures number of variables, >=1 TerminationType completion code: * -5 if distance type is anything different from Euclidean metric * -3 for degenerate dataset: a) less than K distinct points, b) K=0 for non-empty dataset. * +1 for successful completion K number of clusters C array[K,NFeatures], rows of the array store centers CIdx array[NPoints], which contains cluster indexes IterationsCount actual number of iterations performed by clusterizer. If algorithm performed more than one random restart, total number of iterations is returned. Energy merit function, "energy", sum of squared deviations from cluster centers -- ALGLIB -- Copyright 27.11.2012 by Bochkanov Sergey *************************************************************************/class kmeansreport { ae_int_t npoints; ae_int_t nfeatures; ae_int_t terminationtype; ae_int_t iterationscount; double energy; ae_int_t k; real_2d_array c; integer_1d_array cidx; };
clusterizercreate function/************************************************************************* This function initializes clusterizer object. Newly initialized object is empty, i.e. it does not contain dataset. You should use it as follows: 1. creation 2. dataset is added with ClusterizerSetPoints() 3. additional parameters are set 3. clusterization is performed with one of the clustering functions -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizercreate(clusterizerstate& s);
clusterizergetdistances function/************************************************************************* This function returns distance matrix for dataset COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Agglomerative hierarchical clustering algorithm has two phases: ! distance matrix calculation and clustering itself. Only first phase ! (distance matrix calculation) is accelerated by Intel MKL and multi- ! threading. Thus, acceleration is significant only for medium or high- ! dimensional problems. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: XY - array[NPoints,NFeatures], dataset NPoints - number of points, >=0 NFeatures- number of features, >=1 DistType- distance function: * 0 Chebyshev distance (L-inf norm) * 1 city block distance (L1 norm) * 2 Euclidean distance (L2 norm, non-squared) * 10 Pearson correlation: dist(a,b) = 1-corr(a,b) * 11 Absolute Pearson correlation: dist(a,b) = 1-|corr(a,b)| * 12 Uncentered Pearson correlation (cosine of the angle): dist(a,b) = a'*b/(|a|*|b|) * 13 Absolute uncentered Pearson correlation dist(a,b) = |a'*b|/(|a|*|b|) * 20 Spearman rank correlation: dist(a,b) = 1-rankcorr(a,b) * 21 Absolute Spearman rank correlation dist(a,b) = 1-|rankcorr(a,b)| OUTPUT PARAMETERS: D - array[NPoints,NPoints], distance matrix (full matrix is returned, with lower and upper triangles) NOTE: different distance functions have different performance penalty: * Euclidean or Pearson correlation distances are the fastest ones * Spearman correlation distance function is a bit slower * city block and Chebyshev distances are order of magnitude slower The reason behing difference in performance is that correlation-based distance functions are computed using optimized linear algebra kernels, while Chebyshev and city block distance functions are computed using simple nested loops with two branches at each iteration. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizergetdistances( real_2d_array xy, ae_int_t npoints, ae_int_t nfeatures, ae_int_t disttype, real_2d_array& d); void alglib::smp_clusterizergetdistances( real_2d_array xy, ae_int_t npoints, ae_int_t nfeatures, ae_int_t disttype, real_2d_array& d);
clusterizergetkclusters function/************************************************************************* This function takes as input clusterization report Rep, desired clusters count K, and builds top K clusters from hierarchical clusterization tree. It returns assignment of points to clusters (array of cluster indexes). INPUT PARAMETERS: Rep - report from ClusterizerRunAHC() performed on XY K - desired number of clusters, 1<=K<=NPoints. K can be zero only when NPoints=0. OUTPUT PARAMETERS: CIdx - array[NPoints], I-th element contains cluster index (from 0 to K-1) for I-th point of the dataset. CZ - array[K]. This array allows to convert cluster indexes returned by this function to indexes used by Rep.Z. J-th cluster returned by this function corresponds to CZ[J]-th cluster stored in Rep.Z/PZ/PM. It is guaranteed that CZ[I]<CZ[I+1]. NOTE: K clusters built by this subroutine are assumed to have no hierarchy. Although they were obtained by manipulation with top K nodes of dendrogram (i.e. hierarchical decomposition of dataset), this function does not return information about hierarchy. Each of the clusters stand on its own. NOTE: Cluster indexes returned by this function does not correspond to indexes returned in Rep.Z/PZ/PM. Either you work with hierarchical representation of the dataset (dendrogram), or you work with "flat" representation returned by this function. Each of representations has its own clusters indexing system (former uses [0, 2*NPoints-2]), while latter uses [0..K-1]), although it is possible to perform conversion from one system to another by means of CZ array, returned by this function, which allows you to convert indexes stored in CIdx to the numeration system used by Rep.Z. NOTE: this subroutine is optimized for moderate values of K. Say, for K=5 it will perform many times faster than for K=100. Its worst-case performance is O(N*K), although in average case it perform better (up to O(N*log(K))). -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizergetkclusters( ahcreport rep, ae_int_t k, integer_1d_array& cidx, integer_1d_array& cz);
clusterizerrunahc function/************************************************************************* This function performs agglomerative hierarchical clustering COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Agglomerative hierarchical clustering algorithm has two phases: ! distance matrix calculation and clustering itself. Only first phase ! (distance matrix calculation) is accelerated by Intel MKL and multi- ! threading. Thus, acceleration is significant only for medium or high- ! dimensional problems. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() OUTPUT PARAMETERS: Rep - clustering results; see description of AHCReport structure for more information. NOTE 1: hierarchical clustering algorithms require large amounts of memory. In particular, this implementation needs sizeof(double)*NPoints^2 bytes, which are used to store distance matrix. In case we work with user-supplied matrix, this amount is multiplied by 2 (we have to store original matrix and to work with its copy). For example, problem with 10000 points would require 800M of RAM, even when working in a 1-dimensional space. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizerrunahc(clusterizerstate s, ahcreport& rep); void alglib::smp_clusterizerrunahc(clusterizerstate s, ahcreport& rep);
clusterizerrunkmeans function/************************************************************************* This function performs clustering by k-means++ algorithm. You may change algorithm properties by calling: * ClusterizerSetKMeansLimits() to change number of restarts or iterations * ClusterizerSetKMeansInit() to change initialization algorithm By default, one restart and unlimited number of iterations are used. Initialization algorithm is chosen automatically. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (can be used from C# and C++) ! * access to high-performance C++ core (actual for C# users) ! ! K-means clustering algorithm has two phases: selection of initial ! centers and clustering itself. ALGLIB parallelizes both phases. ! Parallel version is optimized for the following scenario: medium or ! high-dimensional problem (20 or more dimensions) with large number of ! points and clusters. However, some speed-up can be obtained even when ! assumptions above are violated. ! ! As for native-vs-managed comparison, working with native core brings ! 30-40% improvement in speed over pure C# version of ALGLIB. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() K - number of clusters, K>=0. K can be zero only when algorithm is called for empty dataset, in this case completion code is set to success (+1). If K=0 and dataset size is non-zero, we can not meaningfully assign points to some center (there are no centers because K=0) and return -3 as completion code (failure). OUTPUT PARAMETERS: Rep - clustering results; see description of KMeansReport structure for more information. NOTE 1: k-means clustering can be performed only for datasets with Euclidean distance function. Algorithm will return negative completion code in Rep.TerminationType in case dataset was added to clusterizer with DistType other than Euclidean (or dataset was specified by distance matrix instead of explicitly given points). -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizerrunkmeans( clusterizerstate s, ae_int_t k, kmeansreport& rep); void alglib::smp_clusterizerrunkmeans( clusterizerstate s, ae_int_t k, kmeansreport& rep);
clusterizerseparatedbycorr function/************************************************************************* This function accepts AHC report Rep, desired maximum intercluster correlation and returns top clusters from hierarchical clusterization tree which are separated by correlation R or LOWER. It returns assignment of points to clusters (array of cluster indexes). There is one more function with similar name - ClusterizerSeparatedByDist, which returns clusters with intercluster distance equal to R or HIGHER (note: higher for distance, lower for correlation). INPUT PARAMETERS: Rep - report from ClusterizerRunAHC() performed on XY R - desired maximum intercluster correlation, -1<=R<=+1 OUTPUT PARAMETERS: K - number of clusters, 1<=K<=NPoints CIdx - array[NPoints], I-th element contains cluster index (from 0 to K-1) for I-th point of the dataset. CZ - array[K]. This array allows to convert cluster indexes returned by this function to indexes used by Rep.Z. J-th cluster returned by this function corresponds to CZ[J]-th cluster stored in Rep.Z/PZ/PM. It is guaranteed that CZ[I]<CZ[I+1]. NOTE: K clusters built by this subroutine are assumed to have no hierarchy. Although they were obtained by manipulation with top K nodes of dendrogram (i.e. hierarchical decomposition of dataset), this function does not return information about hierarchy. Each of the clusters stand on its own. NOTE: Cluster indexes returned by this function does not correspond to indexes returned in Rep.Z/PZ/PM. Either you work with hierarchical representation of the dataset (dendrogram), or you work with "flat" representation returned by this function. Each of representations has its own clusters indexing system (former uses [0, 2*NPoints-2]), while latter uses [0..K-1]), although it is possible to perform conversion from one system to another by means of CZ array, returned by this function, which allows you to convert indexes stored in CIdx to the numeration system used by Rep.Z. NOTE: this subroutine is optimized for moderate values of K. Say, for K=5 it will perform many times faster than for K=100. Its worst-case performance is O(N*K), although in average case it perform better (up to O(N*log(K))). -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizerseparatedbycorr( ahcreport rep, double r, ae_int_t& k, integer_1d_array& cidx, integer_1d_array& cz);
clusterizerseparatedbydist function/************************************************************************* This function accepts AHC report Rep, desired minimum intercluster distance and returns top clusters from hierarchical clusterization tree which are separated by distance R or HIGHER. It returns assignment of points to clusters (array of cluster indexes). There is one more function with similar name - ClusterizerSeparatedByCorr, which returns clusters with intercluster correlation equal to R or LOWER (note: higher for distance, lower for correlation). INPUT PARAMETERS: Rep - report from ClusterizerRunAHC() performed on XY R - desired minimum intercluster distance, R>=0 OUTPUT PARAMETERS: K - number of clusters, 1<=K<=NPoints CIdx - array[NPoints], I-th element contains cluster index (from 0 to K-1) for I-th point of the dataset. CZ - array[K]. This array allows to convert cluster indexes returned by this function to indexes used by Rep.Z. J-th cluster returned by this function corresponds to CZ[J]-th cluster stored in Rep.Z/PZ/PM. It is guaranteed that CZ[I]<CZ[I+1]. NOTE: K clusters built by this subroutine are assumed to have no hierarchy. Although they were obtained by manipulation with top K nodes of dendrogram (i.e. hierarchical decomposition of dataset), this function does not return information about hierarchy. Each of the clusters stand on its own. NOTE: Cluster indexes returned by this function does not correspond to indexes returned in Rep.Z/PZ/PM. Either you work with hierarchical representation of the dataset (dendrogram), or you work with "flat" representation returned by this function. Each of representations has its own clusters indexing system (former uses [0, 2*NPoints-2]), while latter uses [0..K-1]), although it is possible to perform conversion from one system to another by means of CZ array, returned by this function, which allows you to convert indexes stored in CIdx to the numeration system used by Rep.Z. NOTE: this subroutine is optimized for moderate values of K. Say, for K=5 it will perform many times faster than for K=100. Its worst-case performance is O(N*K), although in average case it perform better (up to O(N*log(K))). -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizerseparatedbydist( ahcreport rep, double r, ae_int_t& k, integer_1d_array& cidx, integer_1d_array& cz);
clusterizersetahcalgo function/************************************************************************* This function sets agglomerative hierarchical clustering algorithm INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() Algo - algorithm type: * 0 complete linkage (default algorithm) * 1 single linkage * 2 unweighted average linkage * 3 weighted average linkage * 4 Ward's method NOTE: Ward's method works correctly only with Euclidean distance, that's why algorithm will return negative termination code (failure) for any other distance type. It is possible, however, to use this method with user-supplied distance matrix. It is your responsibility to pass one which was calculated with Euclidean distance function. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizersetahcalgo(clusterizerstate s, ae_int_t algo);
clusterizersetdistances function/************************************************************************* This function adds dataset given by distance matrix to the clusterizer structure. It is important that dataset is not given explicitly - only distance matrix is given. This function overrides all previous calls of ClusterizerSetPoints() or ClusterizerSetDistances(). INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() D - array[NPoints,NPoints], distance matrix given by its upper or lower triangle (main diagonal is ignored because its entries are expected to be zero). NPoints - number of points IsUpper - whether upper or lower triangle of D is given. NOTE 1: different clustering algorithms have different limitations: * agglomerative hierarchical clustering algorithms may be used with any kind of distance metric, including one which is given by distance matrix * k-means++ clustering algorithm may be used only with Euclidean distance function and explicitly given points - it can not be used with dataset given by distance matrix Thus, if you call this function, you will be unable to use k-means clustering algorithm to process your problem. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizersetdistances( clusterizerstate s, real_2d_array d, bool isupper); void alglib::clusterizersetdistances( clusterizerstate s, real_2d_array d, ae_int_t npoints, bool isupper);
Examples: [1]
clusterizersetkmeansinit function/************************************************************************* This function sets k-means initialization algorithm. Several different algorithms can be chosen, including k-means++. INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() InitAlgo- initialization algorithm: * 0 automatic selection ( different versions of ALGLIB may select different algorithms) * 1 random initialization * 2 k-means++ initialization (best quality of initial centers, but long non-parallelizable initialization phase with bad cache locality) * 3 "fast-greedy" algorithm with efficient, easy to parallelize initialization. Quality of initial centers is somewhat worse than that of k-means++. This algorithm is a default one in the current version of ALGLIB. *-1 "debug" algorithm which always selects first K rows of dataset; this algorithm is used for debug purposes only. Do not use it in the industrial code! -- ALGLIB -- Copyright 21.01.2015 by Bochkanov Sergey *************************************************************************/void alglib::clusterizersetkmeansinit( clusterizerstate s, ae_int_t initalgo);
clusterizersetkmeanslimits function/************************************************************************* This function sets k-means properties: number of restarts and maximum number of iterations per one run. INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() Restarts- restarts count, >=1. k-means++ algorithm performs several restarts and chooses best set of centers (one with minimum squared distance). MaxIts - maximum number of k-means iterations performed during one run. >=0, zero value means that algorithm performs unlimited number of iterations. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizersetkmeanslimits( clusterizerstate s, ae_int_t restarts, ae_int_t maxits);
clusterizersetpoints function/************************************************************************* This function adds dataset to the clusterizer structure. This function overrides all previous calls of ClusterizerSetPoints() or ClusterizerSetDistances(). INPUT PARAMETERS: S - clusterizer state, initialized by ClusterizerCreate() XY - array[NPoints,NFeatures], dataset NPoints - number of points, >=0 NFeatures- number of features, >=1 DistType- distance function: * 0 Chebyshev distance (L-inf norm) * 1 city block distance (L1 norm) * 2 Euclidean distance (L2 norm), non-squared * 10 Pearson correlation: dist(a,b) = 1-corr(a,b) * 11 Absolute Pearson correlation: dist(a,b) = 1-|corr(a,b)| * 12 Uncentered Pearson correlation (cosine of the angle): dist(a,b) = a'*b/(|a|*|b|) * 13 Absolute uncentered Pearson correlation dist(a,b) = |a'*b|/(|a|*|b|) * 20 Spearman rank correlation: dist(a,b) = 1-rankcorr(a,b) * 21 Absolute Spearman rank correlation dist(a,b) = 1-|rankcorr(a,b)| NOTE 1: different distance functions have different performance penalty: * Euclidean or Pearson correlation distances are the fastest ones * Spearman correlation distance function is a bit slower * city block and Chebyshev distances are order of magnitude slower The reason behing difference in performance is that correlation-based distance functions are computed using optimized linear algebra kernels, while Chebyshev and city block distance functions are computed using simple nested loops with two branches at each iteration. NOTE 2: different clustering algorithms have different limitations: * agglomerative hierarchical clustering algorithms may be used with any kind of distance metric * k-means++ clustering algorithm may be used only with Euclidean distance function Thus, list of specific clustering algorithms you may use depends on distance function you specify when you set your dataset. -- ALGLIB -- Copyright 10.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::clusterizersetpoints( clusterizerstate s, real_2d_array xy, ae_int_t disttype); void alglib::clusterizersetpoints( clusterizerstate s, real_2d_array xy, ae_int_t npoints, ae_int_t nfeatures, ae_int_t disttype);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// The very simple clusterization example
//
// We have a set of points in 2D space:
// (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
//
// |
// | P3
// |
// | P1
// | P4
// | P0 P2
// |-------------------------
//
// We want to perform Agglomerative Hierarchic Clusterization (AHC),
// using complete linkage (default algorithm) and Euclidean distance
// (default metric).
//
// In order to do that, we:
// * create clusterizer with clusterizercreate()
// * set points XY and metric (2=Euclidean) with clusterizersetpoints()
// * run AHC algorithm with clusterizerrunahc
//
// You may see that clusterization itself is a minor part of the example,
// most of which is dominated by comments :)
//
clusterizerstate s;
ahcreport rep;
real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";
clusterizercreate(s);
clusterizersetpoints(s, xy, 2);
clusterizerrunahc(s, rep);
//
// Now we've built our clusterization tree. Rep.z contains information which
// is required to build dendrogram. I-th row of rep.z represents one merge
// operation, with first cluster to merge having index rep.z[I,0] and second
// one having index rep.z[I,1]. Merge result has index NPoints+I.
//
// Clusters with indexes less than NPoints are single-point initial clusters,
// while ones with indexes from NPoints to 2*NPoints-2 are multi-point
// clusters created during merges.
//
// In our example, Z=[[2,4], [0,1], [3,6], [5,7]]
//
// It means that:
// * first, we merge C2=(P2) and C4=(P4), and create C5=(P2,P4)
// * then, we merge C2=(P0) and C1=(P1), and create C6=(P0,P1)
// * then, we merge C3=(P3) and C6=(P0,P1), and create C7=(P0,P1,P3)
// * finally, we merge C5 and C7 and create C8=(P0,P1,P2,P3,P4)
//
// Thus, we have following dendrogram:
//
// ------8-----
// | |
// | ----7----
// | | |
// ---5--- | ---6---
// | | | | |
// P2 P4 P3 P0 P1
//
printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[2,4],[0,1],[3,6],[5,7]]
//
// We've built dendrogram above by reordering our dataset.
//
// Without such reordering it would be impossible to build dendrogram without
// intersections. Luckily, ahcreport structure contains two additional fields
// which help to build dendrogram from your data:
// * rep.p, which contains permutation applied to dataset
// * rep.pm, which contains another representation of merges
//
// In our example we have:
// * P=[3,4,0,2,1]
// * PZ=[[0,0,1,1,0,0],[3,3,4,4,0,0],[2,2,3,4,0,1],[0,1,2,4,1,2]]
//
// Permutation array P tells us that P0 should be moved to position 3,
// P1 moved to position 4, P2 moved to position 0 and so on:
//
// (P0 P1 P2 P3 P4) => (P2 P4 P3 P0 P1)
//
// Merges array PZ tells us how to perform merges on the sorted dataset.
// One row of PZ corresponds to one merge operations, with first pair of
// elements denoting first of the clusters to merge (start index, end
// index) and next pair of elements denoting second of the clusters to
// merge. Clusters being merged are always adjacent, with first one on
// the left and second one on the right.
//
// For example, first row of PZ tells us that clusters [0,0] and [1,1] are
// merged (single-point clusters, with first one containing P2 and second
// one containing P4). Third row of PZ tells us that we merge one single-
// point cluster [2,2] with one two-point cluster [3,4].
//
// There are two more elements in each row of PZ. These are the helper
// elements, which denote HEIGHT (not size) of left and right subdendrograms.
// For example, according to PZ, first two merges are performed on clusterization
// trees of height 0, while next two merges are performed on 0-1 and 1-2
// pairs of trees correspondingly.
//
printf("%s\n", rep.p.tostring().c_str()); // EXPECTED: [3,4,0,2,1]
printf("%s\n", rep.pm.tostring().c_str()); // EXPECTED: [[0,0,1,1,0,0],[3,3,4,4,0,0],[2,2,3,4,0,1],[0,1,2,4,1,2]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We have three points in 4D space:
// (P0,P1,P2) = ((1, 2, 1, 2), (6, 7, 6, 7), (7, 6, 7, 6))
//
// We want to try clustering them with different distance functions.
// Distance function is chosen when we add dataset to the clusterizer.
// We can choose several distance types - Euclidean, city block, Chebyshev,
// several correlation measures or user-supplied distance matrix.
//
// Here we'll try three distances: Euclidean, Pearson correlation,
// user-supplied distance matrix. Different distance functions lead
// to different choices being made by algorithm during clustering.
//
clusterizerstate s;
ahcreport rep;
ae_int_t disttype;
real_2d_array xy = "[[1, 2, 1, 2], [6, 7, 6, 7], [7, 6, 7, 6]]";
clusterizercreate(s);
// With Euclidean distance function (disttype=2) two closest points
// are P1 and P2, thus:
// * first, we merge P1 and P2 to form C3=[P1,P2]
// * second, we merge P0 and C3 to form C4=[P0,P1,P2]
disttype = 2;
clusterizersetpoints(s, xy, disttype);
clusterizerrunahc(s, rep);
printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[1,2],[0,3]]
// With Pearson correlation distance function (disttype=10) situation
// is different - distance between P0 and P1 is zero, thus:
// * first, we merge P0 and P1 to form C3=[P0,P1]
// * second, we merge P2 and C3 to form C4=[P0,P1,P2]
disttype = 10;
clusterizersetpoints(s, xy, disttype);
clusterizerrunahc(s, rep);
printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[0,1],[2,3]]
// Finally, we try clustering with user-supplied distance matrix:
// [ 0 3 1 ]
// P = [ 3 0 3 ], where P[i,j] = dist(Pi,Pj)
// [ 1 3 0 ]
//
// * first, we merge P0 and P2 to form C3=[P0,P2]
// * second, we merge P1 and C3 to form C4=[P0,P1,P2]
real_2d_array d = "[[0,3,1],[3,0,3],[1,3,0]]";
clusterizersetdistances(s, d, true);
clusterizerrunahc(s, rep);
printf("%s\n", rep.z.tostring().c_str()); // EXPECTED: [[0,2],[1,3]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We have a set of points in 2D space:
// (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
//
// |
// | P3
// |
// | P1
// | P4
// | P0 P2
// |-------------------------
//
// We perform Agglomerative Hierarchic Clusterization (AHC) and we want
// to get top K clusters from clusterization tree for different K.
//
clusterizerstate s;
ahcreport rep;
real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";
integer_1d_array cidx;
integer_1d_array cz;
clusterizercreate(s);
clusterizersetpoints(s, xy, 2);
clusterizerrunahc(s, rep);
// with K=5, every points is assigned to its own cluster:
// C0=P0, C1=P1 and so on...
clusterizergetkclusters(rep, 5, cidx, cz);
printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,1,2,3,4]
// with K=1 we have one large cluster C0=[P0,P1,P2,P3,P4,P5]
clusterizergetkclusters(rep, 1, cidx, cz);
printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,0,0,0,0]
// with K=3 we have three clusters C0=[P3], C1=[P2,P4], C2=[P0,P1]
clusterizergetkclusters(rep, 3, cidx, cz);
printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [2,2,1,0,1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// The very simple clusterization example
//
// We have a set of points in 2D space:
// (P0,P1,P2,P3,P4) = ((1,1),(1,2),(4,1),(2,3),(4,1.5))
//
// |
// | P3
// |
// | P1
// | P4
// | P0 P2
// |-------------------------
//
// We want to perform k-means++ clustering with K=2.
//
// In order to do that, we:
// * create clusterizer with clusterizercreate()
// * set points XY and metric (must be Euclidean, distype=2) with clusterizersetpoints()
// * (optional) set number of restarts from random positions to 5
// * run k-means algorithm with clusterizerrunkmeans()
//
// You may see that clusterization itself is a minor part of the example,
// most of which is dominated by comments :)
//
clusterizerstate s;
kmeansreport rep;
real_2d_array xy = "[[1,1],[1,2],[4,1],[2,3],[4,1.5]]";
clusterizercreate(s);
clusterizersetpoints(s, xy, 2);
clusterizersetkmeanslimits(s, 5, 0);
clusterizerrunkmeans(s, 2, rep);
//
// We've performed clusterization, and it succeeded (completion code is +1).
//
// Now first center is stored in the first row of rep.c, second one is stored
// in the second row. rep.cidx can be used to determine which center is
// closest to some specific point of the dataset.
//
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
// We called clusterizersetpoints() with disttype=2 because k-means++
// algorithm does NOT support metrics other than Euclidean. But what if we
// try to use some other metric?
//
// We change metric type by calling clusterizersetpoints() one more time,
// and try to run k-means algo again. It fails.
//
clusterizersetpoints(s, xy, 0);
clusterizerrunkmeans(s, 2, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: -5
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We have a set of points in 1D space:
// (P0,P1,P2,P3,P4) = (1, 3, 10, 16, 20)
//
// We want to perform Agglomerative Hierarchic Clusterization (AHC),
// using either complete or single linkage and Euclidean distance
// (default metric).
//
// First two steps merge P0/P1 and P3/P4 independently of the linkage type.
// However, third step depends on linkage type being used:
// * in case of complete linkage P2=10 is merged with [P0,P1]
// * in case of single linkage P2=10 is merged with [P3,P4]
//
clusterizerstate s;
ahcreport rep;
real_2d_array xy = "[[1],[3],[10],[16],[20]]";
integer_1d_array cidx;
integer_1d_array cz;
clusterizercreate(s);
clusterizersetpoints(s, xy, 2);
// use complete linkage, reduce set down to 2 clusters.
// print clusterization with clusterizergetkclusters(2).
// P2 must belong to [P0,P1]
clusterizersetahcalgo(s, 0);
clusterizerrunahc(s, rep);
clusterizergetkclusters(rep, 2, cidx, cz);
printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [1,1,1,0,0]
// use single linkage, reduce set down to 2 clusters.
// print clusterization with clusterizergetkclusters(2).
// P2 must belong to [P2,P3]
clusterizersetahcalgo(s, 1);
clusterizerrunahc(s, rep);
clusterizergetkclusters(rep, 2, cidx, cz);
printf("%s\n", cidx.tostring().c_str()); // EXPECTED: [0,0,1,1,1]
return 0;
}
conv subpackageconvc1d function/************************************************************************* 1-dimensional complex convolution. For given A/B returns conv(A,B) (non-circular). Subroutine can automatically choose between three implementations: straightforward O(M*N) formula for very small N (or M), overlap-add algorithm for cases where max(M,N) is significantly larger than min(M,N), but O(M*N) algorithm is too slow, and general FFT-based formula for cases where two previois algorithms are too slow. Algorithm has max(M,N)*log(max(M,N)) complexity for any M/N. INPUT PARAMETERS A - array[0..M-1] - complex function to be transformed M - problem size B - array[0..N-1] - complex function to be transformed N - problem size OUTPUT PARAMETERS R - convolution: A*B. array[0..N+M-2]. NOTE: It is assumed that A is zero at T<0, B is zero too. If one or both functions have non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convc1d( complex_1d_array a, ae_int_t m, complex_1d_array b, ae_int_t n, complex_1d_array& r);
convc1dcircular function/************************************************************************* 1-dimensional circular complex convolution. For given S/R returns conv(S,R) (circular). Algorithm has linearithmic complexity for any M/N. IMPORTANT: normal convolution is commutative, i.e. it is symmetric - conv(A,B)=conv(B,A). Cyclic convolution IS NOT. One function - S - is a signal, periodic function, and another - R - is a response, non-periodic function with limited length. INPUT PARAMETERS S - array[0..M-1] - complex periodic signal M - problem size B - array[0..N-1] - complex non-periodic response N - problem size OUTPUT PARAMETERS R - convolution: A*B. array[0..M-1]. NOTE: It is assumed that B is zero at T<0. If it has non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convc1dcircular( complex_1d_array s, ae_int_t m, complex_1d_array r, ae_int_t n, complex_1d_array& c);
convc1dcircularinv function/************************************************************************* 1-dimensional circular complex deconvolution (inverse of ConvC1DCircular()). Algorithm has M*log(M)) complexity for any M (composite or prime). INPUT PARAMETERS A - array[0..M-1] - convolved periodic signal, A = conv(R, B) M - convolved signal length B - array[0..N-1] - non-periodic response N - response length OUTPUT PARAMETERS R - deconvolved signal. array[0..M-1]. NOTE: deconvolution is unstable process and may result in division by zero (if your response function is degenerate, i.e. has zero Fourier coefficient). NOTE: It is assumed that B is zero at T<0. If it has non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convc1dcircularinv( complex_1d_array a, ae_int_t m, complex_1d_array b, ae_int_t n, complex_1d_array& r);
convc1dinv function/************************************************************************* 1-dimensional complex non-circular deconvolution (inverse of ConvC1D()). Algorithm has M*log(M)) complexity for any M (composite or prime). INPUT PARAMETERS A - array[0..M-1] - convolved signal, A = conv(R, B) M - convolved signal length B - array[0..N-1] - response N - response length, N<=M OUTPUT PARAMETERS R - deconvolved signal. array[0..M-N]. NOTE: deconvolution is unstable process and may result in division by zero (if your response function is degenerate, i.e. has zero Fourier coefficient). NOTE: It is assumed that A is zero at T<0, B is zero too. If one or both functions have non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convc1dinv( complex_1d_array a, ae_int_t m, complex_1d_array b, ae_int_t n, complex_1d_array& r);
convr1d function/************************************************************************* 1-dimensional real convolution. Analogous to ConvC1D(), see ConvC1D() comments for more details. INPUT PARAMETERS A - array[0..M-1] - real function to be transformed M - problem size B - array[0..N-1] - real function to be transformed N - problem size OUTPUT PARAMETERS R - convolution: A*B. array[0..N+M-2]. NOTE: It is assumed that A is zero at T<0, B is zero too. If one or both functions have non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convr1d( real_1d_array a, ae_int_t m, real_1d_array b, ae_int_t n, real_1d_array& r);
convr1dcircular function/************************************************************************* 1-dimensional circular real convolution. Analogous to ConvC1DCircular(), see ConvC1DCircular() comments for more details. INPUT PARAMETERS S - array[0..M-1] - real signal M - problem size B - array[0..N-1] - real response N - problem size OUTPUT PARAMETERS R - convolution: A*B. array[0..M-1]. NOTE: It is assumed that B is zero at T<0. If it has non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convr1dcircular( real_1d_array s, ae_int_t m, real_1d_array r, ae_int_t n, real_1d_array& c);
convr1dcircularinv function/************************************************************************* 1-dimensional complex deconvolution (inverse of ConvC1D()). Algorithm has M*log(M)) complexity for any M (composite or prime). INPUT PARAMETERS A - array[0..M-1] - convolved signal, A = conv(R, B) M - convolved signal length B - array[0..N-1] - response N - response length OUTPUT PARAMETERS R - deconvolved signal. array[0..M-N]. NOTE: deconvolution is unstable process and may result in division by zero (if your response function is degenerate, i.e. has zero Fourier coefficient). NOTE: It is assumed that B is zero at T<0. If it has non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convr1dcircularinv( real_1d_array a, ae_int_t m, real_1d_array b, ae_int_t n, real_1d_array& r);
convr1dinv function/************************************************************************* 1-dimensional real deconvolution (inverse of ConvC1D()). Algorithm has M*log(M)) complexity for any M (composite or prime). INPUT PARAMETERS A - array[0..M-1] - convolved signal, A = conv(R, B) M - convolved signal length B - array[0..N-1] - response N - response length, N<=M OUTPUT PARAMETERS R - deconvolved signal. array[0..M-N]. NOTE: deconvolution is unstable process and may result in division by zero (if your response function is degenerate, i.e. has zero Fourier coefficient). NOTE: It is assumed that A is zero at T<0, B is zero too. If one or both functions have non-zero values at negative T's, you can still use this subroutine - just shift its result correspondingly. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::convr1dinv( real_1d_array a, ae_int_t m, real_1d_array b, ae_int_t n, real_1d_array& r);
corr subpackagecorrc1d function/************************************************************************* 1-dimensional complex cross-correlation. For given Pattern/Signal returns corr(Pattern,Signal) (non-circular). Correlation is calculated using reduction to convolution. Algorithm with max(N,N)*log(max(N,N)) complexity is used (see ConvC1D() for more info about performance). IMPORTANT: for historical reasons subroutine accepts its parameters in reversed order: CorrC1D(Signal, Pattern) = Pattern x Signal (using traditional definition of cross-correlation, denoting cross-correlation as "x"). INPUT PARAMETERS Signal - array[0..N-1] - complex function to be transformed, signal containing pattern N - problem size Pattern - array[0..M-1] - complex function to be transformed, pattern to search withing signal M - problem size OUTPUT PARAMETERS R - cross-correlation, array[0..N+M-2]: * positive lags are stored in R[0..N-1], R[i] = sum(conj(pattern[j])*signal[i+j] * negative lags are stored in R[N..N+M-2], R[N+M-1-i] = sum(conj(pattern[j])*signal[-i+j] NOTE: It is assumed that pattern domain is [0..M-1]. If Pattern is non-zero on [-K..M-1], you can still use this subroutine, just shift result by K. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::corrc1d( complex_1d_array signal, ae_int_t n, complex_1d_array pattern, ae_int_t m, complex_1d_array& r);
corrc1dcircular function/************************************************************************* 1-dimensional circular complex cross-correlation. For given Pattern/Signal returns corr(Pattern,Signal) (circular). Algorithm has linearithmic complexity for any M/N. IMPORTANT: for historical reasons subroutine accepts its parameters in reversed order: CorrC1DCircular(Signal, Pattern) = Pattern x Signal (using traditional definition of cross-correlation, denoting cross-correlation as "x"). INPUT PARAMETERS Signal - array[0..N-1] - complex function to be transformed, periodic signal containing pattern N - problem size Pattern - array[0..M-1] - complex function to be transformed, non-periodic pattern to search withing signal M - problem size OUTPUT PARAMETERS R - convolution: A*B. array[0..M-1]. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::corrc1dcircular( complex_1d_array signal, ae_int_t m, complex_1d_array pattern, ae_int_t n, complex_1d_array& c);
corrr1d function/************************************************************************* 1-dimensional real cross-correlation. For given Pattern/Signal returns corr(Pattern,Signal) (non-circular). Correlation is calculated using reduction to convolution. Algorithm with max(N,N)*log(max(N,N)) complexity is used (see ConvC1D() for more info about performance). IMPORTANT: for historical reasons subroutine accepts its parameters in reversed order: CorrR1D(Signal, Pattern) = Pattern x Signal (using traditional definition of cross-correlation, denoting cross-correlation as "x"). INPUT PARAMETERS Signal - array[0..N-1] - real function to be transformed, signal containing pattern N - problem size Pattern - array[0..M-1] - real function to be transformed, pattern to search withing signal M - problem size OUTPUT PARAMETERS R - cross-correlation, array[0..N+M-2]: * positive lags are stored in R[0..N-1], R[i] = sum(pattern[j]*signal[i+j] * negative lags are stored in R[N..N+M-2], R[N+M-1-i] = sum(pattern[j]*signal[-i+j] NOTE: It is assumed that pattern domain is [0..M-1]. If Pattern is non-zero on [-K..M-1], you can still use this subroutine, just shift result by K. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::corrr1d( real_1d_array signal, ae_int_t n, real_1d_array pattern, ae_int_t m, real_1d_array& r);
corrr1dcircular function/************************************************************************* 1-dimensional circular real cross-correlation. For given Pattern/Signal returns corr(Pattern,Signal) (circular). Algorithm has linearithmic complexity for any M/N. IMPORTANT: for historical reasons subroutine accepts its parameters in reversed order: CorrR1DCircular(Signal, Pattern) = Pattern x Signal (using traditional definition of cross-correlation, denoting cross-correlation as "x"). INPUT PARAMETERS Signal - array[0..N-1] - real function to be transformed, periodic signal containing pattern N - problem size Pattern - array[0..M-1] - real function to be transformed, non-periodic pattern to search withing signal M - problem size OUTPUT PARAMETERS R - convolution: A*B. array[0..M-1]. -- ALGLIB -- Copyright 21.07.2009 by Bochkanov Sergey *************************************************************************/void alglib::corrr1dcircular( real_1d_array signal, ae_int_t m, real_1d_array pattern, ae_int_t n, real_1d_array& c);
correlationtests subpackagepearsoncorrelationsignificance function/************************************************************************* Pearson's correlation coefficient significance test This test checks hypotheses about whether X and Y are samples of two continuous distributions having zero correlation or whether their correlation is non-zero. The following tests are performed: * two-tailed test (null hypothesis - X and Y have zero correlation) * left-tailed test (null hypothesis - the correlation coefficient is greater than or equal to 0) * right-tailed test (null hypothesis - the correlation coefficient is less than or equal to 0). Requirements: * the number of elements in each sample is not less than 5 * normality of distributions of X and Y. Input parameters: R - Pearson's correlation coefficient for X and Y N - number of elements in samples, N>=5. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/void alglib::pearsoncorrelationsignificance( double r, ae_int_t n, double& bothtails, double& lefttail, double& righttail);
spearmanrankcorrelationsignificance function/************************************************************************* Spearman's rank correlation coefficient significance test This test checks hypotheses about whether X and Y are samples of two continuous distributions having zero correlation or whether their correlation is non-zero. The following tests are performed: * two-tailed test (null hypothesis - X and Y have zero correlation) * left-tailed test (null hypothesis - the correlation coefficient is greater than or equal to 0) * right-tailed test (null hypothesis - the correlation coefficient is less than or equal to 0). Requirements: * the number of elements in each sample is not less than 5. The test is non-parametric and doesn't require distributions X and Y to be normal. Input parameters: R - Spearman's rank correlation coefficient for X and Y N - number of elements in samples, N>=5. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/void alglib::spearmanrankcorrelationsignificance( double r, ae_int_t n, double& bothtails, double& lefttail, double& righttail);
datacomp subpackagekmeansgenerate function/************************************************************************* k-means++ clusterization. Backward compatibility function, we recommend to use CLUSTERING subpackage as better replacement. -- ALGLIB -- Copyright 21.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::kmeansgenerate( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t k, ae_int_t restarts, ae_int_t& info, real_2d_array& c, integer_1d_array& xyc);
dawson subpackagedawsonintegral function/************************************************************************* Dawson's Integral Approximates the integral x - 2 | | 2 dawsn(x) = exp( -x ) | exp( t ) dt | | - 0 Three different rational approximations are employed, for the intervals 0 to 3.25; 3.25 to 6.25; and 6.25 up. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,10 10000 6.9e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/double alglib::dawsonintegral(double x);
densesolver subpackagedensesolverlsreport class/************************************************************************* *************************************************************************/class densesolverlsreport { double r2; real_2d_array cx; ae_int_t n; ae_int_t k; };
densesolverreport class/************************************************************************* *************************************************************************/class densesolverreport { double r1; double rinf; };
cmatrixlusolve function/************************************************************************* Complex dense linear solver for A*x=b with complex N*N A given by its LU decomposition and N*1 vectors x and b. This is "slow-but-robust" version of the complex linear solver with additional features which add significant performance overhead. Faster version is CMatrixLUSolveFast() function. Algorithm features: * automatic detection of degenerate cases * O(N^2) complexity * condition number estimation No iterative refinement is provided because exact form of original matrix is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in 10-15x performance penalty when compared ! with "fast" version which just calls triangular solver. ! ! This performance penalty is insignificant when compared with ! cost of large LU decomposition. However, if you call this ! function many times for the same left side, this overhead ! BECOMES significant. It also becomes significant for small- ! scale problems. ! ! In such cases we strongly recommend you to use faster solver, ! CMatrixLUSolveFast() function. INPUT PARAMETERS LUA - array[0..N-1,0..N-1], LU decomposition, CMatrixLU result P - array[0..N-1], pivots array, CMatrixLU result N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixlusolve( complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x);
cmatrixlusolvefast function/************************************************************************* Complex dense linear solver for A*x=b with N*N complex A given by its LU decomposition and N*1 vectors x and b. This is fast lightweight version of solver, which is significantly faster than CMatrixLUSolve(), but does not provide additional information (like condition numbers). Algorithm features: * O(N^2) complexity * no additional time-consuming features, just triangular solver INPUT PARAMETERS LUA - array[0..N-1,0..N-1], LU decomposition, CMatrixLU result P - array[0..N-1], pivots array, CMatrixLU result N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). * -1 N<=0 was passed * 1 task is solved B - array[N]: * info>0 => overwritten by solution * info=-3 => filled by zeros NOTE: unlike CMatrixLUSolve(), this function does NOT check for near-degeneracy of input matrix. It checks for EXACT degeneracy, because this check is easy to do. However, very badly conditioned matrices may went unnoticed. -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixlusolvefast( complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_1d_array& b, ae_int_t& info);
cmatrixlusolvem function/************************************************************************* Dense solver for A*X=B with N*N complex A given by its LU decomposition, and N*M matrices X and B (multiple right sides). "Slow-but-feature-rich" version of the solver. Algorithm features: * automatic detection of degenerate cases * O(M*N^2) complexity * condition number estimation No iterative refinement is provided because exact form of original matrix is not known to subroutine. Use CMatrixSolve or CMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just calls triangular ! solver. ! ! This performance penalty is especially apparent when you use ! ALGLIB parallel capabilities (condition number estimation is ! inherently sequential). It also becomes significant for ! small-scale problems. ! ! In such cases we strongly recommend you to use faster solver, ! CMatrixLUSolveMFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Triangular solver is relatively easy to parallelize. ! However, parallelization will be efficient only for large number of ! right parts M. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS LUA - array[0..N-1,0..N-1], LU decomposition, RMatrixLU result P - array[0..N-1], pivots array, RMatrixLU result N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixlusolvem( complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x); void alglib::smp_cmatrixlusolvem( complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x);
cmatrixlusolvemfast function/************************************************************************* Dense solver for A*X=B with N*N complex A given by its LU decomposition, and N*M matrices X and B (multiple right sides). "Fast-but-lightweight" version of the solver. Algorithm features: * O(M*N^2) complexity * no additional time-consuming features COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Triangular solver is relatively easy to parallelize. ! However, parallelization will be efficient only for large number of ! right parts M. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS LUA - array[0..N-1,0..N-1], LU decomposition, RMatrixLU result P - array[0..N-1], pivots array, RMatrixLU result N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). * -1 N<=0 was passed * 1 task is solved B - array[N,M]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixlusolvemfast( complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_cmatrixlusolvemfast( complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_2d_array& b, ae_int_t m, ae_int_t& info);
cmatrixmixedsolve function/************************************************************************* Dense solver. Same as RMatrixMixedSolve(), but for complex matrices. Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(N^2) complexity INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix LUA - array[0..N-1,0..N-1], LU decomposition, CMatrixLU result P - array[0..N-1], pivots array, CMatrixLU result N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixmixedsolve( complex_2d_array a, complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x);
cmatrixmixedsolvem function/************************************************************************* Dense solver. Same as RMatrixMixedSolveM(), but for complex matrices. Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(M*N^2) complexity INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix LUA - array[0..N-1,0..N-1], LU decomposition, CMatrixLU result P - array[0..N-1], pivots array, CMatrixLU result N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixmixedsolvem( complex_2d_array a, complex_2d_array lua, integer_1d_array p, ae_int_t n, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x);
cmatrixsolve function/************************************************************************* Complex dense solver for A*x=B with N*N complex matrix A and N*1 complex vectors x and b. "Slow-but-feature-rich" version of the solver. Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(N^3) complexity IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system ! and performs iterative refinement, which results in ! significant performance penalty when compared with "fast" ! version which just performs LU decomposition and calls ! triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, CMatrixSolveFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixsolve( complex_2d_array a, ae_int_t n, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x); void alglib::smp_cmatrixsolve( complex_2d_array a, ae_int_t n, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x);
cmatrixsolvefast function/************************************************************************* Complex dense solver for A*x=B with N*N complex matrix A and N*1 complex vectors x and b. "Fast-but-lightweight" version of the solver. Algorithm features: * O(N^3) complexity * no additional time consuming features, just triangular solver COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS: Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). * -1 N<=0 was passed * 1 task is solved B - array[N]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixsolvefast( complex_2d_array a, ae_int_t n, complex_1d_array& b, ae_int_t& info); void alglib::smp_cmatrixsolvefast( complex_2d_array a, ae_int_t n, complex_1d_array& b, ae_int_t& info);
cmatrixsolvem function/************************************************************************* Complex dense solver for A*X=B with N*N complex matrix A, N*M complex matrices X and B. "Slow-but-feature-rich" version which provides additional functions, at the cost of slower performance. Faster version may be invoked with CMatrixSolveMFast() function. Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(N^3+M*N^2) complexity IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system ! and performs iterative refinement, which results in ! significant performance penalty when compared with "fast" ! version which just performs LU decomposition and calls ! triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, CMatrixSolveMFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1,0..M-1], right part M - right part size RFS - iterative refinement switch: * True - refinement is used. Less performance, more precision. * False - refinement is not used. More performance, less precision. OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixsolvem( complex_2d_array a, ae_int_t n, complex_2d_array b, ae_int_t m, bool rfs, ae_int_t& info, densesolverreport& rep, complex_2d_array& x); void alglib::smp_cmatrixsolvem( complex_2d_array a, ae_int_t n, complex_2d_array b, ae_int_t m, bool rfs, ae_int_t& info, densesolverreport& rep, complex_2d_array& x);
cmatrixsolvemfast function/************************************************************************* Complex dense solver for A*X=B with N*N complex matrix A, N*M complex matrices X and B. "Fast-but-lightweight" version which provides just triangular solver - and no additional functions like iterative refinement or condition number estimation. Algorithm features: * O(N^3+M*N^2) complexity * no additional time consuming functions COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS: Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). * -1 N<=0 was passed * 1 task is solved B - array[N,M]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 16.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixsolvemfast( complex_2d_array a, ae_int_t n, complex_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_cmatrixsolvemfast( complex_2d_array a, ae_int_t n, complex_2d_array& b, ae_int_t m, ae_int_t& info);
hpdmatrixcholeskysolve function/************************************************************************* Dense solver for A*x=b with N*N Hermitian positive definite matrix A given by its Cholesky decomposition, and N*1 complex vectors x and b. This is "slow-but-feature-rich" version of the solver which estimates condition number of the system. Algorithm features: * automatic detection of degenerate cases * O(N^2) complexity * condition number estimation * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in 10-15x performance penalty when compared ! with "fast" version which just calls triangular solver. ! ! This performance penalty is insignificant when compared with ! cost of large LU decomposition. However, if you call this ! function many times for the same left side, this overhead ! BECOMES significant. It also becomes significant for small- ! scale problems (N<50). ! ! In such cases we strongly recommend you to use faster solver, ! HPDMatrixCholeskySolveFast() function. INPUT PARAMETERS CHA - array[0..N-1,0..N-1], Cholesky decomposition, SPDMatrixCholesky result N - size of A IsUpper - what half of CHA is provided B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or ill conditioned X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N]: * for info>0 - solution * for info=-3 - filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixcholeskysolve( complex_2d_array cha, ae_int_t n, bool isupper, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x);
hpdmatrixcholeskysolvefast function/************************************************************************* Dense solver for A*x=b with N*N Hermitian positive definite matrix A given by its Cholesky decomposition, and N*1 complex vectors x and b. This is "fast-but-lightweight" version of the solver. Algorithm features: * O(N^2) complexity * matrix is represented by its upper or lower triangle * no additional time-consuming features INPUT PARAMETERS CHA - array[0..N-1,0..N-1], Cholesky decomposition, SPDMatrixCholesky result N - size of A IsUpper - what half of CHA is provided B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or ill conditioned B is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved B - array[N]: * for info>0 - overwritten by solution * for info=-3 - filled by zeros -- ALGLIB -- Copyright 18.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixcholeskysolvefast( complex_2d_array cha, ae_int_t n, bool isupper, complex_1d_array& b, ae_int_t& info);
hpdmatrixcholeskysolvem function/************************************************************************* Dense solver for A*X=B with N*N Hermitian positive definite matrix A given by its Cholesky decomposition and N*M complex matrices X and B. This is "slow-but-feature-rich" version of the solver which, in addition to the solution, estimates condition number of the system. Algorithm features: * automatic detection of degenerate cases * O(M*N^2) complexity * condition number estimation * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just calls triangular ! solver. Amount of overhead introduced depends on M (the ! larger - the more efficient). ! ! This performance penalty is insignificant when compared with ! cost of large Cholesky decomposition. However, if you call ! this function many times for the same left side, this ! overhead BECOMES significant. It also becomes significant ! for small-scale problems (N<50). ! ! In such cases we strongly recommend you to use faster solver, ! HPDMatrixCholeskySolveMFast() function. INPUT PARAMETERS CHA - array[N,N], Cholesky decomposition, HPDMatrixCholesky result N - size of CHA IsUpper - what half of CHA is provided B - array[N,M], right part M - right part size OUTPUT PARAMETERS: Info - return code: * -3 A is singular, or VERY close to singular. X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task was solved Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N]: * for info>0 contains solution * for info=-3 filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixcholeskysolvem( complex_2d_array cha, ae_int_t n, bool isupper, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x); void alglib::smp_hpdmatrixcholeskysolvem( complex_2d_array cha, ae_int_t n, bool isupper, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x);
hpdmatrixcholeskysolvemfast function/************************************************************************* Dense solver for A*X=B with N*N Hermitian positive definite matrix A given by its Cholesky decomposition and N*M complex matrices X and B. This is "fast-but-lightweight" version of the solver. Algorithm features: * O(M*N^2) complexity * matrix is represented by its upper or lower triangle * no additional time-consuming features INPUT PARAMETERS CHA - array[N,N], Cholesky decomposition, HPDMatrixCholesky result N - size of CHA IsUpper - what half of CHA is provided B - array[N,M], right part M - right part size OUTPUT PARAMETERS: Info - return code: * -3 A is singular, or VERY close to singular. X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task was solved B - array[N]: * for info>0 overwritten by solution * for info=-3 filled by zeros -- ALGLIB -- Copyright 18.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixcholeskysolvemfast( complex_2d_array cha, ae_int_t n, bool isupper, complex_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_hpdmatrixcholeskysolvemfast( complex_2d_array cha, ae_int_t n, bool isupper, complex_2d_array& b, ae_int_t m, ae_int_t& info);
hpdmatrixsolve function/************************************************************************* Dense solver for A*x=b, with N*N Hermitian positive definite matrix A, and N*1 complex vectors x and b. "Slow-but-feature-rich" version of the solver. Algorithm features: * automatic detection of degenerate cases * condition number estimation * O(N^3) complexity * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just performs Cholesky ! decomposition and calls triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, HPDMatrixSolveFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1], right part OUTPUT PARAMETERS Info - same as in RMatrixSolve Returns -3 for non-HPD matrices. Rep - same as in RMatrixSolve X - same as in RMatrixSolve -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixsolve( complex_2d_array a, ae_int_t n, bool isupper, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x); void alglib::smp_hpdmatrixsolve( complex_2d_array a, ae_int_t n, bool isupper, complex_1d_array b, ae_int_t& info, densesolverreport& rep, complex_1d_array& x);
hpdmatrixsolvefast function/************************************************************************* Dense solver for A*x=b, with N*N Hermitian positive definite matrix A, and N*1 complex vectors x and b. "Fast-but-lightweight" version of the solver without additional functions. Algorithm features: * O(N^3) complexity * matrix is represented by its upper or lower triangle * no additional time consuming functions COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or not positive definite X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task was solved B - array[0..N-1]: * overwritten by solution * zeros, if A is exactly singular (diagonal of its LU decomposition has exact zeros). -- ALGLIB -- Copyright 17.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixsolvefast( complex_2d_array a, ae_int_t n, bool isupper, complex_1d_array& b, ae_int_t& info); void alglib::smp_hpdmatrixsolvefast( complex_2d_array a, ae_int_t n, bool isupper, complex_1d_array& b, ae_int_t& info);
hpdmatrixsolvem function/************************************************************************* Dense solver for A*X=B, with N*N Hermitian positive definite matrix A and N*M complex matrices X and B. "Slow-but-feature-rich" version of the solver. Algorithm features: * automatic detection of degenerate cases * condition number estimation * O(N^3+M*N^2) complexity * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just calls triangular ! solver. ! ! This performance penalty is especially apparent when you use ! ALGLIB parallel capabilities (condition number estimation is ! inherently sequential). It also becomes significant for ! small-scale problems (N<100). ! ! In such cases we strongly recommend you to use faster solver, ! HPDMatrixSolveMFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - same as in RMatrixSolve. Returns -3 for non-HPD matrices. Rep - same as in RMatrixSolve X - same as in RMatrixSolve -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixsolvem( complex_2d_array a, ae_int_t n, bool isupper, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x); void alglib::smp_hpdmatrixsolvem( complex_2d_array a, ae_int_t n, bool isupper, complex_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, complex_2d_array& x);
hpdmatrixsolvemfast function/************************************************************************* Dense solver for A*X=B, with N*N Hermitian positive definite matrix A and N*M complex matrices X and B. "Fast-but-lightweight" version of the solver. Algorithm features: * O(N^3+M*N^2) complexity * matrix is represented by its upper or lower triangle * no additional time consuming features like condition number estimation COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or is not positive definite. B is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved B - array[0..N-1]: * overwritten by solution * zeros, if problem was not solved -- ALGLIB -- Copyright 17.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixsolvemfast( complex_2d_array a, ae_int_t n, bool isupper, complex_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_hpdmatrixsolvemfast( complex_2d_array a, ae_int_t n, bool isupper, complex_2d_array& b, ae_int_t m, ae_int_t& info);
rmatrixlusolve function/************************************************************************* Dense solver. This subroutine solves a system A*x=b, where A is NxN non-denegerate real matrix given by its LU decomposition, x and b are real vectors. This is "slow-but-robust" version of the linear LU-based solver. Faster version is RMatrixLUSolveFast() function. Algorithm features: * automatic detection of degenerate cases * O(N^2) complexity * condition number estimation No iterative refinement is provided because exact form of original matrix is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in 10-15x performance penalty when compared ! with "fast" version which just calls triangular solver. ! ! This performance penalty is insignificant when compared with ! cost of large LU decomposition. However, if you call this ! function many times for the same left side, this overhead ! BECOMES significant. It also becomes significant for small- ! scale problems. ! ! In such cases we strongly recommend you to use faster solver, ! RMatrixLUSolveFast() function. INPUT PARAMETERS LUA - array[N,N], LU decomposition, RMatrixLU result P - array[N], pivots array, RMatrixLU result N - size of A B - array[N], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixlusolve( real_2d_array lua, integer_1d_array p, ae_int_t n, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x);
rmatrixlusolvefast function/************************************************************************* Dense solver. This subroutine solves a system A*x=b, where A is NxN non-denegerate real matrix given by its LU decomposition, x and b are real vectors. This is "fast-without-any-checks" version of the linear LU-based solver. Slower but more robust version is RMatrixLUSolve() function. Algorithm features: * O(N^2) complexity * fast algorithm without ANY additional checks, just triangular solver INPUT PARAMETERS LUA - array[0..N-1,0..N-1], LU decomposition, RMatrixLU result P - array[0..N-1], pivots array, RMatrixLU result N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved B - array[N]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 18.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixlusolvefast( real_2d_array lua, integer_1d_array p, ae_int_t n, real_1d_array& b, ae_int_t& info);
rmatrixlusolvem function/************************************************************************* Dense solver. Similar to RMatrixLUSolve() but solves task with multiple right parts (where b and x are NxM matrices). This is "robust-but-slow" version of LU-based solver which performs additional checks for non-degeneracy of inputs (condition number estimation). If you need best performance, use "fast-without-any-checks" version, RMatrixLUSolveMFast(). Algorithm features: * automatic detection of degenerate cases * O(M*N^2) complexity * condition number estimation No iterative refinement is provided because exact form of original matrix is not known to subroutine. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just calls triangular ! solver. ! ! This performance penalty is especially apparent when you use ! ALGLIB parallel capabilities (condition number estimation is ! inherently sequential). It also becomes significant for ! small-scale problems. ! ! In such cases we strongly recommend you to use faster solver, ! RMatrixLUSolveMFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Triangular solver is relatively easy to parallelize. ! However, parallelization will be efficient only for large number of ! right parts M. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS LUA - array[N,N], LU decomposition, RMatrixLU result P - array[N], pivots array, RMatrixLU result N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixlusolvem( real_2d_array lua, integer_1d_array p, ae_int_t n, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x); void alglib::smp_rmatrixlusolvem( real_2d_array lua, integer_1d_array p, ae_int_t n, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x);
rmatrixlusolvemfast function/************************************************************************* Dense solver. Similar to RMatrixLUSolve() but solves task with multiple right parts, where b and x are NxM matrices. This is "fast-without-any-checks" version of LU-based solver. It does not estimate condition number of a system, so it is extremely fast. If you need better detection of near-degenerate cases, use RMatrixLUSolveM() function. Algorithm features: * O(M*N^2) complexity * fast algorithm without ANY additional checks, just triangular solver COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. Triangular solver is relatively easy to parallelize. ! However, parallelization will be efficient only for large number of ! right parts M. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: LUA - array[0..N-1,0..N-1], LU decomposition, RMatrixLU result P - array[0..N-1], pivots array, RMatrixLU result N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS: Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). * -1 N<=0 was passed * 1 task is solved B - array[N,M]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 18.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixlusolvemfast( real_2d_array lua, integer_1d_array p, ae_int_t n, real_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_rmatrixlusolvemfast( real_2d_array lua, integer_1d_array p, ae_int_t n, real_2d_array& b, ae_int_t m, ae_int_t& info);
rmatrixmixedsolve function/************************************************************************* Dense solver. This subroutine solves a system A*x=b, where BOTH ORIGINAL A AND ITS LU DECOMPOSITION ARE KNOWN. You can use it if for some reasons you have both A and its LU decomposition. Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(N^2) complexity INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix LUA - array[0..N-1,0..N-1], LU decomposition, RMatrixLU result P - array[0..N-1], pivots array, RMatrixLU result N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixmixedsolve( real_2d_array a, real_2d_array lua, integer_1d_array p, ae_int_t n, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x);
rmatrixmixedsolvem function/************************************************************************* Dense solver. Similar to RMatrixMixedSolve() but solves task with multiple right parts (where b and x are NxM matrices). Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(M*N^2) complexity INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix LUA - array[0..N-1,0..N-1], LU decomposition, RMatrixLU result P - array[0..N-1], pivots array, RMatrixLU result N - size of A B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixmixedsolvem( real_2d_array a, real_2d_array lua, integer_1d_array p, ae_int_t n, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x);
rmatrixsolve function/************************************************************************* Dense solver for A*x=b with N*N real matrix A and N*1 real vectorx x and b. This is "slow-but-feature rich" version of the linear solver. Faster version is RMatrixSolveFast() function. Algorithm features: * automatic detection of degenerate cases * condition number estimation * iterative refinement * O(N^3) complexity IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system ! and performs iterative refinement, which results in ! significant performance penalty when compared with "fast" ! version which just performs LU decomposition and calls ! triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. It also very significant on small matrices. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, RMatrixSolveFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or exactly singular. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixsolve( real_2d_array a, ae_int_t n, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x); void alglib::smp_rmatrixsolve( real_2d_array a, ae_int_t n, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x);
rmatrixsolvefast function/************************************************************************* Dense solver. This subroutine solves a system A*x=b, where A is NxN non-denegerate real matrix, x and b are vectors. This is a "fast" version of linear solver which does NOT provide any additional functions like condition number estimation or iterative refinement. Algorithm features: * efficient algorithm O(N^3) complexity * no performance overhead from additional functionality If you need condition number estimation or iterative refinement, use more feature-rich version - RMatrixSolve(). COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). * -1 N<=0 was passed * 1 task is solved B - array[N]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 16.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixsolvefast( real_2d_array a, ae_int_t n, real_1d_array& b, ae_int_t& info); void alglib::smp_rmatrixsolvefast( real_2d_array a, ae_int_t n, real_1d_array& b, ae_int_t& info);
rmatrixsolvels function/************************************************************************* Dense solver. This subroutine finds solution of the linear system A*X=B with non-square, possibly degenerate A. System is solved in the least squares sense, and general least squares solution X = X0 + CX*y which minimizes |A*X-B| is returned. If A is non-degenerate, solution in the usual sense is returned. Algorithm features: * automatic detection (and correct handling!) of degenerate cases * iterative refinement * O(N^3) complexity COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is only partially supported (some parts are ! optimized, but most - are not). ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..NRows-1,0..NCols-1], system matrix NRows - vertical size of A NCols - horizontal size of A B - array[0..NCols-1], right part Threshold- a number in [0,1]. Singular values beyond Threshold are considered zero. Set it to 0.0, if you don't understand what it means, so the solver will choose good value on its own. OUTPUT PARAMETERS Info - return code: * -4 SVD subroutine failed * -1 if NRows<=0 or NCols<=0 or Threshold<0 was passed * 1 if task is solved Rep - solver report, see below for more info X - array[0..N-1,0..M-1], it contains: * solution of A*X=B (even for singular A) * zeros, if SVD subroutine failed SOLVER REPORT Subroutine sets following fields of the Rep structure: * R2 reciprocal of condition number: 1/cond(A), 2-norm. * N = NCols * K dim(Null(A)) * CX array[0..N-1,0..K-1], kernel of A. Columns of CX store such vectors that A*CX[i]=0. -- ALGLIB -- Copyright 24.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixsolvels( real_2d_array a, ae_int_t nrows, ae_int_t ncols, real_1d_array b, double threshold, ae_int_t& info, densesolverlsreport& rep, real_1d_array& x); void alglib::smp_rmatrixsolvels( real_2d_array a, ae_int_t nrows, ae_int_t ncols, real_1d_array b, double threshold, ae_int_t& info, densesolverlsreport& rep, real_1d_array& x);
rmatrixsolvem function/************************************************************************* Dense solver. Similar to RMatrixSolve() but solves task with multiple right parts (where b and x are NxM matrices). This is "slow-but-robust" version of linear solver with additional functionality like condition number estimation. There also exists faster version - RMatrixSolveMFast(). Algorithm features: * automatic detection of degenerate cases * condition number estimation * optional iterative refinement * O(N^3+M*N^2) complexity IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system ! and performs iterative refinement, which results in ! significant performance penalty when compared with "fast" ! version which just performs LU decomposition and calls ! triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. It also very significant on small matrices. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, RMatrixSolveMFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1,0..M-1], right part M - right part size RFS - iterative refinement switch: * True - refinement is used. Less performance, more precision. * False - refinement is not used. More performance, less precision. OUTPUT PARAMETERS Info - return code: * -3 A is ill conditioned or singular. X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixsolvem( real_2d_array a, ae_int_t n, real_2d_array b, ae_int_t m, bool rfs, ae_int_t& info, densesolverreport& rep, real_2d_array& x); void alglib::smp_rmatrixsolvem( real_2d_array a, ae_int_t n, real_2d_array b, ae_int_t m, bool rfs, ae_int_t& info, densesolverreport& rep, real_2d_array& x);
rmatrixsolvemfast function/************************************************************************* Dense solver. Similar to RMatrixSolve() but solves task with multiple right parts (where b and x are NxM matrices). This is "fast" version of linear solver which does NOT offer additional functions like condition number estimation or iterative refinement. Algorithm features: * O(N^3+M*N^2) complexity * no additional functionality, highest performance COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A B - array[0..N-1,0..M-1], right part M - right part size RFS - iterative refinement switch: * True - refinement is used. Less performance, more precision. * False - refinement is not used. More performance, less precision. OUTPUT PARAMETERS Info - return code: * -3 matrix is exactly singular (ill conditioned matrices are not recognized). X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm B - array[N]: * info>0 => overwritten by solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixsolvemfast( real_2d_array a, ae_int_t n, real_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_rmatrixsolvemfast( real_2d_array a, ae_int_t n, real_2d_array& b, ae_int_t m, ae_int_t& info);
spdmatrixcholeskysolve function/************************************************************************* Dense solver for A*x=b with N*N symmetric positive definite matrix A given by its Cholesky decomposition, and N*1 real vectors x and b. This is "slow- but-feature-rich" version of the solver which, in addition to the solution, performs condition number estimation. Algorithm features: * automatic detection of degenerate cases * O(N^2) complexity * condition number estimation * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in 10-15x performance penalty when compared ! with "fast" version which just calls triangular solver. ! ! This performance penalty is insignificant when compared with ! cost of large LU decomposition. However, if you call this ! function many times for the same left side, this overhead ! BECOMES significant. It also becomes significant for small- ! scale problems (N<50). ! ! In such cases we strongly recommend you to use faster solver, ! SPDMatrixCholeskySolveFast() function. INPUT PARAMETERS CHA - array[N,N], Cholesky decomposition, SPDMatrixCholesky result N - size of A IsUpper - what half of CHA is provided B - array[N], right part OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or ill conditioned X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N]: * for info>0 - solution * for info=-3 - filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixcholeskysolve( real_2d_array cha, ae_int_t n, bool isupper, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x);
spdmatrixcholeskysolvefast function/************************************************************************* Dense solver for A*x=b with N*N symmetric positive definite matrix A given by its Cholesky decomposition, and N*1 real vectors x and b. This is "fast- but-lightweight" version of the solver. Algorithm features: * O(N^2) complexity * matrix is represented by its upper or lower triangle * no additional features INPUT PARAMETERS CHA - array[N,N], Cholesky decomposition, SPDMatrixCholesky result N - size of A IsUpper - what half of CHA is provided B - array[N], right part OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or ill conditioned X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task is solved B - array[N]: * for info>0 - overwritten by solution * for info=-3 - filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixcholeskysolvefast( real_2d_array cha, ae_int_t n, bool isupper, real_1d_array& b, ae_int_t& info);
spdmatrixcholeskysolvem function/************************************************************************* Dense solver for A*X=B with N*N symmetric positive definite matrix A given by its Cholesky decomposition, and N*M vectors X and B. It is "slow-but- feature-rich" version of the solver which estimates condition number of the system. Algorithm features: * automatic detection of degenerate cases * O(M*N^2) complexity * condition number estimation * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just calls triangular ! solver. Amount of overhead introduced depends on M (the ! larger - the more efficient). ! ! This performance penalty is insignificant when compared with ! cost of large LU decomposition. However, if you call this ! function many times for the same left side, this overhead ! BECOMES significant. It also becomes significant for small- ! scale problems (N<50). ! ! In such cases we strongly recommend you to use faster solver, ! SPDMatrixCholeskySolveMFast() function. INPUT PARAMETERS CHA - array[0..N-1,0..N-1], Cholesky decomposition, SPDMatrixCholesky result N - size of CHA IsUpper - what half of CHA is provided B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or badly conditioned X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task was solved Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N]: * for info>0 contains solution * for info=-3 filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixcholeskysolvem( real_2d_array cha, ae_int_t n, bool isupper, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x); void alglib::smp_spdmatrixcholeskysolvem( real_2d_array cha, ae_int_t n, bool isupper, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x);
spdmatrixcholeskysolvemfast function/************************************************************************* Dense solver for A*X=B with N*N symmetric positive definite matrix A given by its Cholesky decomposition, and N*M vectors X and B. It is "fast-but- lightweight" version of the solver which just solves linear system, without any additional functions. Algorithm features: * O(M*N^2) complexity * matrix is represented by its upper or lower triangle * no additional functionality INPUT PARAMETERS CHA - array[N,N], Cholesky decomposition, SPDMatrixCholesky result N - size of CHA IsUpper - what half of CHA is provided B - array[N,M], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or badly conditioned X is filled by zeros in such cases. * -1 N<=0 was passed * 1 task was solved B - array[N]: * for info>0 overwritten by solution * for info=-3 filled by zeros -- ALGLIB -- Copyright 18.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixcholeskysolvemfast( real_2d_array cha, ae_int_t n, bool isupper, real_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_spdmatrixcholeskysolvemfast( real_2d_array cha, ae_int_t n, bool isupper, real_2d_array& b, ae_int_t m, ae_int_t& info);
spdmatrixsolve function/************************************************************************* Dense linear solver for A*x=b with N*N real symmetric positive definite matrix A, N*1 vectors x and b. "Slow-but-feature-rich" version of the solver. Algorithm features: * automatic detection of degenerate cases * condition number estimation * O(N^3) complexity * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just performs Cholesky ! decomposition and calls triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, SPDMatrixSolveFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or non-SPD. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixsolve( real_2d_array a, ae_int_t n, bool isupper, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x); void alglib::smp_spdmatrixsolve( real_2d_array a, ae_int_t n, bool isupper, real_1d_array b, ae_int_t& info, densesolverreport& rep, real_1d_array& x);
spdmatrixsolvefast function/************************************************************************* Dense linear solver for A*x=b with N*N real symmetric positive definite matrix A, N*1 vectors x and b. "Fast-but-lightweight" version of the solver. Algorithm features: * O(N^3) complexity * matrix is represented by its upper or lower triangle * no additional time consuming features like condition number estimation COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1], right part OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular or non-SPD * -1 N<=0 was passed * 1 task was solved B - array[N], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 17.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixsolvefast( real_2d_array a, ae_int_t n, bool isupper, real_1d_array& b, ae_int_t& info); void alglib::smp_spdmatrixsolvefast( real_2d_array a, ae_int_t n, bool isupper, real_1d_array& b, ae_int_t& info);
spdmatrixsolvem function/************************************************************************* Dense solver for A*X=B with N*N symmetric positive definite matrix A, and N*M vectors X and B. It is "slow-but-feature-rich" version of the solver. Algorithm features: * automatic detection of degenerate cases * condition number estimation * O(N^3+M*N^2) complexity * matrix is represented by its upper or lower triangle No iterative refinement is provided because such partial representation of matrix does not allow efficient calculation of extra-precise matrix-vector products for large matrices. Use RMatrixSolve or RMatrixMixedSolve if you need iterative refinement. IMPORTANT: ! this function is NOT the most efficient linear solver provided ! by ALGLIB. It estimates condition number of linear system, ! which results in significant performance penalty when ! compared with "fast" version which just performs Cholesky ! decomposition and calls triangular solver. ! ! This performance penalty is especially visible in the ! multithreaded mode, because both condition number estimation ! and iterative refinement are inherently sequential ! calculations. ! ! Thus, if you need high performance and if you are pretty sure ! that your system is well conditioned, we strongly recommend ! you to use faster solver, SPDMatrixSolveMFast() function. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 matrix is very badly conditioned or non-SPD. * -1 N<=0 was passed * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - additional report, following fields are set: * rep.r1 condition number in 1-norm * rep.rinf condition number in inf-norm X - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 27.01.2010 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixsolvem( real_2d_array a, ae_int_t n, bool isupper, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x); void alglib::smp_spdmatrixsolvem( real_2d_array a, ae_int_t n, bool isupper, real_2d_array b, ae_int_t m, ae_int_t& info, densesolverreport& rep, real_2d_array& x);
spdmatrixsolvemfast function/************************************************************************* Dense solver for A*X=B with N*N symmetric positive definite matrix A, and N*M vectors X and B. It is "fast-but-lightweight" version of the solver. Algorithm features: * O(N^3+M*N^2) complexity * matrix is represented by its upper or lower triangle * no additional time consuming features COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS A - array[0..N-1,0..N-1], system matrix N - size of A IsUpper - what half of A is provided B - array[0..N-1,0..M-1], right part M - right part size OUTPUT PARAMETERS Info - return code: * -3 A is is exactly singular * -1 N<=0 was passed * 1 task was solved B - array[N,M], it contains: * info>0 => solution * info=-3 => filled by zeros -- ALGLIB -- Copyright 17.03.2015 by Bochkanov Sergey *************************************************************************/void alglib::spdmatrixsolvemfast( real_2d_array a, ae_int_t n, bool isupper, real_2d_array& b, ae_int_t m, ae_int_t& info); void alglib::smp_spdmatrixsolvemfast( real_2d_array a, ae_int_t n, bool isupper, real_2d_array& b, ae_int_t m, ae_int_t& info);
dforest subpackagedecisionforest class/************************************************************************* *************************************************************************/class decisionforest { };
dfreport class/************************************************************************* *************************************************************************/class dfreport { double relclserror; double avgce; double rmserror; double avgerror; double avgrelerror; double oobrelclserror; double oobavgce; double oobrmserror; double oobavgerror; double oobavgrelerror; };
dfavgce function/************************************************************************* Average cross-entropy (in bits per element) on the test set INPUT PARAMETERS: DF - decision forest model XY - test set NPoints - test set size RESULT: CrossEntropy/(NPoints*LN(2)). Zero if model solves regression task. -- ALGLIB -- Copyright 16.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::dfavgce( decisionforest df, real_2d_array xy, ae_int_t npoints);
dfavgerror function/************************************************************************* Average error on the test set INPUT PARAMETERS: DF - decision forest model XY - test set NPoints - test set size RESULT: Its meaning for regression task is obvious. As for classification task, it means average error when estimating posterior probabilities. -- ALGLIB -- Copyright 16.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::dfavgerror( decisionforest df, real_2d_array xy, ae_int_t npoints);
dfavgrelerror function/************************************************************************* Average relative error on the test set INPUT PARAMETERS: DF - decision forest model XY - test set NPoints - test set size RESULT: Its meaning for regression task is obvious. As for classification task, it means average relative error when estimating posterior probability of belonging to the correct class. -- ALGLIB -- Copyright 16.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::dfavgrelerror( decisionforest df, real_2d_array xy, ae_int_t npoints);
dfbuildrandomdecisionforest function/************************************************************************* This subroutine builds random decision forest. INPUT PARAMETERS: XY - training set NPoints - training set size, NPoints>=1 NVars - number of independent variables, NVars>=1 NClasses - task type: * NClasses=1 - regression task with one dependent variable * NClasses>1 - classification task with NClasses classes. NTrees - number of trees in a forest, NTrees>=1. recommended values: 50-100. R - percent of a training set used to build individual trees. 0<R<=1. recommended values: 0.1 <= R <= 0.66. OUTPUT PARAMETERS: Info - return code: * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<1, NVars<1, NClasses<1, NTrees<1, R<=0 or R>1). * 1, if task has been solved DF - model built Rep - training report, contains error on a training set and out-of-bag estimates of generalization error. -- ALGLIB -- Copyright 19.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::dfbuildrandomdecisionforest( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t nclasses, ae_int_t ntrees, double r, ae_int_t& info, decisionforest& df, dfreport& rep);
dfbuildrandomdecisionforestx1 function/************************************************************************* This subroutine builds random decision forest. This function gives ability to tune number of variables used when choosing best split. INPUT PARAMETERS: XY - training set NPoints - training set size, NPoints>=1 NVars - number of independent variables, NVars>=1 NClasses - task type: * NClasses=1 - regression task with one dependent variable * NClasses>1 - classification task with NClasses classes. NTrees - number of trees in a forest, NTrees>=1. recommended values: 50-100. NRndVars - number of variables used when choosing best split R - percent of a training set used to build individual trees. 0<R<=1. recommended values: 0.1 <= R <= 0.66. OUTPUT PARAMETERS: Info - return code: * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<1, NVars<1, NClasses<1, NTrees<1, R<=0 or R>1). * 1, if task has been solved DF - model built Rep - training report, contains error on a training set and out-of-bag estimates of generalization error. -- ALGLIB -- Copyright 19.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::dfbuildrandomdecisionforestx1( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t nclasses, ae_int_t ntrees, ae_int_t nrndvars, double r, ae_int_t& info, decisionforest& df, dfreport& rep);
dfprocess function/************************************************************************* Procesing INPUT PARAMETERS: DF - decision forest model X - input vector, array[0..NVars-1]. OUTPUT PARAMETERS: Y - result. Regression estimate when solving regression task, vector of posterior probabilities for classification task. See also DFProcessI. -- ALGLIB -- Copyright 16.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::dfprocess( decisionforest df, real_1d_array x, real_1d_array& y);
dfprocessi function/************************************************************************* 'interactive' variant of DFProcess for languages like Python which support constructs like "Y = DFProcessI(DF,X)" and interactive mode of interpreter This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::dfprocessi( decisionforest df, real_1d_array x, real_1d_array& y);
dfrelclserror function/************************************************************************* Relative classification error on the test set INPUT PARAMETERS: DF - decision forest model XY - test set NPoints - test set size RESULT: percent of incorrectly classified cases. Zero if model solves regression task. -- ALGLIB -- Copyright 16.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::dfrelclserror( decisionforest df, real_2d_array xy, ae_int_t npoints);
dfrmserror function/************************************************************************* RMS error on the test set INPUT PARAMETERS: DF - decision forest model XY - test set NPoints - test set size RESULT: root mean square error. Its meaning for regression task is obvious. As for classification task, RMS error means error when estimating posterior probabilities. -- ALGLIB -- Copyright 16.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::dfrmserror( decisionforest df, real_2d_array xy, ae_int_t npoints);
dfserialize function/************************************************************************* This function serializes data structure to string. Important properties of s_out: * it contains alphanumeric characters, dots, underscores, minus signs * these symbols are grouped into words, which are separated by spaces and Windows-style (CR+LF) newlines * although serializer uses spaces and CR+LF as separators, you can replace any separator character by arbitrary combination of spaces, tabs, Windows or Unix newlines. It allows flexible reformatting of the string in case you want to include it into text or XML file. But you should not insert separators into the middle of the "words" nor you should change case of letters. * s_out can be freely moved between 32-bit and 64-bit systems, little and big endian machines, and so on. You can serialize structure on 32-bit machine and unserialize it on 64-bit one (or vice versa), or serialize it on SPARC and unserialize on x86. You can also serialize it in C++ version of ALGLIB and unserialize in C# one, and vice versa. *************************************************************************/void dfserialize(decisionforest &obj, std::string &s_out); void dfserialize(decisionforest &obj, std::ostream &s_out);
dfunserialize function/************************************************************************* This function unserializes data structure from string. *************************************************************************/void dfunserialize(const std::string &s_in, decisionforest &obj); void dfunserialize(const std::istream &s_in, decisionforest &obj);
elliptic subpackageellipticintegrale function/************************************************************************* Complete elliptic integral of the second kind Approximates the integral pi/2 - | | 2 E(m) = | sqrt( 1 - m sin t ) dt | | - 0 using the approximation P(x) - x log x Q(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 1 10000 2.1e-16 7.3e-17 Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/double alglib::ellipticintegrale(double m);
ellipticintegralk function/************************************************************************* Complete elliptic integral of the first kind Approximates the integral pi/2 - | | | dt K(m) = | ------------------ | 2 | | sqrt( 1 - m sin t ) - 0 using the approximation P(x) - log x Q(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 2.5e-16 6.8e-17 Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::ellipticintegralk(double m);
ellipticintegralkhighprecision function/************************************************************************* Complete elliptic integral of the first kind Approximates the integral pi/2 - | | | dt K(m) = | ------------------ | 2 | | sqrt( 1 - m sin t ) - 0 where m = 1 - m1, using the approximation P(x) - log x Q(x). The argument m1 is used rather than m so that the logarithmic singularity at m = 1 will be shifted to the origin; this preserves maximum accuracy. K(0) = pi/2. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 2.5e-16 6.8e-17 Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::ellipticintegralkhighprecision(double m1);
incompleteellipticintegrale function/************************************************************************* Incomplete elliptic integral of the second kind Approximates the integral phi - | | | 2 E(phi_\m) = | sqrt( 1 - m sin t ) dt | | | - 0 of amplitude phi and modulus m, using the arithmetic - geometric mean algorithm. ACCURACY: Tested at random arguments with phi in [-10, 10] and m in [0, 1]. Relative error: arithmetic domain # trials peak rms IEEE -10,10 150000 3.3e-15 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1993, 2000 by Stephen L. Moshier *************************************************************************/double alglib::incompleteellipticintegrale(double phi, double m);
incompleteellipticintegralk function/************************************************************************* Incomplete elliptic integral of the first kind F(phi|m) Approximates the integral phi - | | | dt F(phi_\m) = | ------------------ | 2 | | sqrt( 1 - m sin t ) - 0 of amplitude phi and modulus m, using the arithmetic - geometric mean algorithm. ACCURACY: Tested at random points with m in [0, 1] and phi as indicated. Relative error: arithmetic domain # trials peak rms IEEE -10,10 200000 7.4e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::incompleteellipticintegralk(double phi, double m);
evd subpackageeigsubspacereport class/************************************************************************* This object stores state of the subspace iteration algorithm. You should use ALGLIB functions to work with this object. *************************************************************************/class eigsubspacereport { ae_int_t iterationscount; };
eigsubspacestate class/************************************************************************* This object stores state of the subspace iteration algorithm. You should use ALGLIB functions to work with this object. *************************************************************************/class eigsubspacestate { };
eigsubspacecreate function/************************************************************************* This function initializes subspace iteration solver. This solver is used to solve symmetric real eigenproblems where just a few (top K) eigenvalues and corresponding eigenvectors is required. This solver can be significantly faster than complete EVD decomposition in the following case: * when only just a small fraction of top eigenpairs of dense matrix is required. When K approaches N, this solver is slower than complete dense EVD * when problem matrix is sparse (and/or is not known explicitly, i.e. only matrix-matrix product can be performed) USAGE (explicit dense/sparse matrix): 1. User initializes algorithm state with eigsubspacecreate() call 2. [optional] User tunes solver parameters by calling eigsubspacesetcond() or other functions 3. User calls eigsubspacesolvedense() or eigsubspacesolvesparse() methods, which take algorithm state and 2D array or alglib.sparsematrix object. USAGE (out-of-core mode): 1. User initializes algorithm state with eigsubspacecreate() call 2. [optional] User tunes solver parameters by calling eigsubspacesetcond() or other functions 3. User activates out-of-core mode of the solver and repeatedly calls communication functions in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) INPUT PARAMETERS: N - problem dimensionality, N>0 K - number of top eigenvector to calculate, 0<K<=N. OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspacecreate( ae_int_t n, ae_int_t k, eigsubspacestate& state);
eigsubspacecreatebuf function/************************************************************************* Buffered version of constructor which aims to reuse previously allocated memory as much as possible. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspacecreatebuf( ae_int_t n, ae_int_t k, eigsubspacestate state);
eigsubspaceooccontinue function/************************************************************************* This function performs subspace iteration in the out-of-core mode. It should be used in conjunction with other out-of-core-related functions of this subspackage in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/bool alglib::eigsubspaceooccontinue(eigsubspacestate state);
eigsubspaceoocgetrequestdata function/************************************************************************* This function is used to retrieve information about out-of-core request sent by solver to user code: matrix X (array[N,RequestSize) which have to be multiplied by out-of-core matrix A in a product A*X. This function returns just request data; in order to get size of the data prior to processing requestm, use eigsubspaceoocgetrequestinfo(). It should be used in conjunction with other out-of-core-related functions of this subspackage in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) INPUT PARAMETERS: State - solver running in out-of-core mode X - possibly preallocated storage; reallocated if needed, left unchanged, if large enough to store request data. OUTPUT PARAMETERS: X - array[N,RequestSize] or larger, leading rectangle is filled with dense matrix X. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspaceoocgetrequestdata( eigsubspacestate state, real_2d_array& x);
eigsubspaceoocgetrequestinfo function/************************************************************************* This function is used to retrieve information about out-of-core request sent by solver to user code: request type (current version of the solver sends only requests for matrix-matrix products) and request size (size of the matrices being multiplied). This function returns just request metrics; in order to get contents of the matrices being multiplied, use eigsubspaceoocgetrequestdata(). It should be used in conjunction with other out-of-core-related functions of this subspackage in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) INPUT PARAMETERS: State - solver running in out-of-core mode OUTPUT PARAMETERS: RequestType - type of the request to process: * 0 - for matrix-matrix product A*X, with A being NxN matrix whose eigenvalues/vectors are needed, and X being NxREQUESTSIZE one which is returned by the eigsubspaceoocgetrequestdata(). RequestSize - size of the X matrix (number of columns), usually it is several times larger than number of vectors K requested by user. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspaceoocgetrequestinfo( eigsubspacestate state, ae_int_t& requesttype, ae_int_t& requestsize);
eigsubspaceoocsendresult function/************************************************************************* This function is used to send user reply to out-of-core request sent by solver. Usually it is product A*X for returned by solver matrix X. It should be used in conjunction with other out-of-core-related functions of this subspackage in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) INPUT PARAMETERS: State - solver running in out-of-core mode AX - array[N,RequestSize] or larger, leading rectangle is filled with product A*X. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspaceoocsendresult( eigsubspacestate state, real_2d_array ax);
eigsubspaceoocstart function/************************************************************************* This function initiates out-of-core mode of subspace eigensolver. It should be used in conjunction with other out-of-core-related functions of this subspackage in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) INPUT PARAMETERS: State - solver object MType - matrix type: * 0 for real symmetric matrix (solver assumes that matrix being processed is symmetric; symmetric direct eigensolver is used for smaller subproblems arising during solution of larger "full" task) Future versions of ALGLIB may introduce support for other matrix types; for now, only symmetric eigenproblems are supported. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspaceoocstart(eigsubspacestate state, ae_int_t mtype);
eigsubspaceoocstop function/************************************************************************* This function finalizes out-of-core mode of subspace eigensolver. It should be used in conjunction with other out-of-core-related functions of this subspackage in a loop like below: > alglib.eigsubspaceoocstart(state) > while alglib.eigsubspaceooccontinue(state) do > alglib.eigsubspaceoocgetrequestinfo(state, out RequestType, out M) > alglib.eigsubspaceoocgetrequestdata(state, out X) > [calculate Y=A*X, with X=R^NxM] > alglib.eigsubspaceoocsendresult(state, in Y) > alglib.eigsubspaceoocstop(state, out W, out Z, out Report) INPUT PARAMETERS: State - solver state OUTPUT PARAMETERS: W - array[K], depending on solver settings: * top K eigenvalues ordered by descending - if eigenvectors are returned in Z * zeros - if invariant subspace is returned in Z Z - array[N,K], depending on solver settings either: * matrix of eigenvectors found * orthogonal basis of K-dimensional invariant subspace Rep - report with additional parameters -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspaceoocstop( eigsubspacestate state, real_1d_array& w, real_2d_array& z, eigsubspacereport& rep);
eigsubspacesetcond function/************************************************************************* This function sets stopping critera for the solver: * error in eigenvector/value allowed by solver * maximum number of iterations to perform INPUT PARAMETERS: State - solver structure Eps - eps>=0, with non-zero value used to tell solver that it can stop after all eigenvalues converged with error roughly proportional to eps*MAX(LAMBDA_MAX), where LAMBDA_MAX is a maximum eigenvalue. Zero value means that no check for precision is performed. MaxIts - maxits>=0, with non-zero value used to tell solver that it can stop after maxits steps (no matter how precise current estimate is) NOTE: passing eps=0 and maxits=0 results in automatic selection of moderate eps as stopping criteria (1.0E-6 in current implementation, but it may change without notice). NOTE: very small values of eps are possible (say, 1.0E-12), although the larger problem you solve (N and/or K), the harder it is to find precise eigenvectors because rounding errors tend to accumulate. NOTE: passing non-zero eps results in some performance penalty, roughly equal to 2N*(2K)^2 FLOPs per iteration. These additional computations are required in order to estimate current error in eigenvalues via Rayleigh-Ritz process. Most of this additional time is spent in construction of ~2Kx2K symmetric subproblem whose eigenvalues are checked with exact eigensolver. This additional time is negligible if you search for eigenvalues of the large dense matrix, but may become noticeable on highly sparse EVD problems, where cost of matrix-matrix product is low. If you set eps to exactly zero, Rayleigh-Ritz phase is completely turned off. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspacesetcond( eigsubspacestate state, double eps, ae_int_t maxits);
eigsubspacesolvedenses function/************************************************************************* This function runs eigensolver for dense NxN symmetric matrix A, given by upper or lower triangle. This function can not process nonsymmetric matrices. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * multithreading support ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! For a situation when you need just a few eigenvectors (~1-10), ! multithreading typically gives sublinear (wrt to cores count) speedup. ! For larger problems it may give you nearly linear increase in ! performance. ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. Best results are achieved for high-dimensional problems ! (NVars is at least 256). ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: State - solver state A - array[N,N], symmetric NxN matrix given by one of its triangles IsUpper - whether upper or lower triangle of A is given (the other one is not referenced at all). OUTPUT PARAMETERS: W - array[K], top K eigenvalues ordered by descending of their absolute values Z - array[N,K], matrix of eigenvectors found Rep - report with additional parameters NOTE: internally this function allocates a copy of NxN dense A. You should take it into account when working with very large matrices occupying almost all RAM. -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspacesolvedenses( eigsubspacestate state, real_2d_array a, bool isupper, real_1d_array& w, real_2d_array& z, eigsubspacereport& rep); void alglib::smp_eigsubspacesolvedenses( eigsubspacestate state, real_2d_array a, bool isupper, real_1d_array& w, real_2d_array& z, eigsubspacereport& rep);
eigsubspacesolvesparses function/************************************************************************* This function runs eigensolver for dense NxN symmetric matrix A, given by upper or lower triangle. This function can not process nonsymmetric matrices. INPUT PARAMETERS: State - solver state A - NxN symmetric matrix given by one of its triangles IsUpper - whether upper or lower triangle of A is given (the other one is not referenced at all). OUTPUT PARAMETERS: W - array[K], top K eigenvalues ordered by descending of their absolute values Z - array[N,K], matrix of eigenvectors found Rep - report with additional parameters -- ALGLIB -- Copyright 16.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::eigsubspacesolvesparses( eigsubspacestate state, sparsematrix a, bool isupper, real_1d_array& w, real_2d_array& z, eigsubspacereport& rep);
hmatrixevd function/************************************************************************* Finding the eigenvalues and eigenvectors of a Hermitian matrix The algorithm finds eigen pairs of a Hermitian matrix by reducing it to real tridiagonal form and using the QL/QR algorithm. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - Hermitian matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - storage format. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. Output parameters: D - eigenvalues in ascending order. Array whose index ranges within [0..N-1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains the eigenvectors. Array whose indexes range within [0..N-1, 0..N-1]. The eigenvectors are stored in the matrix columns. Result: True, if the algorithm has converged. False, if the algorithm hasn't converged (rare case). Note: eigenvectors of Hermitian matrix are defined up to multiplication by a complex number L, such that |L|=1. -- ALGLIB -- Copyright 2005, 23 March 2007 by Bochkanov Sergey *************************************************************************/bool alglib::hmatrixevd( complex_2d_array a, ae_int_t n, ae_int_t zneeded, bool isupper, real_1d_array& d, complex_2d_array& z);
hmatrixevdi function/************************************************************************* Subroutine for finding the eigenvalues and eigenvectors of a Hermitian matrix with given indexes by using bisection and inverse iteration methods Input parameters: A - Hermitian matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. IsUpperA - storage format of matrix A. I1, I2 - index interval for searching (from I1 to I2). 0 <= I1 <= I2 <= N-1. Output parameters: W - array of the eigenvalues found. Array whose index ranges within [0..I2-I1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains eigenvectors. Array whose indexes range within [0..N-1, 0..I2-I1]. In that case, the eigenvectors are stored in the matrix columns. Result: True, if successful. W contains the eigenvalues, Z contains the eigenvectors (if needed). False, if the bisection method subroutine wasn't able to find the eigenvalues in the given interval or if the inverse iteration subroutine wasn't able to find all the corresponding eigenvectors. In that case, the eigenvalues and eigenvectors are not returned. Note: eigen vectors of Hermitian matrix are defined up to multiplication by a complex number L, such as |L|=1. -- ALGLIB -- Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey. *************************************************************************/bool alglib::hmatrixevdi( complex_2d_array a, ae_int_t n, ae_int_t zneeded, bool isupper, ae_int_t i1, ae_int_t i2, real_1d_array& w, complex_2d_array& z);
hmatrixevdr function/************************************************************************* Subroutine for finding the eigenvalues (and eigenvectors) of a Hermitian matrix in a given half-interval (A, B] by using a bisection and inverse iteration Input parameters: A - Hermitian matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. IsUpperA - storage format of matrix A. B1, B2 - half-interval (B1, B2] to search eigenvalues in. Output parameters: M - number of eigenvalues found in a given half-interval, M>=0 W - array of the eigenvalues found. Array whose index ranges within [0..M-1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains eigenvectors. Array whose indexes range within [0..N-1, 0..M-1]. The eigenvectors are stored in the matrix columns. Result: True, if successful. M contains the number of eigenvalues in the given half-interval (could be equal to 0), W contains the eigenvalues, Z contains the eigenvectors (if needed). False, if the bisection method subroutine wasn't able to find the eigenvalues in the given interval or if the inverse iteration subroutine wasn't able to find all the corresponding eigenvectors. In that case, the eigenvalues and eigenvectors are not returned, M is equal to 0. Note: eigen vectors of Hermitian matrix are defined up to multiplication by a complex number L, such as |L|=1. -- ALGLIB -- Copyright 07.01.2006, 24.03.2007 by Bochkanov Sergey. *************************************************************************/bool alglib::hmatrixevdr( complex_2d_array a, ae_int_t n, ae_int_t zneeded, bool isupper, double b1, double b2, ae_int_t& m, real_1d_array& w, complex_2d_array& z);
rmatrixevd function/************************************************************************* Finding eigenvalues and eigenvectors of a general (unsymmetric) matrix COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. Speed-up provided by MKL for this particular problem (EVD) ! is really high, because MKL uses combination of (a) better low-level ! optimizations, and (b) better EVD algorithms. ! ! On one particular SSE-capable machine for N=1024, commercial MKL- ! -capable ALGLIB was: ! * 7-10 times faster than open source "generic C" version ! * 15-18 times faster than "pure C#" version ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. The algorithm finds eigenvalues and eigenvectors of a general matrix by using the QR algorithm with multiple shifts. The algorithm can find eigenvalues and both left and right eigenvectors. The right eigenvector is a vector x such that A*x = w*x, and the left eigenvector is a vector y such that y'*A = w*y' (here y' implies a complex conjugate transposition of vector y). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. VNeeded - flag controlling whether eigenvectors are needed or not. If VNeeded is equal to: * 0, eigenvectors are not returned; * 1, right eigenvectors are returned; * 2, left eigenvectors are returned; * 3, both left and right eigenvectors are returned. Output parameters: WR - real parts of eigenvalues. Array whose index ranges within [0..N-1]. WR - imaginary parts of eigenvalues. Array whose index ranges within [0..N-1]. VL, VR - arrays of left and right eigenvectors (if they are needed). If WI[i]=0, the respective eigenvalue is a real number, and it corresponds to the column number I of matrices VL/VR. If WI[i]>0, we have a pair of complex conjugate numbers with positive and negative imaginary parts: the first eigenvalue WR[i] + sqrt(-1)*WI[i]; the second eigenvalue WR[i+1] + sqrt(-1)*WI[i+1]; WI[i]>0 WI[i+1] = -WI[i] < 0 In that case, the eigenvector corresponding to the first eigenvalue is located in i and i+1 columns of matrices VL/VR (the column number i contains the real part, and the column number i+1 contains the imaginary part), and the vector corresponding to the second eigenvalue is a complex conjugate to the first vector. Arrays whose indexes range within [0..N-1, 0..N-1]. Result: True, if the algorithm has converged. False, if the algorithm has not converged. Note 1: Some users may ask the following question: what if WI[N-1]>0? WI[N] must contain an eigenvalue which is complex conjugate to the N-th eigenvalue, but the array has only size N? The answer is as follows: such a situation cannot occur because the algorithm finds a pairs of eigenvalues, therefore, if WI[i]>0, I is strictly less than N-1. Note 2: The algorithm performance depends on the value of the internal parameter NS of the InternalSchurDecomposition subroutine which defines the number of shifts in the QR algorithm (similarly to the block width in block-matrix algorithms of linear algebra). If you require maximum performance on your machine, it is recommended to adjust this parameter manually. See also the InternalTREVC subroutine. The algorithm is based on the LAPACK 3.0 library. *************************************************************************/bool alglib::rmatrixevd( real_2d_array a, ae_int_t n, ae_int_t vneeded, real_1d_array& wr, real_1d_array& wi, real_2d_array& vl, real_2d_array& vr);
smatrixevd function/************************************************************************* Finding the eigenvalues and eigenvectors of a symmetric matrix The algorithm finds eigen pairs of a symmetric matrix by reducing it to tridiagonal form and using the QL/QR algorithm. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - symmetric matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. IsUpper - storage format. Output parameters: D - eigenvalues in ascending order. Array whose index ranges within [0..N-1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains the eigenvectors. Array whose indexes range within [0..N-1, 0..N-1]. The eigenvectors are stored in the matrix columns. Result: True, if the algorithm has converged. False, if the algorithm hasn't converged (rare case). -- ALGLIB -- Copyright 2005-2008 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixevd( real_2d_array a, ae_int_t n, ae_int_t zneeded, bool isupper, real_1d_array& d, real_2d_array& z);
smatrixevdi function/************************************************************************* Subroutine for finding the eigenvalues and eigenvectors of a symmetric matrix with given indexes by using bisection and inverse iteration methods. Input parameters: A - symmetric matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. IsUpperA - storage format of matrix A. I1, I2 - index interval for searching (from I1 to I2). 0 <= I1 <= I2 <= N-1. Output parameters: W - array of the eigenvalues found. Array whose index ranges within [0..I2-I1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains eigenvectors. Array whose indexes range within [0..N-1, 0..I2-I1]. In that case, the eigenvectors are stored in the matrix columns. Result: True, if successful. W contains the eigenvalues, Z contains the eigenvectors (if needed). False, if the bisection method subroutine wasn't able to find the eigenvalues in the given interval or if the inverse iteration subroutine wasn't able to find all the corresponding eigenvectors. In that case, the eigenvalues and eigenvectors are not returned. -- ALGLIB -- Copyright 07.01.2006 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixevdi( real_2d_array a, ae_int_t n, ae_int_t zneeded, bool isupper, ae_int_t i1, ae_int_t i2, real_1d_array& w, real_2d_array& z);
smatrixevdr function/************************************************************************* Subroutine for finding the eigenvalues (and eigenvectors) of a symmetric matrix in a given half open interval (A, B] by using a bisection and inverse iteration Input parameters: A - symmetric matrix which is given by its upper or lower triangular part. Array [0..N-1, 0..N-1]. N - size of matrix A. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. IsUpperA - storage format of matrix A. B1, B2 - half open interval (B1, B2] to search eigenvalues in. Output parameters: M - number of eigenvalues found in a given half-interval (M>=0). W - array of the eigenvalues found. Array whose index ranges within [0..M-1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains eigenvectors. Array whose indexes range within [0..N-1, 0..M-1]. The eigenvectors are stored in the matrix columns. Result: True, if successful. M contains the number of eigenvalues in the given half-interval (could be equal to 0), W contains the eigenvalues, Z contains the eigenvectors (if needed). False, if the bisection method subroutine wasn't able to find the eigenvalues in the given interval or if the inverse iteration subroutine wasn't able to find all the corresponding eigenvectors. In that case, the eigenvalues and eigenvectors are not returned, M is equal to 0. -- ALGLIB -- Copyright 07.01.2006 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixevdr( real_2d_array a, ae_int_t n, ae_int_t zneeded, bool isupper, double b1, double b2, ae_int_t& m, real_1d_array& w, real_2d_array& z);
smatrixtdevd function/************************************************************************* Finding the eigenvalues and eigenvectors of a tridiagonal symmetric matrix The algorithm finds the eigen pairs of a tridiagonal symmetric matrix by using an QL/QR algorithm with implicit shifts. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: D - the main diagonal of a tridiagonal matrix. Array whose index ranges within [0..N-1]. E - the secondary diagonal of a tridiagonal matrix. Array whose index ranges within [0..N-2]. N - size of matrix A. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not needed; * 1, the eigenvectors of a tridiagonal matrix are multiplied by the square matrix Z. It is used if the tridiagonal matrix is obtained by the similarity transformation of a symmetric matrix; * 2, the eigenvectors of a tridiagonal matrix replace the square matrix Z; * 3, matrix Z contains the first row of the eigenvectors matrix. Z - if ZNeeded=1, Z contains the square matrix by which the eigenvectors are multiplied. Array whose indexes range within [0..N-1, 0..N-1]. Output parameters: D - eigenvalues in ascending order. Array whose index ranges within [0..N-1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains the product of a given matrix (from the left) and the eigenvectors matrix (from the right); * 2, Z contains the eigenvectors. * 3, Z contains the first row of the eigenvectors matrix. If ZNeeded<3, Z is the array whose indexes range within [0..N-1, 0..N-1]. In that case, the eigenvectors are stored in the matrix columns. If ZNeeded=3, Z is the array whose indexes range within [0..0, 0..N-1]. Result: True, if the algorithm has converged. False, if the algorithm hasn't converged. -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994 *************************************************************************/bool alglib::smatrixtdevd( real_1d_array& d, real_1d_array e, ae_int_t n, ae_int_t zneeded, real_2d_array& z);
smatrixtdevdi function/************************************************************************* Subroutine for finding tridiagonal matrix eigenvalues/vectors with given indexes (in ascending order) by using the bisection and inverse iteraion. Input parameters: D - the main diagonal of a tridiagonal matrix. Array whose index ranges within [0..N-1]. E - the secondary diagonal of a tridiagonal matrix. Array whose index ranges within [0..N-2]. N - size of matrix. N>=0. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not needed; * 1, the eigenvectors of a tridiagonal matrix are multiplied by the square matrix Z. It is used if the tridiagonal matrix is obtained by the similarity transformation of a symmetric matrix. * 2, the eigenvectors of a tridiagonal matrix replace matrix Z. I1, I2 - index interval for searching (from I1 to I2). 0 <= I1 <= I2 <= N-1. Z - if ZNeeded is equal to: * 0, Z isn't used and remains unchanged; * 1, Z contains the square matrix (array whose indexes range within [0..N-1, 0..N-1]) which reduces the given symmetric matrix to tridiagonal form; * 2, Z isn't used (but changed on the exit). Output parameters: D - array of the eigenvalues found. Array whose index ranges within [0..I2-I1]. Z - if ZNeeded is equal to: * 0, doesn't contain any information; * 1, contains the product of a given NxN matrix Z (from the left) and Nx(I2-I1) matrix of the eigenvectors found (from the right). Array whose indexes range within [0..N-1, 0..I2-I1]. * 2, contains the matrix of the eigenvalues found. Array whose indexes range within [0..N-1, 0..I2-I1]. Result: True, if successful. In that case, D contains the eigenvalues, Z contains the eigenvectors (if needed). It should be noted that the subroutine changes the size of arrays D and Z. False, if the bisection method subroutine wasn't able to find the eigenvalues in the given interval or if the inverse iteration subroutine wasn't able to find all the corresponding eigenvectors. In that case, the eigenvalues and eigenvectors are not returned. -- ALGLIB -- Copyright 25.12.2005 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixtdevdi( real_1d_array& d, real_1d_array e, ae_int_t n, ae_int_t zneeded, ae_int_t i1, ae_int_t i2, real_2d_array& z);
smatrixtdevdr function/************************************************************************* Subroutine for finding the tridiagonal matrix eigenvalues/vectors in a given half-interval (A, B] by using bisection and inverse iteration. Input parameters: D - the main diagonal of a tridiagonal matrix. Array whose index ranges within [0..N-1]. E - the secondary diagonal of a tridiagonal matrix. Array whose index ranges within [0..N-2]. N - size of matrix, N>=0. ZNeeded - flag controlling whether the eigenvectors are needed or not. If ZNeeded is equal to: * 0, the eigenvectors are not needed; * 1, the eigenvectors of a tridiagonal matrix are multiplied by the square matrix Z. It is used if the tridiagonal matrix is obtained by the similarity transformation of a symmetric matrix. * 2, the eigenvectors of a tridiagonal matrix replace matrix Z. A, B - half-interval (A, B] to search eigenvalues in. Z - if ZNeeded is equal to: * 0, Z isn't used and remains unchanged; * 1, Z contains the square matrix (array whose indexes range within [0..N-1, 0..N-1]) which reduces the given symmetric matrix to tridiagonal form; * 2, Z isn't used (but changed on the exit). Output parameters: D - array of the eigenvalues found. Array whose index ranges within [0..M-1]. M - number of eigenvalues found in the given half-interval (M>=0). Z - if ZNeeded is equal to: * 0, doesn't contain any information; * 1, contains the product of a given NxN matrix Z (from the left) and NxM matrix of the eigenvectors found (from the right). Array whose indexes range within [0..N-1, 0..M-1]. * 2, contains the matrix of the eigenvectors found. Array whose indexes range within [0..N-1, 0..M-1]. Result: True, if successful. In that case, M contains the number of eigenvalues in the given half-interval (could be equal to 0), D contains the eigenvalues, Z contains the eigenvectors (if needed). It should be noted that the subroutine changes the size of arrays D and Z. False, if the bisection method subroutine wasn't able to find the eigenvalues in the given interval or if the inverse iteration subroutine wasn't able to find all the corresponding eigenvectors. In that case, the eigenvalues and eigenvectors are not returned, M is equal to 0. -- ALGLIB -- Copyright 31.03.2008 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixtdevdr( real_1d_array& d, real_1d_array e, ae_int_t n, ae_int_t zneeded, double a, double b, ae_int_t& m, real_2d_array& z);
expintegrals subpackageexponentialintegralei function/************************************************************************* Exponential integral Ei(x) x - t | | e Ei(x) = -|- --- dt . | | t - -inf Not defined for x <= 0. See also expn.c. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,100 50000 8.6e-16 1.3e-16 Cephes Math Library Release 2.8: May, 1999 Copyright 1999 by Stephen L. Moshier *************************************************************************/double alglib::exponentialintegralei(double x);
exponentialintegralen function/************************************************************************* Exponential integral En(x) Evaluates the exponential integral inf. - | | -xt | e E (x) = | ---- dt. n | n | | t - 1 Both n and x must be nonnegative. The routine employs either a power series, a continued fraction, or an asymptotic formula depending on the relative values of n and x. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0, 30 10000 1.7e-15 3.6e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 2000 by Stephen L. Moshier *************************************************************************/double alglib::exponentialintegralen(double x, ae_int_t n);
fdistr subpackagefcdistribution function/************************************************************************* Complemented F distribution Returns the area from x to infinity under the F density function (also known as Snedcor's density or the variance ratio density). inf. - 1 | | a-1 b-1 1-P(x) = ------ | t (1-t) dt B(a,b) | | - x The incomplete beta integral is used, according to the formula P(x) = incbet( df2/2, df1/2, (df2/(df2 + df1*x) ). ACCURACY: Tested at random points (a,b,x) in the indicated intervals. x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 1,100 100000 3.7e-14 5.9e-16 IEEE 1,5 1,100 100000 8.0e-15 1.6e-15 IEEE 0,1 1,10000 100000 1.8e-11 3.5e-13 IEEE 1,5 1,10000 100000 2.0e-11 3.0e-12 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::fcdistribution(ae_int_t a, ae_int_t b, double x);
fdistribution function/************************************************************************* F distribution Returns the area from zero to x under the F density function (also known as Snedcor's density or the variance ratio density). This is the density of x = (u1/df1)/(u2/df2), where u1 and u2 are random variables having Chi square distributions with df1 and df2 degrees of freedom, respectively. The incomplete beta integral is used, according to the formula P(x) = incbet( df1/2, df2/2, (df1*x/(df2 + df1*x) ). The arguments a and b are greater than zero, and x is nonnegative. ACCURACY: Tested at random points (a,b,x). x a,b Relative error: arithmetic domain domain # trials peak rms IEEE 0,1 0,100 100000 9.8e-15 1.7e-15 IEEE 1,5 0,100 100000 6.5e-15 3.5e-16 IEEE 0,1 1,10000 100000 2.2e-11 3.3e-12 IEEE 1,5 1,10000 100000 1.1e-11 1.7e-13 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::fdistribution(ae_int_t a, ae_int_t b, double x);
invfdistribution function/************************************************************************* Inverse of complemented F distribution Finds the F density argument x such that the integral from x to infinity of the F density is equal to the given probability p. This is accomplished using the inverse beta integral function and the relations z = incbi( df2/2, df1/2, p ) x = df2 (1-z) / (df1 z). Note: the following relations hold for the inverse of the uncomplemented F distribution: z = incbi( df1/2, df2/2, p ) x = df2 z / (df1 (1-z)). ACCURACY: Tested at random points (a,b,p). a,b Relative error: arithmetic domain # trials peak rms For p between .001 and 1: IEEE 1,100 100000 8.3e-15 4.7e-16 IEEE 1,10000 100000 2.1e-11 1.4e-13 For p between 10^-6 and 10^-3: IEEE 1,100 50000 1.3e-12 8.4e-15 IEEE 1,10000 50000 3.0e-12 4.8e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invfdistribution(ae_int_t a, ae_int_t b, double y);
fft subpackage| fft_complex_d1 | Complex FFT: simple example | |
| fft_complex_d2 | Complex FFT: advanced example | |
| fft_real_d1 | Real FFT: simple example | |
| fft_real_d2 | Real FFT: advanced example |
fftc1d function/************************************************************************* 1-dimensional complex FFT. Array size N may be arbitrary number (composite or prime). Composite N's are handled with cache-oblivious variation of a Cooley-Tukey algorithm. Small prime-factors are transformed using hard coded codelets (similar to FFTW codelets, but without low-level optimization), large prime-factors are handled with Bluestein's algorithm. Fastests transforms are for smooth N's (prime factors are 2, 3, 5 only), most fast for powers of 2. When N have prime factors larger than these, but orders of magnitude smaller than N, computations will be about 4 times slower than for nearby highly composite N's. When N itself is prime, speed will be 6 times lower. Algorithm has O(N*logN) complexity for any N (composite or prime). INPUT PARAMETERS A - array[0..N-1] - complex function to be transformed N - problem size OUTPUT PARAMETERS A - DFT of a input array, array[0..N-1] A_out[j] = SUM(A_in[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1) -- ALGLIB -- Copyright 29.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::fftc1d(complex_1d_array& a); void alglib::fftc1d(complex_1d_array& a, ae_int_t n);
fftc1dinv function/************************************************************************* 1-dimensional complex inverse FFT. Array size N may be arbitrary number (composite or prime). Algorithm has O(N*logN) complexity for any N (composite or prime). See FFTC1D() description for more information about algorithm performance. INPUT PARAMETERS A - array[0..N-1] - complex array to be transformed N - problem size OUTPUT PARAMETERS A - inverse DFT of a input array, array[0..N-1] A_out[j] = SUM(A_in[k]/N*exp(+2*pi*sqrt(-1)*j*k/N), k = 0..N-1) -- ALGLIB -- Copyright 29.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::fftc1dinv(complex_1d_array& a); void alglib::fftc1dinv(complex_1d_array& a, ae_int_t n);
fftr1d function/************************************************************************* 1-dimensional real FFT. Algorithm has O(N*logN) complexity for any N (composite or prime). INPUT PARAMETERS A - array[0..N-1] - real function to be transformed N - problem size OUTPUT PARAMETERS F - DFT of a input array, array[0..N-1] F[j] = SUM(A[k]*exp(-2*pi*sqrt(-1)*j*k/N), k = 0..N-1) NOTE: F[] satisfies symmetry property F[k] = conj(F[N-k]), so just one half of array is usually needed. But for convinience subroutine returns full complex array (with frequencies above N/2), so its result may be used by other FFT-related subroutines. -- ALGLIB -- Copyright 01.06.2009 by Bochkanov Sergey *************************************************************************/void alglib::fftr1d(real_1d_array a, complex_1d_array& f); void alglib::fftr1d(real_1d_array a, ae_int_t n, complex_1d_array& f);
fftr1dinv function/************************************************************************* 1-dimensional real inverse FFT. Algorithm has O(N*logN) complexity for any N (composite or prime). INPUT PARAMETERS F - array[0..floor(N/2)] - frequencies from forward real FFT N - problem size OUTPUT PARAMETERS A - inverse DFT of a input array, array[0..N-1] NOTE: F[] should satisfy symmetry property F[k] = conj(F[N-k]), so just one half of frequencies array is needed - elements from 0 to floor(N/2). F[0] is ALWAYS real. If N is even F[floor(N/2)] is real too. If N is odd, then F[floor(N/2)] has no special properties. Relying on properties noted above, FFTR1DInv subroutine uses only elements from 0th to floor(N/2)-th. It ignores imaginary part of F[0], and in case N is even it ignores imaginary part of F[floor(N/2)] too. When you call this function using full arguments list - "FFTR1DInv(F,N,A)" - you can pass either either frequencies array with N elements or reduced array with roughly N/2 elements - subroutine will successfully transform both. If you call this function using reduced arguments list - "FFTR1DInv(F,A)" - you must pass FULL array with N elements (although higher N/2 are still not used) because array size is used to automatically determine FFT length -- ALGLIB -- Copyright 01.06.2009 by Bochkanov Sergey *************************************************************************/void alglib::fftr1dinv(complex_1d_array f, real_1d_array& a); void alglib::fftr1dinv(complex_1d_array f, ae_int_t n, real_1d_array& a);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// first we demonstrate forward FFT:
// [1i,1i,1i,1i] is converted to [4i, 0, 0, 0]
//
complex_1d_array z = "[1i,1i,1i,1i]";
fftc1d(z);
printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [4i,0,0,0]
//
// now we convert [4i, 0, 0, 0] back to [1i,1i,1i,1i]
// with backward FFT
//
fftc1dinv(z);
printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [1i,1i,1i,1i]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// first we demonstrate forward FFT:
// [0,1,0,1i] is converted to [1+1i, -1-1i, -1-1i, 1+1i]
//
complex_1d_array z = "[0,1,0,1i]";
fftc1d(z);
printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [1+1i, -1-1i, -1-1i, 1+1i]
//
// now we convert result back with backward FFT
//
fftc1dinv(z);
printf("%s\n", z.tostring(3).c_str()); // EXPECTED: [0,1,0,1i]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// first we demonstrate forward FFT:
// [1,1,1,1] is converted to [4, 0, 0, 0]
//
real_1d_array x = "[1,1,1,1]";
complex_1d_array f;
real_1d_array x2;
fftr1d(x, f);
printf("%s\n", f.tostring(3).c_str()); // EXPECTED: [4,0,0,0]
//
// now we convert [4, 0, 0, 0] back to [1,1,1,1]
// with backward FFT
//
fftr1dinv(f, x2);
printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,1,1,1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "fasttransforms.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// first we demonstrate forward FFT:
// [1,2,3,4] is converted to [10, -2+2i, -2, -2-2i]
//
// note that output array is self-adjoint:
// * f[0] = conj(f[0])
// * f[1] = conj(f[3])
// * f[2] = conj(f[2])
//
real_1d_array x = "[1,2,3,4]";
complex_1d_array f;
real_1d_array x2;
fftr1d(x, f);
printf("%s\n", f.tostring(3).c_str()); // EXPECTED: [10, -2+2i, -2, -2-2i]
//
// now we convert [10, -2+2i, -2, -2-2i] back to [1,2,3,4]
//
fftr1dinv(f, x2);
printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,2,3,4]
//
// remember that F is self-adjoint? It means that we can pass just half
// (slightly larger than half) of F to inverse real FFT and still get our result.
//
// I.e. instead [10, -2+2i, -2, -2-2i] we pass just [10, -2+2i, -2] and everything works!
//
// NOTE: in this case we should explicitly pass array length (which is 4) to ALGLIB;
// if not, it will automatically use array length to determine FFT size and
// will erroneously make half-length FFT.
//
f = "[10, -2+2i, -2]";
fftr1dinv(f, 4, x2);
printf("%s\n", x2.tostring(3).c_str()); // EXPECTED: [1,2,3,4]
return 0;
}
fht subpackagefhtr1d function/************************************************************************* 1-dimensional Fast Hartley Transform. Algorithm has O(N*logN) complexity for any N (composite or prime). INPUT PARAMETERS A - array[0..N-1] - real function to be transformed N - problem size OUTPUT PARAMETERS A - FHT of a input array, array[0..N-1], A_out[k] = sum(A_in[j]*(cos(2*pi*j*k/N)+sin(2*pi*j*k/N)), j=0..N-1) -- ALGLIB -- Copyright 04.06.2009 by Bochkanov Sergey *************************************************************************/void alglib::fhtr1d(real_1d_array& a, ae_int_t n);
fhtr1dinv function/************************************************************************* 1-dimensional inverse FHT. Algorithm has O(N*logN) complexity for any N (composite or prime). INPUT PARAMETERS A - array[0..N-1] - complex array to be transformed N - problem size OUTPUT PARAMETERS A - inverse FHT of a input array, array[0..N-1] -- ALGLIB -- Copyright 29.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::fhtr1dinv(real_1d_array& a, ae_int_t n);
filters subpackage| filters_d_ema | EMA(alpha) filter | |
| filters_d_lrma | LRMA(k) filter | |
| filters_d_sma | SMA(k) filter |
filterema function/************************************************************************* Filters: exponential moving averages. This filter replaces array by results of EMA(alpha) filter. EMA(alpha) is defined as filter which replaces X[] by S[]: S[0] = X[0] S[t] = alpha*X[t] + (1-alpha)*S[t-1] INPUT PARAMETERS: X - array[N], array to process. It can be larger than N, in this case only first N points are processed. N - points count, N>=0 alpha - 0<alpha<=1, smoothing parameter. OUTPUT PARAMETERS: X - array, whose first N elements were processed with EMA(alpha) NOTE 1: this function uses efficient in-place algorithm which does not allocate temporary arrays. NOTE 2: this algorithm uses BOTH previous points and current one, i.e. new value of X[i] depends on BOTH previous point and X[i] itself. NOTE 3: technical analytis users quite often work with EMA coefficient expressed in DAYS instead of fractions. If you want to calculate EMA(N), where N is a number of days, you can use alpha=2/(N+1). -- ALGLIB -- Copyright 25.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::filterema(real_1d_array& x, double alpha); void alglib::filterema(real_1d_array& x, ae_int_t n, double alpha);
Examples: [1]
filterlrma function/************************************************************************* Filters: linear regression moving averages. This filter replaces array by results of LRMA(K) filter. LRMA(K) is defined as filter which, for each data point, builds linear regression model using K prevous points (point itself is included in these K points) and calculates value of this linear model at the point in question. INPUT PARAMETERS: X - array[N], array to process. It can be larger than N, in this case only first N points are processed. N - points count, N>=0 K - K>=1 (K can be larger than N , such cases will be correctly handled). Window width. K=1 corresponds to identity transformation (nothing changes). OUTPUT PARAMETERS: X - array, whose first N elements were processed with SMA(K) NOTE 1: this function uses efficient in-place algorithm which does not allocate temporary arrays. NOTE 2: this algorithm makes only one pass through array and uses running sum to speed-up calculation of the averages. Additional measures are taken to ensure that running sum on a long sequence of zero elements will be correctly reset to zero even in the presence of round-off error. NOTE 3: this is unsymmetric version of the algorithm, which does NOT averages points after the current one. Only X[i], X[i-1], ... are used when calculating new value of X[i]. We should also note that this algorithm uses BOTH previous points and current one, i.e. new value of X[i] depends on BOTH previous point and X[i] itself. -- ALGLIB -- Copyright 25.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::filterlrma(real_1d_array& x, ae_int_t k); void alglib::filterlrma(real_1d_array& x, ae_int_t n, ae_int_t k);
Examples: [1]
filtersma function/************************************************************************* Filters: simple moving averages (unsymmetric). This filter replaces array by results of SMA(K) filter. SMA(K) is defined as filter which averages at most K previous points (previous - not points AROUND central point) - or less, in case of the first K-1 points. INPUT PARAMETERS: X - array[N], array to process. It can be larger than N, in this case only first N points are processed. N - points count, N>=0 K - K>=1 (K can be larger than N , such cases will be correctly handled). Window width. K=1 corresponds to identity transformation (nothing changes). OUTPUT PARAMETERS: X - array, whose first N elements were processed with SMA(K) NOTE 1: this function uses efficient in-place algorithm which does not allocate temporary arrays. NOTE 2: this algorithm makes only one pass through array and uses running sum to speed-up calculation of the averages. Additional measures are taken to ensure that running sum on a long sequence of zero elements will be correctly reset to zero even in the presence of round-off error. NOTE 3: this is unsymmetric version of the algorithm, which does NOT averages points after the current one. Only X[i], X[i-1], ... are used when calculating new value of X[i]. We should also note that this algorithm uses BOTH previous points and current one, i.e. new value of X[i] depends on BOTH previous point and X[i] itself. -- ALGLIB -- Copyright 25.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::filtersma(real_1d_array& x, ae_int_t k); void alglib::filtersma(real_1d_array& x, ae_int_t n, ae_int_t k);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Here we demonstrate EMA(0.5) filtering for time series.
//
real_1d_array x = "[5,6,7,8]";
//
// Apply filter.
// We should get [5, 5.5, 6.25, 7.125] as result
//
filterema(x, 0.5);
printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [5,5.5,6.25,7.125]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Here we demonstrate LRMA(3) filtering for time series.
//
real_1d_array x = "[7,8,8,9,12,12]";
//
// Apply filter.
// We should get [7.0000, 8.0000, 8.1667, 8.8333, 11.6667, 12.5000] as result
//
filterlrma(x, 3);
printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [7.0000,8.0000,8.1667,8.8333,11.6667,12.5000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Here we demonstrate SMA(k) filtering for time series.
//
real_1d_array x = "[5,6,7,8]";
//
// Apply filter.
// We should get [5, 5.5, 6.5, 7.5] as result
//
filtersma(x, 2);
printf("%s\n", x.tostring(4).c_str()); // EXPECTED: [5,5.5,6.5,7.5]
return 0;
}
fresnel subpackagefresnelintegral function/************************************************************************* Fresnel integral Evaluates the Fresnel integrals x - | | C(x) = | cos(pi/2 t**2) dt, | | - 0 x - | | S(x) = | sin(pi/2 t**2) dt. | | - 0 The integrals are evaluated by a power series for x < 1. For x >= 1 auxiliary functions f(x) and g(x) are employed such that C(x) = 0.5 + f(x) sin( pi/2 x**2 ) - g(x) cos( pi/2 x**2 ) S(x) = 0.5 - f(x) cos( pi/2 x**2 ) - g(x) sin( pi/2 x**2 ) ACCURACY: Relative error. Arithmetic function domain # trials peak rms IEEE S(x) 0, 10 10000 2.0e-15 3.2e-16 IEEE C(x) 0, 10 10000 1.8e-15 3.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier *************************************************************************/void alglib::fresnelintegral(double x, double& c, double& s);
gammafunc subpackagegammafunction function/************************************************************************* Gamma function Input parameters: X - argument Domain: 0 < X < 171.6 -170 < X < 0, X is not an integer. Relative error: arithmetic domain # trials peak rms IEEE -170,-33 20000 2.3e-15 3.3e-16 IEEE -33, 33 20000 9.4e-16 2.2e-16 IEEE 33, 171.6 20000 2.3e-15 3.2e-16 Cephes Math Library Release 2.8: June, 2000 Original copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007). *************************************************************************/double alglib::gammafunction(double x);
lngamma function/************************************************************************* Natural logarithm of gamma function Input parameters: X - argument Result: logarithm of the absolute value of the Gamma(X). Output parameters: SgnGam - sign(Gamma(X)) Domain: 0 < X < 2.55e305 -2.55e305 < X < 0, X is not an integer. ACCURACY: arithmetic domain # trials peak rms IEEE 0, 3 28000 5.4e-16 1.1e-16 IEEE 2.718, 2.556e305 40000 3.5e-16 8.3e-17 The error criterion was relative when the function magnitude was greater than one but absolute when it was less than one. The following test used the relative error criterion, though at certain points the relative error could be much higher than indicated. IEEE -200, -4 10000 4.8e-16 1.3e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier Translated to AlgoPascal by Bochkanov Sergey (2005, 2006, 2007). *************************************************************************/double alglib::lngamma(double x, double& sgngam);
gkq subpackagegkqgenerategaussjacobi function/************************************************************************* Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Jacobi quadrature on [-1,1] with weight function W(x)=Power(1-x,Alpha)*Power(1+x,Beta). INPUT PARAMETERS: N - number of Kronrod nodes, must be odd number, >=3. Alpha - power-law coefficient, Alpha>-1 Beta - power-law coefficient, Beta>-1 OUTPUT PARAMETERS: Info - error code: * -5 no real and positive Gauss-Kronrod formula can be created for such a weight function with a given number of nodes. * -4 an error was detected when calculating weights/nodes. Alpha or Beta are too close to -1 to obtain weights/nodes with high enough accuracy, or, may be, N is too large. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N was passed * +1 OK * +2 OK, but quadrature rule have exterior nodes, x[0]<-1 or x[n-1]>+1 X - array[0..N-1] - array of quadrature nodes, ordered in ascending order. WKronrod - array[0..N-1] - Kronrod weights WGauss - array[0..N-1] - Gauss weights (interleaved with zeros corresponding to extended Kronrod nodes). -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gkqgenerategaussjacobi( ae_int_t n, double alpha, double beta, ae_int_t& info, real_1d_array& x, real_1d_array& wkronrod, real_1d_array& wgauss);
gkqgenerategausslegendre function/************************************************************************* Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Legendre quadrature with N points. GKQLegendreCalc (calculation) or GKQLegendreTbl (precomputed table) is used depending on machine precision and number of nodes. INPUT PARAMETERS: N - number of Kronrod nodes, must be odd number, >=3. OUTPUT PARAMETERS: Info - error code: * -4 an error was detected when calculating weights/nodes. N is too large to obtain weights/nodes with high enough accuracy. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, ordered in ascending order. WKronrod - array[0..N-1] - Kronrod weights WGauss - array[0..N-1] - Gauss weights (interleaved with zeros corresponding to extended Kronrod nodes). -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gkqgenerategausslegendre( ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& wkronrod, real_1d_array& wgauss);
gkqgeneraterec function/************************************************************************* Computation of nodes and weights of a Gauss-Kronrod quadrature formula The algorithm generates the N-point Gauss-Kronrod quadrature formula with weight function given by coefficients alpha and beta of a recurrence relation which generates a system of orthogonal polynomials: P-1(x) = 0 P0(x) = 1 Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x) and zero moment Mu0 Mu0 = integral(W(x)dx,a,b) INPUT PARAMETERS: Alpha - alpha coefficients, array[0..floor(3*K/2)]. Beta - beta coefficients, array[0..ceil(3*K/2)]. Beta[0] is not used and may be arbitrary. Beta[I]>0. Mu0 - zeroth moment of the weight function. N - number of nodes of the Gauss-Kronrod quadrature formula, N >= 3, N = 2*K+1. OUTPUT PARAMETERS: Info - error code: * -5 no real and positive Gauss-Kronrod formula can be created for such a weight function with a given number of nodes. * -4 N is too large, task may be ill conditioned - x[i]=x[i+1] found. * -3 internal eigenproblem solver hasn't converged * -2 Beta[i]<=0 * -1 incorrect N was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. WKronrod - array[0..N-1] - Kronrod weights WGauss - array[0..N-1] - Gauss weights (interleaved with zeros corresponding to extended Kronrod nodes). -- ALGLIB -- Copyright 08.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gkqgeneraterec( real_1d_array alpha, real_1d_array beta, double mu0, ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& wkronrod, real_1d_array& wgauss);
gkqlegendrecalc function/************************************************************************* Returns Gauss and Gauss-Kronrod nodes for quadrature with N points. Reduction to tridiagonal eigenproblem is used. INPUT PARAMETERS: N - number of Kronrod nodes, must be odd number, >=3. OUTPUT PARAMETERS: Info - error code: * -4 an error was detected when calculating weights/nodes. N is too large to obtain weights/nodes with high enough accuracy. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, ordered in ascending order. WKronrod - array[0..N-1] - Kronrod weights WGauss - array[0..N-1] - Gauss weights (interleaved with zeros corresponding to extended Kronrod nodes). -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gkqlegendrecalc( ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& wkronrod, real_1d_array& wgauss);
gkqlegendretbl function/************************************************************************* Returns Gauss and Gauss-Kronrod nodes for quadrature with N points using pre-calculated table. Nodes/weights were computed with accuracy up to 1.0E-32 (if MPFR version of ALGLIB is used). In standard double precision accuracy reduces to something about 2.0E-16 (depending on your compiler's handling of long floating point constants). INPUT PARAMETERS: N - number of Kronrod nodes. N can be 15, 21, 31, 41, 51, 61. OUTPUT PARAMETERS: X - array[0..N-1] - array of quadrature nodes, ordered in ascending order. WKronrod - array[0..N-1] - Kronrod weights WGauss - array[0..N-1] - Gauss weights (interleaved with zeros corresponding to extended Kronrod nodes). -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gkqlegendretbl( ae_int_t n, real_1d_array& x, real_1d_array& wkronrod, real_1d_array& wgauss, double& eps);
gq subpackagegqgenerategausshermite function/************************************************************************* Returns nodes/weights for Gauss-Hermite quadrature on (-inf,+inf) with weight function W(x)=Exp(-x*x) INPUT PARAMETERS: N - number of nodes, >=1 OUTPUT PARAMETERS: Info - error code: * -4 an error was detected when calculating weights/nodes. May be, N is too large. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N/Alpha was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgenerategausshermite( ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& w);
gqgenerategaussjacobi function/************************************************************************* Returns nodes/weights for Gauss-Jacobi quadrature on [-1,1] with weight function W(x)=Power(1-x,Alpha)*Power(1+x,Beta). INPUT PARAMETERS: N - number of nodes, >=1 Alpha - power-law coefficient, Alpha>-1 Beta - power-law coefficient, Beta>-1 OUTPUT PARAMETERS: Info - error code: * -4 an error was detected when calculating weights/nodes. Alpha or Beta are too close to -1 to obtain weights/nodes with high enough accuracy, or, may be, N is too large. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N/Alpha/Beta was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgenerategaussjacobi( ae_int_t n, double alpha, double beta, ae_int_t& info, real_1d_array& x, real_1d_array& w);
gqgenerategausslaguerre function/************************************************************************* Returns nodes/weights for Gauss-Laguerre quadrature on [0,+inf) with weight function W(x)=Power(x,Alpha)*Exp(-x) INPUT PARAMETERS: N - number of nodes, >=1 Alpha - power-law coefficient, Alpha>-1 OUTPUT PARAMETERS: Info - error code: * -4 an error was detected when calculating weights/nodes. Alpha is too close to -1 to obtain weights/nodes with high enough accuracy or, may be, N is too large. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N/Alpha was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgenerategausslaguerre( ae_int_t n, double alpha, ae_int_t& info, real_1d_array& x, real_1d_array& w);
gqgenerategausslegendre function/************************************************************************* Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N nodes. INPUT PARAMETERS: N - number of nodes, >=1 OUTPUT PARAMETERS: Info - error code: * -4 an error was detected when calculating weights/nodes. N is too large to obtain weights/nodes with high enough accuracy. Try to use multiple precision version. * -3 internal eigenproblem solver hasn't converged * -1 incorrect N was passed * +1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 12.05.2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgenerategausslegendre( ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& w);
gqgenerategausslobattorec function/************************************************************************* Computation of nodes and weights for a Gauss-Lobatto quadrature formula The algorithm generates the N-point Gauss-Lobatto quadrature formula with weight function given by coefficients alpha and beta of a recurrence which generates a system of orthogonal polynomials. P-1(x) = 0 P0(x) = 1 Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x) and zeroth moment Mu0 Mu0 = integral(W(x)dx,a,b) INPUT PARAMETERS: Alpha - array[0..N-2], alpha coefficients Beta - array[0..N-2], beta coefficients. Zero-indexed element is not used, may be arbitrary. Beta[I]>0 Mu0 - zeroth moment of the weighting function. A - left boundary of the integration interval. B - right boundary of the integration interval. N - number of nodes of the quadrature formula, N>=3 (including the left and right boundary nodes). OUTPUT PARAMETERS: Info - error code: * -3 internal eigenproblem solver hasn't converged * -2 Beta[i]<=0 * -1 incorrect N was passed * 1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 2005-2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgenerategausslobattorec( real_1d_array alpha, real_1d_array beta, double mu0, double a, double b, ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& w);
gqgenerategaussradaurec function/************************************************************************* Computation of nodes and weights for a Gauss-Radau quadrature formula The algorithm generates the N-point Gauss-Radau quadrature formula with weight function given by the coefficients alpha and beta of a recurrence which generates a system of orthogonal polynomials. P-1(x) = 0 P0(x) = 1 Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x) and zeroth moment Mu0 Mu0 = integral(W(x)dx,a,b) INPUT PARAMETERS: Alpha - array[0..N-2], alpha coefficients. Beta - array[0..N-1], beta coefficients Zero-indexed element is not used. Beta[I]>0 Mu0 - zeroth moment of the weighting function. A - left boundary of the integration interval. N - number of nodes of the quadrature formula, N>=2 (including the left boundary node). OUTPUT PARAMETERS: Info - error code: * -3 internal eigenproblem solver hasn't converged * -2 Beta[i]<=0 * -1 incorrect N was passed * 1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 2005-2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgenerategaussradaurec( real_1d_array alpha, real_1d_array beta, double mu0, double a, ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& w);
gqgeneraterec function/************************************************************************* Computation of nodes and weights for a Gauss quadrature formula The algorithm generates the N-point Gauss quadrature formula with weight function given by coefficients alpha and beta of a recurrence relation which generates a system of orthogonal polynomials: P-1(x) = 0 P0(x) = 1 Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x) and zeroth moment Mu0 Mu0 = integral(W(x)dx,a,b) INPUT PARAMETERS: Alpha - array[0..N-1], alpha coefficients Beta - array[0..N-1], beta coefficients Zero-indexed element is not used and may be arbitrary. Beta[I]>0. Mu0 - zeroth moment of the weight function. N - number of nodes of the quadrature formula, N>=1 OUTPUT PARAMETERS: Info - error code: * -3 internal eigenproblem solver hasn't converged * -2 Beta[i]<=0 * -1 incorrect N was passed * 1 OK X - array[0..N-1] - array of quadrature nodes, in ascending order. W - array[0..N-1] - array of quadrature weights. -- ALGLIB -- Copyright 2005-2009 by Bochkanov Sergey *************************************************************************/void alglib::gqgeneraterec( real_1d_array alpha, real_1d_array beta, double mu0, ae_int_t n, ae_int_t& info, real_1d_array& x, real_1d_array& w);
hermite subpackagehermitecalculate function/************************************************************************* Calculation of the value of the Hermite polynomial. Parameters: n - degree, n>=0 x - argument Result: the value of the Hermite polynomial Hn at x *************************************************************************/double alglib::hermitecalculate(ae_int_t n, double x);
hermitecoefficients function/************************************************************************* Representation of Hn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/void alglib::hermitecoefficients(ae_int_t n, real_1d_array& c);
hermitesum function/************************************************************************* Summation of Hermite polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*H0(x) + c[1]*H1(x) + ... + c[N]*HN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Hermite polynomial at x *************************************************************************/double alglib::hermitesum(real_1d_array c, ae_int_t n, double x);
hqrnd subpackagehqrndstate class/************************************************************************* Portable high quality random number generator state. Initialized with HQRNDRandomize() or HQRNDSeed(). Fields: S1, S2 - seed values V - precomputed value MagicV - 'magic' value used to determine whether State structure was correctly initialized. *************************************************************************/class hqrndstate { };
hqrndcontinuous function/************************************************************************* This function generates random number from continuous distribution given by finite sample X. INPUT PARAMETERS State - high quality random number generator, must be initialized with HQRNDRandomize() or HQRNDSeed(). X - finite sample, array[N] (can be larger, in this case only leading N elements are used). THIS ARRAY MUST BE SORTED BY ASCENDING. N - number of elements to use, N>=1 RESULT this function returns random number from continuous distribution which tries to approximate X as mush as possible. min(X)<=Result<=max(X). -- ALGLIB -- Copyright 08.11.2011 by Bochkanov Sergey *************************************************************************/double alglib::hqrndcontinuous( hqrndstate state, real_1d_array x, ae_int_t n);
hqrnddiscrete function/************************************************************************* This function generates random number from discrete distribution given by finite sample X. INPUT PARAMETERS State - high quality random number generator, must be initialized with HQRNDRandomize() or HQRNDSeed(). X - finite sample N - number of elements to use, N>=1 RESULT this function returns one of the X[i] for random i=0..N-1 -- ALGLIB -- Copyright 08.11.2011 by Bochkanov Sergey *************************************************************************/double alglib::hqrnddiscrete( hqrndstate state, real_1d_array x, ae_int_t n);
hqrndexponential function/************************************************************************* Random number generator: exponential distribution State structure must be initialized with HQRNDRandomize() or HQRNDSeed(). -- ALGLIB -- Copyright 11.08.2007 by Bochkanov Sergey *************************************************************************/double alglib::hqrndexponential(hqrndstate state, double lambdav);
hqrndnormal function/************************************************************************* Random number generator: normal numbers This function generates one random number from normal distribution. Its performance is equal to that of HQRNDNormal2() State structure must be initialized with HQRNDRandomize() or HQRNDSeed(). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/double alglib::hqrndnormal(hqrndstate state);
hqrndnormal2 function/************************************************************************* Random number generator: normal numbers This function generates two independent random numbers from normal distribution. Its performance is equal to that of HQRNDNormal() State structure must be initialized with HQRNDRandomize() or HQRNDSeed(). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::hqrndnormal2(hqrndstate state, double& x1, double& x2);
hqrndrandomize function/************************************************************************* HQRNDState initialization with random values which come from standard RNG. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::hqrndrandomize(hqrndstate& state);
hqrndseed function/************************************************************************* HQRNDState initialization with seed values -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::hqrndseed(ae_int_t s1, ae_int_t s2, hqrndstate& state);
hqrnduniformi function/************************************************************************* This function generates random integer number in [0, N) 1. State structure must be initialized with HQRNDRandomize() or HQRNDSeed() 2. N can be any positive number except for very large numbers: * close to 2^31 on 32-bit systems * close to 2^62 on 64-bit systems An exception will be generated if N is too large. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::hqrnduniformi(hqrndstate state, ae_int_t n);
hqrnduniformr function/************************************************************************* This function generates random real number in (0,1), not including interval boundaries State structure must be initialized with HQRNDRandomize() or HQRNDSeed(). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/double alglib::hqrnduniformr(hqrndstate state);
hqrndunit2 function/************************************************************************* Random number generator: random X and Y such that X^2+Y^2=1 State structure must be initialized with HQRNDRandomize() or HQRNDSeed(). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::hqrndunit2(hqrndstate state, double& x, double& y);
ibetaf subpackageincompletebeta function/************************************************************************* Incomplete beta integral Returns incomplete beta integral of the arguments, evaluated from zero to x. The function is defined as x - - | (a+b) | | a-1 b-1 ----------- | t (1-t) dt. - - | | | (a) | (b) - 0 The domain of definition is 0 <= x <= 1. In this implementation a and b are restricted to positive values. The integral from x to 1 may be obtained by the symmetry relation 1 - incbet( a, b, x ) = incbet( b, a, 1-x ). The integral is evaluated by a continued fraction expansion or, when b*x is small, by a power series. ACCURACY: Tested at uniformly distributed random points (a,b,x) with a and b in "domain" and x between 0 and 1. Relative error arithmetic domain # trials peak rms IEEE 0,5 10000 6.9e-15 4.5e-16 IEEE 0,85 250000 2.2e-13 1.7e-14 IEEE 0,1000 30000 5.3e-12 6.3e-13 IEEE 0,10000 250000 9.3e-11 7.1e-12 IEEE 0,100000 10000 8.7e-10 4.8e-11 Outputs smaller than the IEEE gradual underflow threshold were excluded from these statistics. Cephes Math Library, Release 2.8: June, 2000 Copyright 1984, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::incompletebeta(double a, double b, double x);
invincompletebeta function/************************************************************************* Inverse of imcomplete beta integral Given y, the function finds x such that incbet( a, b, x ) = y . The routine performs interval halving or Newton iterations to find the root of incbet(a,b,x) - y = 0. ACCURACY: Relative error: x a,b arithmetic domain domain # trials peak rms IEEE 0,1 .5,10000 50000 5.8e-12 1.3e-13 IEEE 0,1 .25,100 100000 1.8e-13 3.9e-15 IEEE 0,1 0,5 50000 1.1e-12 5.5e-15 With a and b constrained to half-integer or integer values: IEEE 0,1 .5,10000 50000 5.8e-12 1.1e-13 IEEE 0,1 .5,100 100000 1.7e-14 7.9e-16 With a = .5, b constrained to half-integer or integer values: IEEE 0,1 .5,10000 10000 8.3e-11 1.0e-11 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1996, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invincompletebeta(double a, double b, double y);
idwint subpackageidwinterpolant class/************************************************************************* IDW interpolant. *************************************************************************/class idwinterpolant { };
idwbuildmodifiedshepard function/************************************************************************* IDW interpolant using modified Shepard method for uniform point distributions. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. D - nodal function type, either: * 0 constant model. Just for demonstration only, worst model ever. * 1 linear model, least squares fitting. Simpe model for datasets too small for quadratic models * 2 quadratic model, least squares fitting. Best model available (if your dataset is large enough). * -1 "fast" linear model, use with caution!!! It is significantly faster than linear/quadratic and better than constant model. But it is less robust (especially in the presence of noise). NQ - number of points used to calculate nodal functions (ignored for constant models). NQ should be LARGER than: * max(1.5*(1+NX),2^NX+1) for linear model, * max(3/4*(NX+2)*(NX+1),2^NX+1) for quadratic model. Values less than this threshold will be silently increased. NW - number of points used to calculate weights and to interpolate. Required: >=2^NX+1, values less than this threshold will be silently increased. Recommended value: about 2*NQ OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * best results are obtained with quadratic models, worst - with constant models * when N is large, NQ and NW must be significantly smaller than N both to obtain optimal performance and to obtain optimal accuracy. In 2 or 3-dimensional tasks NQ=15 and NW=25 are good values to start with. * NQ and NW may be greater than N. In such cases they will be automatically decreased. * this subroutine is always succeeds (as long as correct parameters are passed). * see 'Multivariate Interpolation of Large Sets of Scattered Data' by Robert J. Renka for more information on this algorithm. * this subroutine assumes that point distribution is uniform at the small scales. If it isn't - for example, points are concentrated along "lines", but "lines" distribution is uniform at the larger scale - then you should use IDWBuildModifiedShepardR() -- ALGLIB PROJECT -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/void alglib::idwbuildmodifiedshepard( real_2d_array xy, ae_int_t n, ae_int_t nx, ae_int_t d, ae_int_t nq, ae_int_t nw, idwinterpolant& z);
idwbuildmodifiedshepardr function/************************************************************************* IDW interpolant using modified Shepard method for non-uniform datasets. This type of model uses constant nodal functions and interpolates using all nodes which are closer than user-specified radius R. It may be used when points distribution is non-uniform at the small scale, but it is at the distances as large as R. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. R - radius, R>0 OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * if there is less than IDWKMin points within R-ball, algorithm selects IDWKMin closest ones, so that continuity properties of interpolant are preserved even far from points. -- ALGLIB PROJECT -- Copyright 11.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::idwbuildmodifiedshepardr( real_2d_array xy, ae_int_t n, ae_int_t nx, double r, idwinterpolant& z);
idwbuildnoisy function/************************************************************************* IDW model for noisy data. This subroutine may be used to handle noisy data, i.e. data with noise in OUTPUT values. It differs from IDWBuildModifiedShepard() in the following aspects: * nodal functions are not constrained to pass through nodes: Qi(xi)<>yi, i.e. we have fitting instead of interpolation. * weights which are used during least squares fitting stage are all equal to 1.0 (independently of distance) * "fast"-linear or constant nodal functions are not supported (either not robust enough or too rigid) This problem require far more complex tuning than interpolation problems. Below you can find some recommendations regarding this problem: * focus on tuning NQ; it controls noise reduction. As for NW, you can just make it equal to 2*NQ. * you can use cross-validation to determine optimal NQ. * optimal NQ is a result of complex tradeoff between noise level (more noise = larger NQ required) and underlying function complexity (given fixed N, larger NQ means smoothing of compex features in the data). For example, NQ=N will reduce noise to the minimum level possible, but you will end up with just constant/linear/quadratic (depending on D) least squares model for the whole dataset. INPUT PARAMETERS: XY - X and Y values, array[0..N-1,0..NX]. First NX columns contain X-values, last column contain Y-values. N - number of nodes, N>0. NX - space dimension, NX>=1. D - nodal function degree, either: * 1 linear model, least squares fitting. Simpe model for datasets too small for quadratic models (or for very noisy problems). * 2 quadratic model, least squares fitting. Best model available (if your dataset is large enough). NQ - number of points used to calculate nodal functions. NQ should be significantly larger than 1.5 times the number of coefficients in a nodal function to overcome effects of noise: * larger than 1.5*(1+NX) for linear model, * larger than 3/4*(NX+2)*(NX+1) for quadratic model. Values less than this threshold will be silently increased. NW - number of points used to calculate weights and to interpolate. Required: >=2^NX+1, values less than this threshold will be silently increased. Recommended value: about 2*NQ or larger OUTPUT PARAMETERS: Z - IDW interpolant. NOTES: * best results are obtained with quadratic models, linear models are not recommended to use unless you are pretty sure that it is what you want * this subroutine is always succeeds (as long as correct parameters are passed). * see 'Multivariate Interpolation of Large Sets of Scattered Data' by Robert J. Renka for more information on this algorithm. -- ALGLIB PROJECT -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/void alglib::idwbuildnoisy( real_2d_array xy, ae_int_t n, ae_int_t nx, ae_int_t d, ae_int_t nq, ae_int_t nw, idwinterpolant& z);
idwcalc function/************************************************************************* IDW interpolation INPUT PARAMETERS: Z - IDW interpolant built with one of model building subroutines. X - array[0..NX-1], interpolation point Result: IDW interpolant Z(X) -- ALGLIB -- Copyright 02.03.2010 by Bochkanov Sergey *************************************************************************/double alglib::idwcalc(idwinterpolant z, real_1d_array x);
igammaf subpackageincompletegamma function/************************************************************************* Incomplete gamma integral The function is defined by x - 1 | | -t a-1 igam(a,x) = ----- | e t dt. - | | | (a) - 0 In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,30 200000 3.6e-14 2.9e-15 IEEE 0,100 300000 9.9e-14 1.5e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::incompletegamma(double a, double x);
incompletegammac function/************************************************************************* Complemented incomplete gamma integral The function is defined by igamc(a,x) = 1 - igam(a,x) inf. - 1 | | -t a-1 = ----- | e t dt. - | | | (a) - x In this implementation both arguments must be positive. The integral is evaluated by either a power series or continued fraction expansion, depending on the relative values of a and x. ACCURACY: Tested at random a, x. a x Relative error: arithmetic domain domain # trials peak rms IEEE 0.5,100 0,100 200000 1.9e-14 1.7e-15 IEEE 0.01,0.5 0,100 200000 1.4e-13 1.6e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1985, 1987, 2000 by Stephen L. Moshier *************************************************************************/double alglib::incompletegammac(double a, double x);
invincompletegammac function/************************************************************************* Inverse of complemented imcomplete gamma integral Given p, the function finds x such that igamc( a, x ) = p. Starting with the approximate value 3 x = a t where t = 1 - d - ndtri(p) sqrt(d) and d = 1/9a, the routine performs up to 10 Newton iterations to find the root of igamc(a,x) - p = 0. ACCURACY: Tested at random a, p in the intervals indicated. a p Relative error: arithmetic domain domain # trials peak rms IEEE 0.5,100 0,0.5 100000 1.0e-14 1.7e-15 IEEE 0.01,0.5 0,0.5 100000 9.0e-14 3.4e-15 IEEE 0.5,10000 0,0.5 20000 2.3e-13 3.8e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invincompletegammac(double a, double y0);
inverseupdate subpackagermatrixinvupdatecolumn function/************************************************************************* Inverse matrix update by the Sherman-Morrison formula The algorithm updates matrix A^-1 when adding a vector to a column of matrix A. Input parameters: InvA - inverse of matrix A. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. UpdColumn - the column of A whose vector U was added. 0 <= UpdColumn <= N-1 U - the vector to be added to a column. Array whose index ranges within [0..N-1]. Output parameters: InvA - inverse of modified matrix A. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixinvupdatecolumn( real_2d_array& inva, ae_int_t n, ae_int_t updcolumn, real_1d_array u);
rmatrixinvupdaterow function/************************************************************************* Inverse matrix update by the Sherman-Morrison formula The algorithm updates matrix A^-1 when adding a vector to a row of matrix A. Input parameters: InvA - inverse of matrix A. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. UpdRow - the row of A whose vector V was added. 0 <= Row <= N-1 V - the vector to be added to a row. Array whose index ranges within [0..N-1]. Output parameters: InvA - inverse of modified matrix A. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixinvupdaterow( real_2d_array& inva, ae_int_t n, ae_int_t updrow, real_1d_array v);
rmatrixinvupdatesimple function/************************************************************************* Inverse matrix update by the Sherman-Morrison formula The algorithm updates matrix A^-1 when adding a number to an element of matrix A. Input parameters: InvA - inverse of matrix A. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. UpdRow - row where the element to be updated is stored. UpdColumn - column where the element to be updated is stored. UpdVal - a number to be added to the element. Output parameters: InvA - inverse of modified matrix A. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixinvupdatesimple( real_2d_array& inva, ae_int_t n, ae_int_t updrow, ae_int_t updcolumn, double updval);
rmatrixinvupdateuv function/************************************************************************* Inverse matrix update by the Sherman-Morrison formula The algorithm computes the inverse of matrix A+u*v' by using the given matrix A^-1 and the vectors u and v. Input parameters: InvA - inverse of matrix A. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. U - the vector modifying the matrix. Array whose index ranges within [0..N-1]. V - the vector modifying the matrix. Array whose index ranges within [0..N-1]. Output parameters: InvA - inverse of matrix A + u*v'. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixinvupdateuv( real_2d_array& inva, ae_int_t n, real_1d_array u, real_1d_array v);
jacobianelliptic subpackagejacobianellipticfunctions function/************************************************************************* Jacobian Elliptic Functions Evaluates the Jacobian elliptic functions sn(u|m), cn(u|m), and dn(u|m) of parameter m between 0 and 1, and real argument u. These functions are periodic, with quarter-period on the real axis equal to the complete elliptic integral ellpk(1.0-m). Relation to incomplete elliptic integral: If u = ellik(phi,m), then sn(u|m) = sin(phi), and cn(u|m) = cos(phi). Phi is called the amplitude of u. Computation is by means of the arithmetic-geometric mean algorithm, except when m is within 1e-9 of 0 or 1. In the latter case with m close to 1, the approximation applies only for phi < pi/2. ACCURACY: Tested at random points with u between 0 and 10, m between 0 and 1. Absolute error (* = relative error): arithmetic function # trials peak rms IEEE phi 10000 9.2e-16* 1.4e-16* IEEE sn 50000 4.1e-15 4.6e-16 IEEE cn 40000 3.6e-15 4.4e-16 IEEE dn 10000 1.3e-12 1.8e-14 Peak error observed in consistency check using addition theorem for sn(u+v) was 4e-16 (absolute). Also tested by the above relation to the incomplete elliptic integral. Accuracy deteriorates when u is large. Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/void alglib::jacobianellipticfunctions( double u, double m, double& sn, double& cn, double& dn, double& ph);
jarquebera subpackagejarqueberatest function/************************************************************************* Jarque-Bera test This test checks hypotheses about the fact that a given sample X is a sample of normal random variable. Requirements: * the number of elements in the sample is not less than 5. Input parameters: X - sample. Array whose index goes from 0 to N-1. N - size of the sample. N>=5 Output parameters: P - p-value for the test Accuracy of the approximation used (5<=N<=1951): p-value relative error (5<=N<=1951) [1, 0.1] < 1% [0.1, 0.01] < 2% [0.01, 0.001] < 6% [0.001, 0] wasn't measured For N>1951 accuracy wasn't measured but it shouldn't be sharply different from table values. -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/void alglib::jarqueberatest(real_1d_array x, ae_int_t n, double& p);
laguerre subpackagelaguerrecalculate function/************************************************************************* Calculation of the value of the Laguerre polynomial. Parameters: n - degree, n>=0 x - argument Result: the value of the Laguerre polynomial Ln at x *************************************************************************/double alglib::laguerrecalculate(ae_int_t n, double x);
laguerrecoefficients function/************************************************************************* Representation of Ln as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/void alglib::laguerrecoefficients(ae_int_t n, real_1d_array& c);
laguerresum function/************************************************************************* Summation of Laguerre polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*L0(x) + c[1]*L1(x) + ... + c[N]*LN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Laguerre polynomial at x *************************************************************************/double alglib::laguerresum(real_1d_array c, ae_int_t n, double x);
lda subpackagefisherlda function/************************************************************************* Multiclass Fisher LDA Subroutine finds coefficients of linear combination which optimally separates training set on classes. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. Best results are achieved for high-dimensional problems ! (NVars is at least 256). ! ! Multithreading is used to accelerate initial phase of LDA, which ! includes calculation of products of large matrices. Again, for best ! efficiency problem must be high-dimensional. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: XY - training set, array[0..NPoints-1,0..NVars]. First NVars columns store values of independent variables, next column stores number of class (from 0 to NClasses-1) which dataset element belongs to. Fractional values are rounded to nearest integer. NPoints - training set size, NPoints>=0 NVars - number of independent variables, NVars>=1 NClasses - number of classes, NClasses>=2 OUTPUT PARAMETERS: Info - return code: * -4, if internal EVD subroutine hasn't converged * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<0, NVars<1, NClasses<2) * 1, if task has been solved * 2, if there was a multicollinearity in training set, but task has been solved. W - linear combination coefficients, array[0..NVars-1] -- ALGLIB -- Copyright 31.05.2008 by Bochkanov Sergey *************************************************************************/void alglib::fisherlda( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t nclasses, ae_int_t& info, real_1d_array& w);
fisherldan function/************************************************************************* N-dimensional multiclass Fisher LDA Subroutine finds coefficients of linear combinations which optimally separates training set on classes. It returns N-dimensional basis whose vector are sorted by quality of training set separation (in descending order). COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. Best results are achieved for high-dimensional problems ! (NVars is at least 256). ! ! Multithreading is used to accelerate initial phase of LDA, which ! includes calculation of products of large matrices. Again, for best ! efficiency problem must be high-dimensional. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: XY - training set, array[0..NPoints-1,0..NVars]. First NVars columns store values of independent variables, next column stores number of class (from 0 to NClasses-1) which dataset element belongs to. Fractional values are rounded to nearest integer. NPoints - training set size, NPoints>=0 NVars - number of independent variables, NVars>=1 NClasses - number of classes, NClasses>=2 OUTPUT PARAMETERS: Info - return code: * -4, if internal EVD subroutine hasn't converged * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<0, NVars<1, NClasses<2) * 1, if task has been solved * 2, if there was a multicollinearity in training set, but task has been solved. W - basis, array[0..NVars-1,0..NVars-1] columns of matrix stores basis vectors, sorted by quality of training set separation (in descending order) -- ALGLIB -- Copyright 31.05.2008 by Bochkanov Sergey *************************************************************************/void alglib::fisherldan( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t nclasses, ae_int_t& info, real_2d_array& w); void alglib::smp_fisherldan( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t nclasses, ae_int_t& info, real_2d_array& w);
legendre subpackagelegendrecalculate function/************************************************************************* Calculation of the value of the Legendre polynomial Pn. Parameters: n - degree, n>=0 x - argument Result: the value of the Legendre polynomial Pn at x *************************************************************************/double alglib::legendrecalculate(ae_int_t n, double x);
legendrecoefficients function/************************************************************************* Representation of Pn as C[0] + C[1]*X + ... + C[N]*X^N Input parameters: N - polynomial degree, n>=0 Output parameters: C - coefficients *************************************************************************/void alglib::legendrecoefficients(ae_int_t n, real_1d_array& c);
legendresum function/************************************************************************* Summation of Legendre polynomials using Clenshaw's recurrence formula. This routine calculates c[0]*P0(x) + c[1]*P1(x) + ... + c[N]*PN(x) Parameters: n - degree, n>=0 x - argument Result: the value of the Legendre polynomial at x *************************************************************************/double alglib::legendresum(real_1d_array c, ae_int_t n, double x);
lincg subpackage| lincg_d_1 | Solution of sparse linear systems with CG |
lincgreport class/************************************************************************* *************************************************************************/class lincgreport { ae_int_t iterationscount; ae_int_t nmv; ae_int_t terminationtype; double r2; };
lincgstate class/************************************************************************* This object stores state of the linear CG method. You should use ALGLIB functions to work with this object. Never try to access its fields directly! *************************************************************************/class lincgstate { };
lincgcreate function/************************************************************************* This function initializes linear CG Solver. This solver is used to solve symmetric positive definite problems. If you want to solve nonsymmetric (or non-positive definite) problem you may use LinLSQR solver provided by ALGLIB. USAGE: 1. User initializes algorithm state with LinCGCreate() call 2. User tunes solver parameters with LinCGSetCond() and other functions 3. Optionally, user sets starting point with LinCGSetStartingPoint() 4. User calls LinCGSolveSparse() function which takes algorithm state and SparseMatrix object. 5. User calls LinCGResults() to get solution 6. Optionally, user may call LinCGSolveSparse() again to solve another problem with different matrix and/or right part without reinitializing LinCGState structure. INPUT PARAMETERS: N - problem dimension, N>0 OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgcreate(ae_int_t n, lincgstate& state);
Examples: [1]
lincgresults function/************************************************************************* CG-solver: results. This function must be called after LinCGSolve INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[N], solution Rep - optimization report: * Rep.TerminationType completetion code: * -5 input matrix is either not positive definite, too large or too small * -4 overflow/underflow during solution (ill conditioned problem) * 1 ||residual||<=EpsF*||b|| * 5 MaxIts steps was taken * 7 rounding errors prevent further progress, best point found is returned * Rep.IterationsCount contains iterations count * NMV countains number of matrix-vector calculations -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgresults( lincgstate state, real_1d_array& x, lincgreport& rep);
Examples: [1]
lincgsetcond function/************************************************************************* This function sets stopping criteria. INPUT PARAMETERS: EpsF - algorithm will be stopped if norm of residual is less than EpsF*||b||. MaxIts - algorithm will be stopped if number of iterations is more than MaxIts. OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: If both EpsF and MaxIts are zero then small EpsF will be set to small value. -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetcond(lincgstate state, double epsf, ae_int_t maxits);
lincgsetprecdiag function/************************************************************************* This function changes preconditioning settings of LinCGSolveSparse() function. LinCGSolveSparse() will use diagonal of the system matrix as preconditioner. This preconditioning mode is active by default. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 19.11.2012 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetprecdiag(lincgstate state);
lincgsetprecunit function/************************************************************************* This function changes preconditioning settings of LinCGSolveSparse() function. By default, SolveSparse() uses diagonal preconditioner, but if you want to use solver without preconditioning, you can call this function which forces solver to use unit matrix for preconditioning. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 19.11.2012 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetprecunit(lincgstate state);
lincgsetrestartfreq function/************************************************************************* This function sets restart frequency. By default, algorithm is restarted after N subsequent iterations. -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetrestartfreq(lincgstate state, ae_int_t srf);
lincgsetrupdatefreq function/************************************************************************* This function sets frequency of residual recalculations. Algorithm updates residual r_k using iterative formula, but recalculates it from scratch after each 10 iterations. It is done to avoid accumulation of numerical errors and to stop algorithm when r_k starts to grow. Such low update frequence (1/10) gives very little overhead, but makes algorithm a bit more robust against numerical errors. However, you may change it INPUT PARAMETERS: Freq - desired update frequency, Freq>=0. Zero value means that no updates will be done. -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetrupdatefreq(lincgstate state, ae_int_t freq);
lincgsetstartingpoint function/************************************************************************* This function sets starting point. By default, zero starting point is used. INPUT PARAMETERS: X - starting point, array[N] OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetstartingpoint(lincgstate state, real_1d_array x);
lincgsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinCGOptimize(). -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgsetxrep(lincgstate state, bool needxrep);
lincgsolvesparse function/************************************************************************* Procedure for solution of A*x=b with sparse A. INPUT PARAMETERS: State - algorithm state A - sparse matrix in the CRS format (you MUST contvert it to CRS format by calling SparseConvertToCRS() function). IsUpper - whether upper or lower triangle of A is used: * IsUpper=True => only upper triangle is used and lower triangle is not referenced at all * IsUpper=False => only lower triangle is used and upper triangle is not referenced at all B - right part, array[N] RESULT: This function returns no result. You can get solution by calling LinCGResults() NOTE: this function uses lightweight preconditioning - multiplication by inverse of diag(A). If you want, you can turn preconditioning off by calling LinCGSetPrecUnit(). However, preconditioning cost is low and preconditioner is very important for solution of badly scaled problems. -- ALGLIB -- Copyright 14.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::lincgsolvesparse( lincgstate state, sparsematrix a, bool isupper, real_1d_array b);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example illustrates solution of sparse linear systems with
// conjugate gradient method.
//
// Suppose that we have linear system A*x=b with sparse symmetric
// positive definite A (represented by sparsematrix object)
// [ 5 1 ]
// [ 1 7 2 ]
// A = [ 2 8 1 ]
// [ 1 4 1 ]
// [ 1 4 ]
// and right part b
// [ 7 ]
// [ 17 ]
// b = [ 14 ]
// [ 10 ]
// [ 6 ]
// and we want to solve this system using sparse linear CG. In order
// to do so, we have to create left part (sparsematrix object) and
// right part (dense array).
//
// Initially, sparse matrix is created in the Hash-Table format,
// which allows easy initialization, but do not allow matrix to be
// used in the linear solvers. So after construction you should convert
// sparse matrix to CRS format (one suited for linear operations).
//
// It is important to note that in our example we initialize full
// matrix A, both lower and upper triangles. However, it is symmetric
// and sparse solver needs just one half of the matrix. So you may
// save about half of the space by filling only one of the triangles.
//
sparsematrix a;
sparsecreate(5, 5, a);
sparseset(a, 0, 0, 5.0);
sparseset(a, 0, 1, 1.0);
sparseset(a, 1, 0, 1.0);
sparseset(a, 1, 1, 7.0);
sparseset(a, 1, 2, 2.0);
sparseset(a, 2, 1, 2.0);
sparseset(a, 2, 2, 8.0);
sparseset(a, 2, 3, 1.0);
sparseset(a, 3, 2, 1.0);
sparseset(a, 3, 3, 4.0);
sparseset(a, 3, 4, 1.0);
sparseset(a, 4, 3, 1.0);
sparseset(a, 4, 4, 4.0);
//
// Now our matrix is fully initialized, but we have to do one more
// step - convert it from Hash-Table format to CRS format (see
// documentation on sparse matrices for more information about these
// formats).
//
// If you omit this call, ALGLIB will generate exception on the first
// attempt to use A in linear operations.
//
sparseconverttocrs(a);
//
// Initialization of the right part
//
real_1d_array b = "[7,17,14,10,6]";
//
// Now we have to create linear solver object and to use it for the
// solution of the linear system.
//
// NOTE: lincgsolvesparse() accepts additional parameter which tells
// what triangle of the symmetric matrix should be used - upper
// or lower. Because we've filled both parts of the matrix, we
// can use any part - upper or lower.
//
lincgstate s;
lincgreport rep;
real_1d_array x;
lincgcreate(5, s);
lincgsolvesparse(s, a, true, b);
lincgresults(s, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.000,2.000,1.000,2.000,1.000]
return 0;
}
linlsqr subpackage| linlsqr_d_1 | Solution of sparse linear systems with CG |
linlsqrreport class/************************************************************************* *************************************************************************/class linlsqrreport { ae_int_t iterationscount; ae_int_t nmv; ae_int_t terminationtype; };
linlsqrstate class/************************************************************************* This object stores state of the LinLSQR method. You should use ALGLIB functions to work with this object. *************************************************************************/class linlsqrstate { };
linlsqrcreate function/************************************************************************* This function initializes linear LSQR Solver. This solver is used to solve non-symmetric (and, possibly, non-square) problems. Least squares solution is returned for non-compatible systems. USAGE: 1. User initializes algorithm state with LinLSQRCreate() call 2. User tunes solver parameters with LinLSQRSetCond() and other functions 3. User calls LinLSQRSolveSparse() function which takes algorithm state and SparseMatrix object. 4. User calls LinLSQRResults() to get solution 5. Optionally, user may call LinLSQRSolveSparse() again to solve another problem with different matrix and/or right part without reinitializing LinLSQRState structure. INPUT PARAMETERS: M - number of rows in A N - number of variables, N>0 OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 30.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrcreate(ae_int_t m, ae_int_t n, linlsqrstate& state);
Examples: [1]
linlsqrresults function/************************************************************************* LSQR solver: results. This function must be called after LinLSQRSolve INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[N], solution Rep - optimization report: * Rep.TerminationType completetion code: * 1 ||Rk||<=EpsB*||B|| * 4 ||A^T*Rk||/(||A||*||Rk||)<=EpsA * 5 MaxIts steps was taken * 7 rounding errors prevent further progress, X contains best point found so far. (sometimes returned on singular systems) * Rep.IterationsCount contains iterations count * NMV countains number of matrix-vector calculations -- ALGLIB -- Copyright 30.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrresults( linlsqrstate state, real_1d_array& x, linlsqrreport& rep);
Examples: [1]
linlsqrsetcond function/************************************************************************* This function sets stopping criteria. INPUT PARAMETERS: EpsA - algorithm will be stopped if ||A^T*Rk||/(||A||*||Rk||)<=EpsA. EpsB - algorithm will be stopped if ||Rk||<=EpsB*||B|| MaxIts - algorithm will be stopped if number of iterations more than MaxIts. OUTPUT PARAMETERS: State - structure which stores algorithm state NOTE: if EpsA,EpsB,EpsC and MaxIts are zero then these variables will be setted as default values. -- ALGLIB -- Copyright 30.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrsetcond( linlsqrstate state, double epsa, double epsb, ae_int_t maxits);
linlsqrsetlambdai function/************************************************************************* This function sets optional Tikhonov regularization coefficient. It is zero by default. INPUT PARAMETERS: LambdaI - regularization factor, LambdaI>=0 OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 30.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrsetlambdai(linlsqrstate state, double lambdai);
linlsqrsetprecdiag function/************************************************************************* This function changes preconditioning settings of LinCGSolveSparse() function. LinCGSolveSparse() will use diagonal of the system matrix as preconditioner. This preconditioning mode is active by default. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 19.11.2012 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrsetprecdiag(linlsqrstate state);
linlsqrsetprecunit function/************************************************************************* This function changes preconditioning settings of LinLSQQSolveSparse() function. By default, SolveSparse() uses diagonal preconditioner, but if you want to use solver without preconditioning, you can call this function which forces solver to use unit matrix for preconditioning. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 19.11.2012 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrsetprecunit(linlsqrstate state);
linlsqrsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinCGOptimize(). -- ALGLIB -- Copyright 30.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrsetxrep(linlsqrstate state, bool needxrep);
linlsqrsolvesparse function/************************************************************************* Procedure for solution of A*x=b with sparse A. INPUT PARAMETERS: State - algorithm state A - sparse M*N matrix in the CRS format (you MUST contvert it to CRS format by calling SparseConvertToCRS() function BEFORE you pass it to this function). B - right part, array[M] RESULT: This function returns no result. You can get solution by calling LinCGResults() NOTE: this function uses lightweight preconditioning - multiplication by inverse of diag(A). If you want, you can turn preconditioning off by calling LinLSQRSetPrecUnit(). However, preconditioning cost is low and preconditioner is very important for solution of badly scaled problems. -- ALGLIB -- Copyright 30.11.2011 by Bochkanov Sergey *************************************************************************/void alglib::linlsqrsolvesparse( linlsqrstate state, sparsematrix a, real_1d_array b);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "solvers.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example illustrates solution of sparse linear least squares problem
// with LSQR algorithm.
//
// Suppose that we have least squares problem min|A*x-b| with sparse A
// represented by sparsematrix object
// [ 1 1 ]
// [ 1 1 ]
// A = [ 2 1 ]
// [ 1 ]
// [ 1 ]
// and right part b
// [ 4 ]
// [ 2 ]
// b = [ 4 ]
// [ 1 ]
// [ 2 ]
// and we want to solve this system in the least squares sense using
// LSQR algorithm. In order to do so, we have to create left part
// (sparsematrix object) and right part (dense array).
//
// Initially, sparse matrix is created in the Hash-Table format,
// which allows easy initialization, but do not allow matrix to be
// used in the linear solvers. So after construction you should convert
// sparse matrix to CRS format (one suited for linear operations).
//
sparsematrix a;
sparsecreate(5, 2, a);
sparseset(a, 0, 0, 1.0);
sparseset(a, 0, 1, 1.0);
sparseset(a, 1, 0, 1.0);
sparseset(a, 1, 1, 1.0);
sparseset(a, 2, 0, 2.0);
sparseset(a, 2, 1, 1.0);
sparseset(a, 3, 0, 1.0);
sparseset(a, 4, 1, 1.0);
//
// Now our matrix is fully initialized, but we have to do one more
// step - convert it from Hash-Table format to CRS format (see
// documentation on sparse matrices for more information about these
// formats).
//
// If you omit this call, ALGLIB will generate exception on the first
// attempt to use A in linear operations.
//
sparseconverttocrs(a);
//
// Initialization of the right part
//
real_1d_array b = "[4,2,4,1,2]";
//
// Now we have to create linear solver object and to use it for the
// solution of the linear system.
//
linlsqrstate s;
linlsqrreport rep;
real_1d_array x;
linlsqrcreate(5, 2, s);
linlsqrsolvesparse(s, a, b);
linlsqrresults(s, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [1.000,2.000]
return 0;
}
linreg subpackage| linreg_d_basic | Linear regression used to build the very basic model and unpack coefficients |
linearmodel class/************************************************************************* *************************************************************************/class linearmodel { };
lrreport class/************************************************************************* LRReport structure contains additional information about linear model: * C - covariation matrix, array[0..NVars,0..NVars]. C[i,j] = Cov(A[i],A[j]) * RMSError - root mean square error on a training set * AvgError - average error on a training set * AvgRelError - average relative error on a training set (excluding observations with zero function value). * CVRMSError - leave-one-out cross-validation estimate of generalization error. Calculated using fast algorithm with O(NVars*NPoints) complexity. * CVAvgError - cross-validation estimate of average error * CVAvgRelError - cross-validation estimate of average relative error All other fields of the structure are intended for internal use and should not be used outside ALGLIB. *************************************************************************/class lrreport { real_2d_array c; double rmserror; double avgerror; double avgrelerror; double cvrmserror; double cvavgerror; double cvavgrelerror; ae_int_t ncvdefects; integer_1d_array cvdefects; };
lravgerror function/************************************************************************* Average error on the test set INPUT PARAMETERS: LM - linear model XY - test set NPoints - test set size RESULT: average error. -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/double alglib::lravgerror( linearmodel lm, real_2d_array xy, ae_int_t npoints);
lravgrelerror function/************************************************************************* RMS error on the test set INPUT PARAMETERS: LM - linear model XY - test set NPoints - test set size RESULT: average relative error. -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/double alglib::lravgrelerror( linearmodel lm, real_2d_array xy, ae_int_t npoints);
lrbuild function/************************************************************************* Linear regression Subroutine builds model: Y = A(0)*X[0] + ... + A(N-1)*X[N-1] + A(N) and model found in ALGLIB format, covariation matrix, training set errors (rms, average, average relative) and leave-one-out cross-validation estimate of the generalization error. CV estimate calculated using fast algorithm with O(NPoints*NVars) complexity. When covariation matrix is calculated standard deviations of function values are assumed to be equal to RMS error on the training set. INPUT PARAMETERS: XY - training set, array [0..NPoints-1,0..NVars]: * NVars columns - independent variables * last column - dependent variable NPoints - training set size, NPoints>NVars+1 NVars - number of independent variables OUTPUT PARAMETERS: Info - return code: * -255, in case of unknown internal error * -4, if internal SVD subroutine haven't converged * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1). * 1, if subroutine successfully finished LM - linear model in the ALGLIB format. Use subroutines of this unit to work with the model. AR - additional results -- ALGLIB -- Copyright 02.08.2008 by Bochkanov Sergey *************************************************************************/void alglib::lrbuild( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t& info, linearmodel& lm, lrreport& ar);
Examples: [1]
lrbuilds function/************************************************************************* Linear regression Variant of LRBuild which uses vector of standatd deviations (errors in function values). INPUT PARAMETERS: XY - training set, array [0..NPoints-1,0..NVars]: * NVars columns - independent variables * last column - dependent variable S - standard deviations (errors in function values) array[0..NPoints-1], S[i]>0. NPoints - training set size, NPoints>NVars+1 NVars - number of independent variables OUTPUT PARAMETERS: Info - return code: * -255, in case of unknown internal error * -4, if internal SVD subroutine haven't converged * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1). * -2, if S[I]<=0 * 1, if subroutine successfully finished LM - linear model in the ALGLIB format. Use subroutines of this unit to work with the model. AR - additional results -- ALGLIB -- Copyright 02.08.2008 by Bochkanov Sergey *************************************************************************/void alglib::lrbuilds( real_2d_array xy, real_1d_array s, ae_int_t npoints, ae_int_t nvars, ae_int_t& info, linearmodel& lm, lrreport& ar);
lrbuildz function/************************************************************************* Like LRBuild but builds model Y = A(0)*X[0] + ... + A(N-1)*X[N-1] i.e. with zero constant term. -- ALGLIB -- Copyright 30.10.2008 by Bochkanov Sergey *************************************************************************/void alglib::lrbuildz( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t& info, linearmodel& lm, lrreport& ar);
lrbuildzs function/************************************************************************* Like LRBuildS, but builds model Y = A(0)*X[0] + ... + A(N-1)*X[N-1] i.e. with zero constant term. -- ALGLIB -- Copyright 30.10.2008 by Bochkanov Sergey *************************************************************************/void alglib::lrbuildzs( real_2d_array xy, real_1d_array s, ae_int_t npoints, ae_int_t nvars, ae_int_t& info, linearmodel& lm, lrreport& ar);
lrpack function/************************************************************************* "Packs" coefficients and creates linear model in ALGLIB format (LRUnpack reversed). INPUT PARAMETERS: V - coefficients, array[0..NVars] NVars - number of independent variables OUTPUT PAREMETERS: LM - linear model. -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/void alglib::lrpack(real_1d_array v, ae_int_t nvars, linearmodel& lm);
lrprocess function/************************************************************************* Procesing INPUT PARAMETERS: LM - linear model X - input vector, array[0..NVars-1]. Result: value of linear model regression estimate -- ALGLIB -- Copyright 03.09.2008 by Bochkanov Sergey *************************************************************************/double alglib::lrprocess(linearmodel lm, real_1d_array x);
lrrmserror function/************************************************************************* RMS error on the test set INPUT PARAMETERS: LM - linear model XY - test set NPoints - test set size RESULT: root mean square error. -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/double alglib::lrrmserror( linearmodel lm, real_2d_array xy, ae_int_t npoints);
lrunpack function/************************************************************************* Unpacks coefficients of linear model. INPUT PARAMETERS: LM - linear model in ALGLIB format OUTPUT PARAMETERS: V - coefficients, array[0..NVars] constant term (intercept) is stored in the V[NVars]. NVars - number of independent variables (one less than number of coefficients) -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/void alglib::lrunpack(linearmodel lm, real_1d_array& v, ae_int_t& nvars);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
//
// We have:
// * xy - matrix of basic function values (exp(0.5*x)) and expected values
//
real_2d_array xy = "[[0.606531,1.133719],[0.670320,1.306522],[0.740818,1.504604],[0.818731,1.554663],[0.904837,1.884638],[1.000000,2.072436],[1.105171,2.257285],[1.221403,2.534068],[1.349859,2.622017],[1.491825,2.897713],[1.648721,3.219371]]";
ae_int_t info;
ae_int_t nvars;
linearmodel model;
lrreport rep;
real_1d_array c;
lrbuildz(xy, 11, 1, info, model, rep);
printf("%d\n", int(info)); // EXPECTED: 1
lrunpack(model, c, nvars);
printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.98650,0.00000]
return 0;
}
logit subpackagelogitmodel class/************************************************************************* *************************************************************************/class logitmodel { };
mnlreport class/************************************************************************* MNLReport structure contains information about training process: * NGrad - number of gradient calculations * NHess - number of Hessian calculations *************************************************************************/class mnlreport { ae_int_t ngrad; ae_int_t nhess; };
mnlavgce function/************************************************************************* Average cross-entropy (in bits per element) on the test set INPUT PARAMETERS: LM - logit model XY - test set NPoints - test set size RESULT: CrossEntropy/(NPoints*ln(2)). -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/double alglib::mnlavgce( logitmodel lm, real_2d_array xy, ae_int_t npoints);
mnlavgerror function/************************************************************************* Average error on the test set INPUT PARAMETERS: LM - logit model XY - test set NPoints - test set size RESULT: average error (error when estimating posterior probabilities). -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/double alglib::mnlavgerror( logitmodel lm, real_2d_array xy, ae_int_t npoints);
mnlavgrelerror function/************************************************************************* Average relative error on the test set INPUT PARAMETERS: LM - logit model XY - test set NPoints - test set size RESULT: average relative error (error when estimating posterior probabilities). -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/double alglib::mnlavgrelerror( logitmodel lm, real_2d_array xy, ae_int_t ssize);
mnlclserror function/************************************************************************* Classification error on test set = MNLRelClsError*NPoints -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mnlclserror( logitmodel lm, real_2d_array xy, ae_int_t npoints);
mnlpack function/************************************************************************* "Packs" coefficients and creates logit model in ALGLIB format (MNLUnpack reversed). INPUT PARAMETERS: A - model (see MNLUnpack) NVars - number of independent variables NClasses - number of classes OUTPUT PARAMETERS: LM - logit model. -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/void alglib::mnlpack( real_2d_array a, ae_int_t nvars, ae_int_t nclasses, logitmodel& lm);
mnlprocess function/************************************************************************* Procesing INPUT PARAMETERS: LM - logit model, passed by non-constant reference (some fields of structure are used as temporaries when calculating model output). X - input vector, array[0..NVars-1]. Y - (possibly) preallocated buffer; if size of Y is less than NClasses, it will be reallocated.If it is large enough, it is NOT reallocated, so we can save some time on reallocation. OUTPUT PARAMETERS: Y - result, array[0..NClasses-1] Vector of posterior probabilities for classification task. -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/void alglib::mnlprocess(logitmodel lm, real_1d_array x, real_1d_array& y);
mnlprocessi function/************************************************************************* 'interactive' variant of MNLProcess for languages like Python which support constructs like "Y = MNLProcess(LM,X)" and interactive mode of the interpreter This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/void alglib::mnlprocessi( logitmodel lm, real_1d_array x, real_1d_array& y);
mnlrelclserror function/************************************************************************* Relative classification error on the test set INPUT PARAMETERS: LM - logit model XY - test set NPoints - test set size RESULT: percent of incorrectly classified cases. -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/double alglib::mnlrelclserror( logitmodel lm, real_2d_array xy, ae_int_t npoints);
mnlrmserror function/************************************************************************* RMS error on the test set INPUT PARAMETERS: LM - logit model XY - test set NPoints - test set size RESULT: root mean square error (error when estimating posterior probabilities). -- ALGLIB -- Copyright 30.08.2008 by Bochkanov Sergey *************************************************************************/double alglib::mnlrmserror( logitmodel lm, real_2d_array xy, ae_int_t npoints);
mnltrainh function/************************************************************************* This subroutine trains logit model. INPUT PARAMETERS: XY - training set, array[0..NPoints-1,0..NVars] First NVars columns store values of independent variables, next column stores number of class (from 0 to NClasses-1) which dataset element belongs to. Fractional values are rounded to nearest integer. NPoints - training set size, NPoints>=1 NVars - number of independent variables, NVars>=1 NClasses - number of classes, NClasses>=2 OUTPUT PARAMETERS: Info - return code: * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<NVars+2, NVars<1, NClasses<2). * 1, if task has been solved LM - model built Rep - training report -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/void alglib::mnltrainh( real_2d_array xy, ae_int_t npoints, ae_int_t nvars, ae_int_t nclasses, ae_int_t& info, logitmodel& lm, mnlreport& rep);
mnlunpack function/************************************************************************* Unpacks coefficients of logit model. Logit model have form: P(class=i) = S(i) / (S(0) + S(1) + ... +S(M-1)) S(i) = Exp(A[i,0]*X[0] + ... + A[i,N-1]*X[N-1] + A[i,N]), when i<M-1 S(M-1) = 1 INPUT PARAMETERS: LM - logit model in ALGLIB format OUTPUT PARAMETERS: V - coefficients, array[0..NClasses-2,0..NVars] NVars - number of independent variables NClasses - number of classes -- ALGLIB -- Copyright 10.09.2008 by Bochkanov Sergey *************************************************************************/void alglib::mnlunpack( logitmodel lm, real_2d_array& a, ae_int_t& nvars, ae_int_t& nclasses);
lsfit subpackage| 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_nlfgh | Nonlinear fitting using gradient and Hessian | |
| lsfit_d_nlscale | Nonlinear fitting with custom scaling and bound constraints | |
| lsfit_d_pol | Unconstrained polynomial fitting | |
| lsfit_d_polc | Constrained polynomial fitting | |
| lsfit_d_spline | Unconstrained fitting by penalized regression spline | |
| lsfit_t_4pl | 4-parameter logistic fitting | |
| lsfit_t_5pl | 5-parameter logistic fitting |
barycentricfitreport class/************************************************************************* Barycentric fitting report: RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error TaskRCond reciprocal of task's condition number *************************************************************************/class barycentricfitreport { double taskrcond; ae_int_t dbest; double rmserror; double avgerror; double avgrelerror; double maxerror; };
lsfitreport class/************************************************************************* Least squares fitting report. This structure contains informational fields which are set by fitting functions provided by this unit. Different functions initialize different sets of fields, so you should read documentation on specific function you used in order to know which fields are initialized. TaskRCond reciprocal of task's condition number IterationsCount number of internal iterations VarIdx if user-supplied gradient contains errors which were detected by nonlinear fitter, this field is set to index of the first component of gradient which is suspected to be spoiled by bugs. RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error WRMSError weighted RMS error CovPar covariance matrix for parameters, filled by some solvers ErrPar vector of errors in parameters, filled by some solvers ErrCurve vector of fit errors - variability of the best-fit curve, filled by some solvers. Noise vector of per-point noise estimates, filled by some solvers. R2 coefficient of determination (non-weighted, non-adjusted), filled by some solvers. *************************************************************************/class lsfitreport { double taskrcond; ae_int_t iterationscount; ae_int_t varidx; double rmserror; double avgerror; double avgrelerror; double maxerror; double wrmserror; real_2d_array covpar; real_1d_array errpar; real_1d_array errcurve; real_1d_array noise; double r2; };
lsfitstate class/************************************************************************* Nonlinear fitter. You should use ALGLIB functions to work with fitter. Never try to access its fields directly! *************************************************************************/class lsfitstate { };
polynomialfitreport class/************************************************************************* Polynomial fitting report: TaskRCond reciprocal of task's condition number RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error *************************************************************************/class polynomialfitreport { double taskrcond; double rmserror; double avgerror; double avgrelerror; double maxerror; };
spline1dfitreport class/************************************************************************* Spline fitting report: RMSError RMS error AvgError average error AvgRelError average relative error (for non-zero Y[I]) MaxError maximum error Fields below are filled by obsolete functions (Spline1DFitCubic, Spline1DFitHermite). Modern fitting functions do NOT fill these fields: TaskRCond reciprocal of task's condition number *************************************************************************/class spline1dfitreport { double taskrcond; double rmserror; double avgerror; double avgrelerror; double maxerror; };
barycentricfitfloaterhormann function/************************************************************************* Rational least squares fitting using Floater-Hormann rational functions with optimal D chosen from [0,9]. Equidistant grid with M node on [min(x),max(x)] is used to build basis functions. Different values of D are tried, optimal D (least root mean square error) is chosen. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2) (mostly dominated by the least squares solver). COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points, N>0. M - number of basis functions ( = number_of_nodes), M>=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentricfitfloaterhormann( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, barycentricinterpolant& b, barycentricfitreport& rep); void alglib::smp_barycentricfitfloaterhormann( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, barycentricinterpolant& b, barycentricfitreport& rep);
barycentricfitfloaterhormannwc function/************************************************************************* Weghted rational least squares fitting using Floater-Hormann rational functions with optimal D chosen from [0,9], with constraints and individual weights. Equidistant grid with M node on [min(x),max(x)] is used to build basis functions. Different values of D are tried, optimal D (least WEIGHTED root mean square error) is chosen. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2) (mostly dominated by the least squares solver). SEE ALSO * BarycentricFitFloaterHormann(), "lightweight" fitting without invididual weights and constraints. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points, N>0. XC - points where function values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints, 0<=K<M. K=0 means no constraints (XC/YC/DC are not used in such cases) M - number of basis functions ( = number_of_nodes), M>=2. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints -1 means another errors in parameters passed (N<=0, for example) B - barycentric interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * DBest best value of the D parameter * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroutine doesn't calculate task's condition number for K<>0. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained barycentric interpolants: * excessive constraints can be inconsistent. Floater-Hormann basis functions aren't as flexible as splines (although they are very smooth). * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints IS NOT GUARANTEED. * in the several special cases, however, we CAN guarantee consistency. * one of this cases is constraints on the function VALUES at the interval boundaries. Note that consustency of the constraints on the function DERIVATIVES is NOT guaranteed (you can use in such cases cubic splines which are more flexible). * another special case is ONE constraint on the function value (OR, but not AND, derivative) anywhere in the interval Our final recommendation is to use constraints WHEN AND ONLY WHEN you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentricfitfloaterhormannwc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, barycentricinterpolant& b, barycentricfitreport& rep); void alglib::smp_barycentricfitfloaterhormannwc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, barycentricinterpolant& b, barycentricfitreport& rep);
logisticcalc4 function/************************************************************************* This function calculates value of four-parameter logistic (4PL) model at specified point X. 4PL model has following form: F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B)) INPUT PARAMETERS: X - current point, X>=0: * zero X is correctly handled even for B<=0 * negative X results in exception. A, B, C, D- parameters of 4PL model: * A is unconstrained * B is unconstrained; zero or negative values are handled correctly. * C>0, non-positive value results in exception * D is unconstrained RESULT: model value at X NOTE: if B=0, denominator is assumed to be equal to 2.0 even for zero X (strictly speaking, 0^0 is undefined). NOTE: this function also throws exception if all input parameters are correct, but overflow was detected during calculations. NOTE: this function performs a lot of checks; if you need really high performance, consider evaluating model yourself, without checking for degenerate cases. -- ALGLIB PROJECT -- Copyright 14.05.2014 by Bochkanov Sergey *************************************************************************/double alglib::logisticcalc4( double x, double a, double b, double c, double d);
Examples: [1]
logisticcalc5 function/************************************************************************* This function calculates value of five-parameter logistic (5PL) model at specified point X. 5PL model has following form: F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G) INPUT PARAMETERS: X - current point, X>=0: * zero X is correctly handled even for B<=0 * negative X results in exception. A, B, C, D, G- parameters of 5PL model: * A is unconstrained * B is unconstrained; zero or negative values are handled correctly. * C>0, non-positive value results in exception * D is unconstrained * G>0, non-positive value results in exception RESULT: model value at X NOTE: if B=0, denominator is assumed to be equal to Power(2.0,G) even for zero X (strictly speaking, 0^0 is undefined). NOTE: this function also throws exception if all input parameters are correct, but overflow was detected during calculations. NOTE: this function performs a lot of checks; if you need really high performance, consider evaluating model yourself, without checking for degenerate cases. -- ALGLIB PROJECT -- Copyright 14.05.2014 by Bochkanov Sergey *************************************************************************/double alglib::logisticcalc5( double x, double a, double b, double c, double d, double g);
Examples: [1]
logisticfit4 function/************************************************************************* This function fits four-parameter logistic (4PL) model to data provided by user. 4PL model has following form: F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B)) Here: * A, D - unconstrained (see LogisticFit4EC() for constrained 4PL) * B>=0 * C>0 IMPORTANT: output of this function is constrained in such way that B>0. Because 4PL model is symmetric with respect to B, there is no need to explore B<0. Constraining B makes algorithm easier to stabilize and debug. Users who for some reason prefer to work with negative B's should transform output themselves (swap A and D, replace B by -B). 4PL fitting is implemented as follows: * we perform small number of restarts from random locations which helps to solve problem of bad local extrema. Locations are only partially random - we use input data to determine good initial guess, but we include controlled amount of randomness. * we perform Levenberg-Marquardt fitting with very tight constraints on parameters B and C - it allows us to find good initial guess for the second stage without risk of running into "flat spot". * second Levenberg-Marquardt round is performed without excessive constraints. Results from the previous round are used as initial guess. * after fitting is done, we compare results with best values found so far, rewrite "best solution" if needed, and move to next random location. Overall algorithm is very stable and is not prone to bad local extrema. Furthermore, it automatically scales when input data have very large or very small range. INPUT PARAMETERS: X - array[N], stores X-values. MUST include only non-negative numbers (but may include zero values). Can be unsorted. Y - array[N], values to fit. N - number of points. If N is less than length of X/Y, only leading N elements are used. OUTPUT PARAMETERS: A, B, C, D- parameters of 4PL model Rep - fitting report. This structure has many fields, but ONLY ONES LISTED BELOW ARE SET: * Rep.IterationsCount - number of iterations performed * Rep.RMSError - root-mean-square error * Rep.AvgError - average absolute error * Rep.AvgRelError - average relative error (calculated for non-zero Y-values) * Rep.MaxError - maximum absolute error * Rep.R2 - coefficient of determination, R-squared. This coefficient is calculated as R2=1-RSS/TSS (in case of nonlinear regression there are multiple ways to define R2, each of them giving different results). NOTE: after you obtained coefficients, you can evaluate model with LogisticCalc4() function. NOTE: if you need better control over fitting process than provided by this function, you may use LogisticFit45X(). NOTE: step is automatically scaled according to scale of parameters being fitted before we compare its length with EpsX. Thus, this function can be used to fit data with very small or very large values without changing EpsX. -- ALGLIB PROJECT -- Copyright 14.02.2014 by Bochkanov Sergey *************************************************************************/void alglib::logisticfit4( real_1d_array x, real_1d_array y, ae_int_t n, double& a, double& b, double& c, double& d, lsfitreport& rep);
Examples: [1]
logisticfit45x function/************************************************************************* This is "expert" 4PL/5PL fitting function, which can be used if you need better control over fitting process than provided by LogisticFit4() or LogisticFit5(). This function fits model of the form F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B)) (4PL model) or F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G) (5PL model) Here: * A, D - unconstrained * B>=0 for 4PL, unconstrained for 5PL * C>0 * G>0 (if present) INPUT PARAMETERS: X - array[N], stores X-values. MUST include only non-negative numbers (but may include zero values). Can be unsorted. Y - array[N], values to fit. N - number of points. If N is less than length of X/Y, only leading N elements are used. CnstrLeft- optional equality constraint for model value at the left boundary (at X=0). Specify NAN (Not-a-Number) if you do not need constraint on the model value at X=0 (in C++ you can pass alglib::fp_nan as parameter, in C# it will be Double.NaN). See below, section "EQUALITY CONSTRAINTS" for more information about constraints. CnstrRight- optional equality constraint for model value at X=infinity. Specify NAN (Not-a-Number) if you do not need constraint on the model value (in C++ you can pass alglib::fp_nan as parameter, in C# it will be Double.NaN). See below, section "EQUALITY CONSTRAINTS" for more information about constraints. Is4PL - whether 4PL or 5PL models are fitted LambdaV - regularization coefficient, LambdaV>=0. Set it to zero unless you know what you are doing. EpsX - stopping condition (step size), EpsX>=0. Zero value means that small step is automatically chosen. See notes below for more information. RsCnt - number of repeated restarts from random points. 4PL/5PL models are prone to problem of bad local extrema. Utilizing multiple random restarts allows us to improve algorithm convergence. RsCnt>=0. Zero value means that function automatically choose small amount of restarts (recommended). OUTPUT PARAMETERS: A, B, C, D- parameters of 4PL model G - parameter of 5PL model; for Is4PL=True, G=1 is returned. Rep - fitting report. This structure has many fields, but ONLY ONES LISTED BELOW ARE SET: * Rep.IterationsCount - number of iterations performed * Rep.RMSError - root-mean-square error * Rep.AvgError - average absolute error * Rep.AvgRelError - average relative error (calculated for non-zero Y-values) * Rep.MaxError - maximum absolute error * Rep.R2 - coefficient of determination, R-squared. This coefficient is calculated as R2=1-RSS/TSS (in case of nonlinear regression there are multiple ways to define R2, each of them giving different results). NOTE: after you obtained coefficients, you can evaluate model with LogisticCalc5() function. NOTE: step is automatically scaled according to scale of parameters being fitted before we compare its length with EpsX. Thus, this function can be used to fit data with very small or very large values without changing EpsX. EQUALITY CONSTRAINTS ON PARAMETERS 4PL/5PL solver supports equality constraints on model values at the left boundary (X=0) and right boundary (X=infinity). These constraints are completely optional and you can specify both of them, only one - or no constraints at all. Parameter CnstrLeft contains left constraint (or NAN for unconstrained fitting), and CnstrRight contains right one. For 4PL, left constraint ALWAYS corresponds to parameter A, and right one is ALWAYS constraint on D. That's because 4PL model is normalized in such way that B>=0. For 5PL model things are different. Unlike 4PL one, 5PL model is NOT symmetric with respect to change in sign of B. Thus, negative B's are possible, and left constraint may constrain parameter A (for positive B's) - or parameter D (for negative B's). Similarly changes meaning of right constraint. You do not have to decide what parameter to constrain - algorithm will automatically determine correct parameters as fitting progresses. However, question highlighted above is important when you interpret fitting results. -- ALGLIB PROJECT -- Copyright 14.02.2014 by Bochkanov Sergey *************************************************************************/void alglib::logisticfit45x( real_1d_array x, real_1d_array y, ae_int_t n, double cnstrleft, double cnstrright, bool is4pl, double lambdav, double epsx, ae_int_t rscnt, double& a, double& b, double& c, double& d, double& g, lsfitreport& rep);
logisticfit4ec function/************************************************************************* This function fits four-parameter logistic (4PL) model to data provided by user, with optional constraints on parameters A and D. 4PL model has following form: F(x|A,B,C,D) = D+(A-D)/(1+Power(x/C,B)) Here: * A, D - with optional equality constraints * B>=0 * C>0 IMPORTANT: output of this function is constrained in such way that B>0. Because 4PL model is symmetric with respect to B, there is no need to explore B<0. Constraining B makes algorithm easier to stabilize and debug. Users who for some reason prefer to work with negative B's should transform output themselves (swap A and D, replace B by -B). 4PL fitting is implemented as follows: * we perform small number of restarts from random locations which helps to solve problem of bad local extrema. Locations are only partially random - we use input data to determine good initial guess, but we include controlled amount of randomness. * we perform Levenberg-Marquardt fitting with very tight constraints on parameters B and C - it allows us to find good initial guess for the second stage without risk of running into "flat spot". * second Levenberg-Marquardt round is performed without excessive constraints. Results from the previous round are used as initial guess. * after fitting is done, we compare results with best values found so far, rewrite "best solution" if needed, and move to next random location. Overall algorithm is very stable and is not prone to bad local extrema. Furthermore, it automatically scales when input data have very large or very small range. INPUT PARAMETERS: X - array[N], stores X-values. MUST include only non-negative numbers (but may include zero values). Can be unsorted. Y - array[N], values to fit. N - number of points. If N is less than length of X/Y, only leading N elements are used. CnstrLeft- optional equality constraint for model value at the left boundary (at X=0). Specify NAN (Not-a-Number) if you do not need constraint on the model value at X=0 (in C++ you can pass alglib::fp_nan as parameter, in C# it will be Double.NaN). See below, section "EQUALITY CONSTRAINTS" for more information about constraints. CnstrRight- optional equality constraint for model value at X=infinity. Specify NAN (Not-a-Number) if you do not need constraint on the model value (in C++ you can pass alglib::fp_nan as parameter, in C# it will be Double.NaN). See below, section "EQUALITY CONSTRAINTS" for more information about constraints. OUTPUT PARAMETERS: A, B, C, D- parameters of 4PL model Rep - fitting report. This structure has many fields, but ONLY ONES LISTED BELOW ARE SET: * Rep.IterationsCount - number of iterations performed * Rep.RMSError - root-mean-square error * Rep.AvgError - average absolute error * Rep.AvgRelError - average relative error (calculated for non-zero Y-values) * Rep.MaxError - maximum absolute error * Rep.R2 - coefficient of determination, R-squared. This coefficient is calculated as R2=1-RSS/TSS (in case of nonlinear regression there are multiple ways to define R2, each of them giving different results). NOTE: after you obtained coefficients, you can evaluate model with LogisticCalc4() function. NOTE: if you need better control over fitting process than provided by this function, you may use LogisticFit45X(). NOTE: step is automatically scaled according to scale of parameters being fitted before we compare its length with EpsX. Thus, this function can be used to fit data with very small or very large values without changing EpsX. EQUALITY CONSTRAINTS ON PARAMETERS 4PL/5PL solver supports equality constraints on model values at the left boundary (X=0) and right boundary (X=infinity). These constraints are completely optional and you can specify both of them, only one - or no constraints at all. Parameter CnstrLeft contains left constraint (or NAN for unconstrained fitting), and CnstrRight contains right one. For 4PL, left constraint ALWAYS corresponds to parameter A, and right one is ALWAYS constraint on D. That's because 4PL model is normalized in such way that B>=0. -- ALGLIB PROJECT -- Copyright 14.02.2014 by Bochkanov Sergey *************************************************************************/void alglib::logisticfit4ec( real_1d_array x, real_1d_array y, ae_int_t n, double cnstrleft, double cnstrright, double& a, double& b, double& c, double& d, lsfitreport& rep);
logisticfit5 function/************************************************************************* This function fits five-parameter logistic (5PL) model to data provided by user. 5PL model has following form: F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G) Here: * A, D - unconstrained * B - unconstrained * C>0 * G>0 IMPORTANT: unlike in 4PL fitting, output of this function is NOT constrained in such way that B is guaranteed to be positive. Furthermore, unlike 4PL, 5PL model is NOT symmetric with respect to B, so you can NOT transform model to equivalent one, with B having desired sign (>0 or <0). 5PL fitting is implemented as follows: * we perform small number of restarts from random locations which helps to solve problem of bad local extrema. Locations are only partially random - we use input data to determine good initial guess, but we include controlled amount of randomness. * we perform Levenberg-Marquardt fitting with very tight constraints on parameters B and C - it allows us to find good initial guess for the second stage without risk of running into "flat spot". Parameter G is fixed at G=1. * second Levenberg-Marquardt round is performed without excessive constraints on B and C, but with G still equal to 1. Results from the previous round are used as initial guess. * third Levenberg-Marquardt round relaxes constraints on G and tries two different models - one with B>0 and one with B<0. * after fitting is done, we compare results with best values found so far, rewrite "best solution" if needed, and move to next random location. Overall algorithm is very stable and is not prone to bad local extrema. Furthermore, it automatically scales when input data have very large or very small range. INPUT PARAMETERS: X - array[N], stores X-values. MUST include only non-negative numbers (but may include zero values). Can be unsorted. Y - array[N], values to fit. N - number of points. If N is less than length of X/Y, only leading N elements are used. OUTPUT PARAMETERS: A,B,C,D,G- parameters of 5PL model Rep - fitting report. This structure has many fields, but ONLY ONES LISTED BELOW ARE SET: * Rep.IterationsCount - number of iterations performed * Rep.RMSError - root-mean-square error * Rep.AvgError - average absolute error * Rep.AvgRelError - average relative error (calculated for non-zero Y-values) * Rep.MaxError - maximum absolute error * Rep.R2 - coefficient of determination, R-squared. This coefficient is calculated as R2=1-RSS/TSS (in case of nonlinear regression there are multiple ways to define R2, each of them giving different results). NOTE: after you obtained coefficients, you can evaluate model with LogisticCalc5() function. NOTE: if you need better control over fitting process than provided by this function, you may use LogisticFit45X(). NOTE: step is automatically scaled according to scale of parameters being fitted before we compare its length with EpsX. Thus, this function can be used to fit data with very small or very large values without changing EpsX. -- ALGLIB PROJECT -- Copyright 14.02.2014 by Bochkanov Sergey *************************************************************************/void alglib::logisticfit5( real_1d_array x, real_1d_array y, ae_int_t n, double& a, double& b, double& c, double& d, double& g, lsfitreport& rep);
Examples: [1]
logisticfit5ec function/************************************************************************* This function fits five-parameter logistic (5PL) model to data provided by user, subject to optional equality constraints on parameters A and D. 5PL model has following form: F(x|A,B,C,D,G) = D+(A-D)/Power(1+Power(x/C,B),G) Here: * A, D - with optional equality constraints * B - unconstrained * C>0 * G>0 IMPORTANT: unlike in 4PL fitting, output of this function is NOT constrained in such way that B is guaranteed to be positive. Furthermore, unlike 4PL, 5PL model is NOT symmetric with respect to B, so you can NOT transform model to equivalent one, with B having desired sign (>0 or <0). 5PL fitting is implemented as follows: * we perform small number of restarts from random locations which helps to solve problem of bad local extrema. Locations are only partially random - we use input data to determine good initial guess, but we include controlled amount of randomness. * we perform Levenberg-Marquardt fitting with very tight constraints on parameters B and C - it allows us to find good initial guess for the second stage without risk of running into "flat spot". Parameter G is fixed at G=1. * second Levenberg-Marquardt round is performed without excessive constraints on B and C, but with G still equal to 1. Results from the previous round are used as initial guess. * third Levenberg-Marquardt round relaxes constraints on G and tries two different models - one with B>0 and one with B<0. * after fitting is done, we compare results with best values found so far, rewrite "best solution" if needed, and move to next random location. Overall algorithm is very stable and is not prone to bad local extrema. Furthermore, it automatically scales when input data have very large or very small range. INPUT PARAMETERS: X - array[N], stores X-values. MUST include only non-negative numbers (but may include zero values). Can be unsorted. Y - array[N], values to fit. N - number of points. If N is less than length of X/Y, only leading N elements are used. CnstrLeft- optional equality constraint for model value at the left boundary (at X=0). Specify NAN (Not-a-Number) if you do not need constraint on the model value at X=0 (in C++ you can pass alglib::fp_nan as parameter, in C# it will be Double.NaN). See below, section "EQUALITY CONSTRAINTS" for more information about constraints. CnstrRight- optional equality constraint for model value at X=infinity. Specify NAN (Not-a-Number) if you do not need constraint on the model value (in C++ you can pass alglib::fp_nan as parameter, in C# it will be Double.NaN). See below, section "EQUALITY CONSTRAINTS" for more information about constraints. OUTPUT PARAMETERS: A,B,C,D,G- parameters of 5PL model Rep - fitting report. This structure has many fields, but ONLY ONES LISTED BELOW ARE SET: * Rep.IterationsCount - number of iterations performed * Rep.RMSError - root-mean-square error * Rep.AvgError - average absolute error * Rep.AvgRelError - average relative error (calculated for non-zero Y-values) * Rep.MaxError - maximum absolute error * Rep.R2 - coefficient of determination, R-squared. This coefficient is calculated as R2=1-RSS/TSS (in case of nonlinear regression there are multiple ways to define R2, each of them giving different results). NOTE: after you obtained coefficients, you can evaluate model with LogisticCalc5() function. NOTE: if you need better control over fitting process than provided by this function, you may use LogisticFit45X(). NOTE: step is automatically scaled according to scale of parameters being fitted before we compare its length with EpsX. Thus, this function can be used to fit data with very small or very large values without changing EpsX. EQUALITY CONSTRAINTS ON PARAMETERS 5PL solver supports equality constraints on model values at the left boundary (X=0) and right boundary (X=infinity). These constraints are completely optional and you can specify both of them, only one - or no constraints at all. Parameter CnstrLeft contains left constraint (or NAN for unconstrained fitting), and CnstrRight contains right one. Unlike 4PL one, 5PL model is NOT symmetric with respect to change in sign of B. Thus, negative B's are possible, and left constraint may constrain parameter A (for positive B's) - or parameter D (for negative B's). Similarly changes meaning of right constraint. You do not have to decide what parameter to constrain - algorithm will automatically determine correct parameters as fitting progresses. However, question highlighted above is important when you interpret fitting results. -- ALGLIB PROJECT -- Copyright 14.02.2014 by Bochkanov Sergey *************************************************************************/void alglib::logisticfit5ec( real_1d_array x, real_1d_array y, ae_int_t n, double cnstrleft, double cnstrright, double& a, double& b, double& c, double& d, double& g, lsfitreport& rep);
lsfitcreatef function/************************************************************************* Nonlinear least squares fitting using function values only. Combination of numerical differentiation and secant updates is used to obtain function Jacobian. Nonlinear task min(F(c)) is solved, where F(c) = (f(c,x[0])-y[0])^2 + ... + (f(c,x[n-1])-y[n-1])^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]). INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted DiffStep- numerical differentiation step; should not be very small or large; large = loss of accuracy small = growth of round-off errors OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 18.10.2008 by Bochkanov Sergey *************************************************************************/void alglib::lsfitcreatef( real_2d_array x, real_1d_array y, real_1d_array c, double diffstep, lsfitstate& state); void alglib::lsfitcreatef( real_2d_array x, real_1d_array y, real_1d_array c, ae_int_t n, ae_int_t m, ae_int_t k, double diffstep, lsfitstate& state);
lsfitcreatefg function/************************************************************************* Nonlinear least squares fitting using gradient only, without individual weights. Nonlinear task min(F(c)) is solved, where F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]) and its gradient. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted CheapFG - boolean flag, which is: * True if both function and gradient calculation complexity are less than O(M^2). An improved algorithm can be used which corresponds to FGJ scheme from MINLM unit. * False otherwise. Standard Jacibian-bases Levenberg-Marquardt algo will be used (FJ scheme). OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitcreatefg( real_2d_array x, real_1d_array y, real_1d_array c, bool cheapfg, lsfitstate& state); void alglib::lsfitcreatefg( real_2d_array x, real_1d_array y, real_1d_array c, ae_int_t n, ae_int_t m, ae_int_t k, bool cheapfg, lsfitstate& state);
Examples: [1]
lsfitcreatefgh function/************************************************************************* Nonlinear least squares fitting using gradient/Hessian, without individial weights. Nonlinear task min(F(c)) is solved, where F(c) = ((f(c,x[0])-y[0]))^2 + ... + ((f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses f(c,x[i]), its gradient and its Hessian. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitcreatefgh( real_2d_array x, real_1d_array y, real_1d_array c, lsfitstate& state); void alglib::lsfitcreatefgh( real_2d_array x, real_1d_array y, real_1d_array c, ae_int_t n, ae_int_t m, ae_int_t k, lsfitstate& state);
Examples: [1]
lsfitcreatewf function/************************************************************************* Weighted nonlinear least squares fitting using function values only. Combination of numerical differentiation and secant updates is used to obtain function Jacobian. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]). INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted DiffStep- numerical differentiation step; should not be very small or large; large = loss of accuracy small = growth of round-off errors OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 18.10.2008 by Bochkanov Sergey *************************************************************************/void alglib::lsfitcreatewf( real_2d_array x, real_1d_array y, real_1d_array w, real_1d_array c, double diffstep, lsfitstate& state); void alglib::lsfitcreatewf( real_2d_array x, real_1d_array y, real_1d_array w, real_1d_array c, ae_int_t n, ae_int_t m, ae_int_t k, double diffstep, lsfitstate& state);
lsfitcreatewfg function/************************************************************************* Weighted nonlinear least squares fitting using gradient only. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses only f(c,x[i]) and its gradient. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted CheapFG - boolean flag, which is: * True if both function and gradient calculation complexity are less than O(M^2). An improved algorithm can be used which corresponds to FGJ scheme from MINLM unit. * False otherwise. Standard Jacibian-bases Levenberg-Marquardt algo will be used (FJ scheme). OUTPUT PARAMETERS: State - structure which stores algorithm state See also: LSFitResults LSFitCreateFG (fitting without weights) LSFitCreateWFGH (fitting using Hessian) LSFitCreateFGH (fitting using Hessian, without weights) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitcreatewfg( real_2d_array x, real_1d_array y, real_1d_array w, real_1d_array c, bool cheapfg, lsfitstate& state); void alglib::lsfitcreatewfg( real_2d_array x, real_1d_array y, real_1d_array w, real_1d_array c, ae_int_t n, ae_int_t m, ae_int_t k, bool cheapfg, lsfitstate& state);
Examples: [1]
lsfitcreatewfgh function/************************************************************************* Weighted nonlinear least squares fitting using gradient/Hessian. Nonlinear task min(F(c)) is solved, where F(c) = (w[0]*(f(c,x[0])-y[0]))^2 + ... + (w[n-1]*(f(c,x[n-1])-y[n-1]))^2, * N is a number of points, * M is a dimension of a space points belong to, * K is a dimension of a space of parameters being fitted, * w is an N-dimensional vector of weight coefficients, * x is a set of N points, each of them is an M-dimensional vector, * c is a K-dimensional vector of parameters being fitted This subroutine uses f(c,x[i]), its gradient and its Hessian. INPUT PARAMETERS: X - array[0..N-1,0..M-1], points (one row = one point) Y - array[0..N-1], function values. W - weights, array[0..N-1] C - array[0..K-1], initial approximation to the solution, N - number of points, N>1 M - dimension of space K - number of parameters being fitted OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitcreatewfgh( real_2d_array x, real_1d_array y, real_1d_array w, real_1d_array c, lsfitstate& state); void alglib::lsfitcreatewfgh( real_2d_array x, real_1d_array y, real_1d_array w, real_1d_array c, ae_int_t n, ae_int_t m, ae_int_t k, lsfitstate& state);
Examples: [1]
lsfitfit function/************************************************************************* This family of functions is used to launcn iterations of nonlinear fitter These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x hess - callback which calculates function (or merit function) value func, gradient grad and Hessian hess at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. this algorithm is somewhat unusual because it works with parameterized function f(C,X), where X is a function argument (we have many points which are characterized by different argument values), and C is a parameter to fit. For example, if we want to do linear fit by f(c0,c1,x) = c0*x+c1, then x will be argument, and {c0,c1} will be parameters. It is important to understand that this algorithm finds minimum in the space of function PARAMETERS (not arguments), so it needs derivatives of f() with respect to C, not X. In the example above it will need f=c0*x+c1 and {df/dc0,df/dc1} = {x,1} instead of {df/dx} = {c0}. 2. Callback functions accept C as the first parameter, and X as the second 3. If state was created with LSFitCreateFG(), algorithm needs just function and its gradient, but if state was created with LSFitCreateFGH(), algorithm will need function, gradient and Hessian. According to the said above, there ase several versions of this function, which accept different sets of callbacks. This flexibility opens way to subtle errors - you may create state with LSFitCreateFGH() (optimization using Hessian), but call function which does not accept Hessian. So when algorithm will request Hessian, there will be no callback to call. In this case exception will be thrown. Be careful to avoid such errors because there is no way to find them at compile time - you can see them at runtime only. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void lsfitfit(lsfitstate &state, void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr), void (*rep)(const real_1d_array &c, double func, void *ptr) = NULL, void *ptr = NULL); void lsfitfit(lsfitstate &state, void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr), void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*rep)(const real_1d_array &c, double func, void *ptr) = NULL, void *ptr = NULL); void lsfitfit(lsfitstate &state, void (*func)(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr), void (*grad)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*hess)(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr), void (*rep)(const real_1d_array &c, double func, void *ptr) = NULL, void *ptr = NULL);
lsfitlinear function/************************************************************************* Linear least squares fitting. QR decomposition is used to reduce task to MxM, then triangular solver or SVD-based solver is used depending on condition number of the system. It allows to maximize speed and retain decent accuracy. IMPORTANT: if you want to perform polynomial fitting, it may be more convenient to use PolynomialFit() function. This function gives best results on polynomial problems and solves numerical stability issues which arise when you fit high-degree polynomials to your data. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I, J] - value of J-th basis function in I-th point. N - number of points used. N>=1. M - number of basis functions, M>=1. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * Rep.TaskRCond reciprocal of condition number * R2 non-adjusted coefficient of determination (non-weighted) * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED ERRORS IN PARAMETERS This solver also calculates different kinds of errors in parameters and fills corresponding fields of report: * Rep.CovPar covariance matrix for parameters, array[K,K]. * Rep.ErrPar errors in parameters, array[K], errpar = sqrt(diag(CovPar)) * Rep.ErrCurve vector of fit errors - standard deviations of empirical best-fit curve from "ideal" best-fit curve built with infinite number of samples, array[N]. errcurve = sqrt(diag(F*CovPar*F')), where F is functions matrix. * Rep.Noise vector of per-point estimates of noise, array[N] NOTE: noise in the data is estimated as follows: * for fitting without user-supplied weights all points are assumed to have same level of noise, which is estimated from the data * for fitting with user-supplied weights we assume that noise level in I-th point is inversely proportional to Ith weight. Coefficient of proportionality is estimated from the data. NOTE: we apply small amount of regularization when we invert squared Jacobian and calculate covariance matrix. It guarantees that algorithm won't divide by zero during inversion, but skews error estimates a bit (fractional error is about 10^-9). However, we believe that this difference is insignificant for all practical purposes except for the situation when you want to compare ALGLIB results with "reference" implementation up to the last significant digit. NOTE: covariance matrix is estimated using correction for degrees of freedom (covariances are divided by N-M instead of dividing by N). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitlinear( real_1d_array y, real_2d_array fmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::lsfitlinear( real_1d_array y, real_2d_array fmatrix, ae_int_t n, ae_int_t m, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinear( real_1d_array y, real_2d_array fmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinear( real_1d_array y, real_2d_array fmatrix, ae_int_t n, ae_int_t m, ae_int_t& info, real_1d_array& c, lsfitreport& rep);
Examples: [1]
lsfitlinearc function/************************************************************************* Constained linear least squares fitting. This is variation of LSFitLinear(), which searchs for min|A*x=b| given that K additional constaints C*x=bc are satisfied. It reduces original task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinear() is called. IMPORTANT: if you want to perform polynomial fitting, it may be more convenient to use PolynomialFit() function. This function gives best results on polynomial problems and solves numerical stability issues which arise when you fit high-degree polynomials to your data. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I,J] - value of J-th basis function in I-th point. CMatrix - a table of constaints, array[0..K-1,0..M]. I-th row of CMatrix corresponds to I-th linear constraint: CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M] N - number of points used. N>=1. M - number of basis functions, M>=1. K - number of constraints, 0 <= K < M K=0 corresponds to absence of constraints. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -3 either too many constraints (M or more), degenerate constraints (some constraints are repetead twice) or inconsistent constraints were specified. * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * R2 non-adjusted coefficient of determination (non-weighted) * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. ERRORS IN PARAMETERS This solver also calculates different kinds of errors in parameters and fills corresponding fields of report: * Rep.CovPar covariance matrix for parameters, array[K,K]. * Rep.ErrPar errors in parameters, array[K], errpar = sqrt(diag(CovPar)) * Rep.ErrCurve vector of fit errors - standard deviations of empirical best-fit curve from "ideal" best-fit curve built with infinite number of samples, array[N]. errcurve = sqrt(diag(F*CovPar*F')), where F is functions matrix. * Rep.Noise vector of per-point estimates of noise, array[N] IMPORTANT: errors in parameters are calculated without taking into account boundary/linear constraints! Presence of constraints changes distribution of errors, but there is no easy way to account for constraints when you calculate covariance matrix. NOTE: noise in the data is estimated as follows: * for fitting without user-supplied weights all points are assumed to have same level of noise, which is estimated from the data * for fitting with user-supplied weights we assume that noise level in I-th point is inversely proportional to Ith weight. Coefficient of proportionality is estimated from the data. NOTE: we apply small amount of regularization when we invert squared Jacobian and calculate covariance matrix. It guarantees that algorithm won't divide by zero during inversion, but skews error estimates a bit (fractional error is about 10^-9). However, we believe that this difference is insignificant for all practical purposes except for the situation when you want to compare ALGLIB results with "reference" implementation up to the last significant digit. NOTE: covariance matrix is estimated using correction for degrees of freedom (covariances are divided by N-M instead of dividing by N). -- ALGLIB -- Copyright 07.09.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitlinearc( real_1d_array y, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::lsfitlinearc( real_1d_array y, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t n, ae_int_t m, ae_int_t k, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinearc( real_1d_array y, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinearc( real_1d_array y, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t n, ae_int_t m, ae_int_t k, ae_int_t& info, real_1d_array& c, lsfitreport& rep);
Examples: [1]
lsfitlinearw function/************************************************************************* Weighted linear least squares fitting. QR decomposition is used to reduce task to MxM, then triangular solver or SVD-based solver is used depending on condition number of the system. It allows to maximize speed and retain decent accuracy. IMPORTANT: if you want to perform polynomial fitting, it may be more convenient to use PolynomialFit() function. This function gives best results on polynomial problems and solves numerical stability issues which arise when you fit high-degree polynomials to your data. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. W - array[0..N-1] Weights corresponding to function values. Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I, J] - value of J-th basis function in I-th point. N - number of points used. N>=1. M - number of basis functions, M>=1. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -1 incorrect N/M were specified * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * Rep.TaskRCond reciprocal of condition number * R2 non-adjusted coefficient of determination (non-weighted) * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED ERRORS IN PARAMETERS This solver also calculates different kinds of errors in parameters and fills corresponding fields of report: * Rep.CovPar covariance matrix for parameters, array[K,K]. * Rep.ErrPar errors in parameters, array[K], errpar = sqrt(diag(CovPar)) * Rep.ErrCurve vector of fit errors - standard deviations of empirical best-fit curve from "ideal" best-fit curve built with infinite number of samples, array[N]. errcurve = sqrt(diag(F*CovPar*F')), where F is functions matrix. * Rep.Noise vector of per-point estimates of noise, array[N] NOTE: noise in the data is estimated as follows: * for fitting without user-supplied weights all points are assumed to have same level of noise, which is estimated from the data * for fitting with user-supplied weights we assume that noise level in I-th point is inversely proportional to Ith weight. Coefficient of proportionality is estimated from the data. NOTE: we apply small amount of regularization when we invert squared Jacobian and calculate covariance matrix. It guarantees that algorithm won't divide by zero during inversion, but skews error estimates a bit (fractional error is about 10^-9). However, we believe that this difference is insignificant for all practical purposes except for the situation when you want to compare ALGLIB results with "reference" implementation up to the last significant digit. NOTE: covariance matrix is estimated using correction for degrees of freedom (covariances are divided by N-M instead of dividing by N). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitlinearw( real_1d_array y, real_1d_array w, real_2d_array fmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::lsfitlinearw( real_1d_array y, real_1d_array w, real_2d_array fmatrix, ae_int_t n, ae_int_t m, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinearw( real_1d_array y, real_1d_array w, real_2d_array fmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinearw( real_1d_array y, real_1d_array w, real_2d_array fmatrix, ae_int_t n, ae_int_t m, ae_int_t& info, real_1d_array& c, lsfitreport& rep);
Examples: [1]
lsfitlinearwc function/************************************************************************* Weighted constained linear least squares fitting. This is variation of LSFitLinearW(), which searchs for min|A*x=b| given that K additional constaints C*x=bc are satisfied. It reduces original task to modified one: min|B*y-d| WITHOUT constraints, then LSFitLinearW() is called. IMPORTANT: if you want to perform polynomial fitting, it may be more convenient to use PolynomialFit() function. This function gives best results on polynomial problems and solves numerical stability issues which arise when you fit high-degree polynomials to your data. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: Y - array[0..N-1] Function values in N points. W - array[0..N-1] Weights corresponding to function values. Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. FMatrix - a table of basis functions values, array[0..N-1, 0..M-1]. FMatrix[I,J] - value of J-th basis function in I-th point. CMatrix - a table of constaints, array[0..K-1,0..M]. I-th row of CMatrix corresponds to I-th linear constraint: CMatrix[I,0]*C[0] + ... + CMatrix[I,M-1]*C[M-1] = CMatrix[I,M] N - number of points used. N>=1. M - number of basis functions, M>=1. K - number of constraints, 0 <= K < M K=0 corresponds to absence of constraints. OUTPUT PARAMETERS: Info - error code: * -4 internal SVD decomposition subroutine failed (very rare and for degenerate systems only) * -3 either too many constraints (M or more), degenerate constraints (some constraints are repetead twice) or inconsistent constraints were specified. * 1 task is solved C - decomposition coefficients, array[0..M-1] Rep - fitting report. Following fields are set: * R2 non-adjusted coefficient of determination (non-weighted) * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. ERRORS IN PARAMETERS This solver also calculates different kinds of errors in parameters and fills corresponding fields of report: * Rep.CovPar covariance matrix for parameters, array[K,K]. * Rep.ErrPar errors in parameters, array[K], errpar = sqrt(diag(CovPar)) * Rep.ErrCurve vector of fit errors - standard deviations of empirical best-fit curve from "ideal" best-fit curve built with infinite number of samples, array[N]. errcurve = sqrt(diag(F*CovPar*F')), where F is functions matrix. * Rep.Noise vector of per-point estimates of noise, array[N] IMPORTANT: errors in parameters are calculated without taking into account boundary/linear constraints! Presence of constraints changes distribution of errors, but there is no easy way to account for constraints when you calculate covariance matrix. NOTE: noise in the data is estimated as follows: * for fitting without user-supplied weights all points are assumed to have same level of noise, which is estimated from the data * for fitting with user-supplied weights we assume that noise level in I-th point is inversely proportional to Ith weight. Coefficient of proportionality is estimated from the data. NOTE: we apply small amount of regularization when we invert squared Jacobian and calculate covariance matrix. It guarantees that algorithm won't divide by zero during inversion, but skews error estimates a bit (fractional error is about 10^-9). However, we believe that this difference is insignificant for all practical purposes except for the situation when you want to compare ALGLIB results with "reference" implementation up to the last significant digit. NOTE: covariance matrix is estimated using correction for degrees of freedom (covariances are divided by N-M instead of dividing by N). -- ALGLIB -- Copyright 07.09.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitlinearwc( real_1d_array y, real_1d_array w, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::lsfitlinearwc( real_1d_array y, real_1d_array w, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t n, ae_int_t m, ae_int_t k, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinearwc( real_1d_array y, real_1d_array w, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t& info, real_1d_array& c, lsfitreport& rep); void alglib::smp_lsfitlinearwc( real_1d_array y, real_1d_array w, real_2d_array fmatrix, real_2d_array cmatrix, ae_int_t n, ae_int_t m, ae_int_t k, ae_int_t& info, real_1d_array& c, lsfitreport& rep);
Examples: [1]
lsfitresults function/************************************************************************* Nonlinear least squares fitting results. Called after return from LSFitFit(). INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: Info - completion code: * -7 gradient verification failed. See LSFitSetGradientCheck() for more information. * 2 relative step is no more than EpsX. * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible C - array[0..K-1], solution Rep - optimization report. On success following fields are set: * R2 non-adjusted coefficient of determination (non-weighted) * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED * WRMSError weighted rms error on the (X,Y). ERRORS IN PARAMETERS This solver also calculates different kinds of errors in parameters and fills corresponding fields of report: * Rep.CovPar covariance matrix for parameters, array[K,K]. * Rep.ErrPar errors in parameters, array[K], errpar = sqrt(diag(CovPar)) * Rep.ErrCurve vector of fit errors - standard deviations of empirical best-fit curve from "ideal" best-fit curve built with infinite number of samples, array[N]. errcurve = sqrt(diag(J*CovPar*J')), where J is Jacobian matrix. * Rep.Noise vector of per-point estimates of noise, array[N] IMPORTANT: errors in parameters are calculated without taking into account boundary/linear constraints! Presence of constraints changes distribution of errors, but there is no easy way to account for constraints when you calculate covariance matrix. NOTE: noise in the data is estimated as follows: * for fitting without user-supplied weights all points are assumed to have same level of noise, which is estimated from the data * for fitting with user-supplied weights we assume that noise level in I-th point is inversely proportional to Ith weight. Coefficient of proportionality is estimated from the data. NOTE: we apply small amount of regularization when we invert squared Jacobian and calculate covariance matrix. It guarantees that algorithm won't divide by zero during inversion, but skews error estimates a bit (fractional error is about 10^-9). However, we believe that this difference is insignificant for all practical purposes except for the situation when you want to compare ALGLIB results with "reference" implementation up to the last significant digit. NOTE: covariance matrix is estimated using correction for degrees of freedom (covariances are divided by N-M instead of dividing by N). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitresults( lsfitstate state, ae_int_t& info, real_1d_array& c, lsfitreport& rep);
lsfitsetbc function/************************************************************************* This function sets boundary constraints for underlying optimizer Boundary constraints are inactive by default (after initial creation). They are preserved until explicitly turned off with another SetBC() call. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[K]. If some (all) variables are unbounded, you may specify very small number or -INF (latter is recommended because it will allow solver to use better algorithm). BndU - upper bounds, array[K]. If some (all) variables are unbounded, you may specify very large number or +INF (latter is recommended because it will allow solver to use better algorithm). NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: unlike other constrained optimization algorithms, this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetbc( lsfitstate state, real_1d_array bndl, real_1d_array bndu);
lsfitsetcond function/************************************************************************* Stopping conditions for nonlinear least squares fitting. INPUT PARAMETERS: State - structure which stores algorithm state EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by LSFitSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Only Levenberg-Marquardt iterations are counted (L-BFGS/CG iterations are NOT counted because their cost is very low compared to that of LM). NOTE Passing EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (according to the scheme used by MINLM unit). -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetcond(lsfitstate state, double epsx, ae_int_t maxits);
lsfitsetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before fitting begins * LSFitFit() is called * prior to actual fitting, for each point in data set X_i and each component of parameters being fited C_j algorithm performs following steps: * two trial steps are made to C_j-TestStep*S[j] and C_j+TestStep*S[j], where C_j is j-th parameter and S[j] is a scale of j-th parameter * if needed, steps are bounded with respect to constraints on C[] * F(X_i|C) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N*K (points count * parameters count) gradient evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with LSFitSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. NOTE 4: this function works only for optimizers created with LSFitCreateWFG() or LSFitCreateFG() constructors. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 15.06.2012 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetgradientcheck(lsfitstate state, double teststep);
lsfitsetlc function/************************************************************************* This function sets linear constraints for underlying optimizer Linear constraints are inactive by default (after initial creation). They are preserved until explicitly turned off with another SetLC() call. INPUT PARAMETERS: State - structure stores algorithm state C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT IMPORTANT: if you have linear constraints, it is strongly recommended to set scale of variables with lsfitsetscale(). QP solver which is used to calculate linearly constrained steps heavily relies on good scaling of input problems. NOTE: linear (non-box) constraints are satisfied only approximately - there always exists some violation due to numerical errors and algorithmic limitations. NOTE: general linear constraints add significant overhead to solution process. Although solver performs roughly same amount of iterations (when compared with similar box-only constrained problem), each iteration now involves solution of linearly constrained QP subproblem, which requires ~3-5 times more Cholesky decompositions. Thus, if you can reformulate your problem in such way this it has only box constraints, it may be beneficial to do so. -- ALGLIB -- Copyright 29.04.2017 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetlc( lsfitstate state, real_2d_array c, integer_1d_array ct); void alglib::lsfitsetlc( lsfitstate state, real_2d_array c, integer_1d_array ct, ae_int_t k);
lsfitsetscale function/************************************************************************* This function sets scaling coefficients for underlying optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Generally, scale is NOT considered to be a form of preconditioner. But LM optimizer is unique in that it uses scaling matrix both in the stopping condition tests and as Marquardt damping factor. Proper scaling is very important for the algorithm performance. It is less important for the quality of results, but still has some influence (it is easier to converge when variables are properly scaled, so premature stopping is possible when very badly scalled variables are combined with relaxed stopping conditions). INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetscale(lsfitstate state, real_1d_array s);
Examples: [1]
lsfitsetstpmax function/************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. NOTE: non-zero StpMax leads to moderate performance degradation because intermediate step of preconditioned L-BFGS optimization is incompatible with limits on step size. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetstpmax(lsfitstate state, double stpmax);
lsfitsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not When reports are needed, State.C (current parameters) and State.F (current value of fitting function) are reported. -- ALGLIB -- Copyright 15.08.2010 by Bochkanov Sergey *************************************************************************/void alglib::lsfitsetxrep(lsfitstate state, bool needxrep);
lstfitpiecewiselinearrdp function/************************************************************************* This subroutine fits piecewise linear curve to points with Ramer-Douglas- Peucker algorithm, which stops after achieving desired precision. IMPORTANT: * it performs non-least-squares fitting; it builds curve, but this curve does not minimize some least squares metric. See description of RDP algorithm (say, in Wikipedia) for more details on WHAT is performed. * this function does NOT work with parametric curves (i.e. curves which can be represented as {X(t),Y(t)}. It works with curves which can be represented as Y(X). Thus, it is impossible to model figures like circles with this functions. If you want to work with parametric curves, you should use ParametricRDPFixed() function provided by "Parametric" subpackage of "Interpolation" package. INPUT PARAMETERS: X - array of X-coordinates: * at least N elements * can be unordered (points are automatically sorted) * this function may accept non-distinct X (see below for more information on handling of such inputs) Y - array of Y-coordinates: * at least N elements N - number of elements in X/Y Eps - positive number, desired precision. OUTPUT PARAMETERS: X2 - X-values of corner points for piecewise approximation, has length NSections+1 or zero (for NSections=0). Y2 - Y-values of corner points, has length NSections+1 or zero (for NSections=0). NSections- number of sections found by algorithm, NSections can be zero for degenerate datasets (N<=1 or all X[] are non-distinct). NOTE: X2/Y2 are ordered arrays, i.e. (X2[0],Y2[0]) is a first point of curve, (X2[NSection-1],Y2[NSection-1]) is the last point. -- ALGLIB -- Copyright 02.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::lstfitpiecewiselinearrdp( real_1d_array x, real_1d_array y, ae_int_t n, double eps, real_1d_array& x2, real_1d_array& y2, ae_int_t& nsections);
lstfitpiecewiselinearrdpfixed function/************************************************************************* This subroutine fits piecewise linear curve to points with Ramer-Douglas- Peucker algorithm, which stops after generating specified number of linear sections. IMPORTANT: * it does NOT perform least-squares fitting; it builds curve, but this curve does not minimize some least squares metric. See description of RDP algorithm (say, in Wikipedia) for more details on WHAT is performed. * this function does NOT work with parametric curves (i.e. curves which can be represented as {X(t),Y(t)}. It works with curves which can be represented as Y(X). Thus, it is impossible to model figures like circles with this functions. If you want to work with parametric curves, you should use ParametricRDPFixed() function provided by "Parametric" subpackage of "Interpolation" package. INPUT PARAMETERS: X - array of X-coordinates: * at least N elements * can be unordered (points are automatically sorted) * this function may accept non-distinct X (see below for more information on handling of such inputs) Y - array of Y-coordinates: * at least N elements N - number of elements in X/Y M - desired number of sections: * at most M sections are generated by this function * less than M sections can be generated if we have N<M (or some X are non-distinct). OUTPUT PARAMETERS: X2 - X-values of corner points for piecewise approximation, has length NSections+1 or zero (for NSections=0). Y2 - Y-values of corner points, has length NSections+1 or zero (for NSections=0). NSections- number of sections found by algorithm, NSections<=M, NSections can be zero for degenerate datasets (N<=1 or all X[] are non-distinct). NOTE: X2/Y2 are ordered arrays, i.e. (X2[0],Y2[0]) is a first point of curve, (X2[NSection-1],Y2[NSection-1]) is the last point. -- ALGLIB -- Copyright 02.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::lstfitpiecewiselinearrdpfixed( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, real_1d_array& x2, real_1d_array& y2, ae_int_t& nsections);
polynomialfit function/************************************************************************* Fitting by polynomials in barycentric form. This function provides simple unterface for unconstrained unweighted fitting. See PolynomialFitWC() if you need constrained fitting. Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO: PolynomialFitWC() COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points, N>0 * if given, only leading N elements of X/Y are used * if not given, automatically determined from sizes of X/Y M - number of basis functions (= polynomial_degree + 1), M>=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD P - interpolant in barycentric form. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED NOTES: you can convert P from barycentric form to the power or Chebyshev basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from POLINT subpackage. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::polynomialfit( real_1d_array x, real_1d_array y, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep); void alglib::polynomialfit( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep); void alglib::smp_polynomialfit( real_1d_array x, real_1d_array y, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep); void alglib::smp_polynomialfit( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep);
Examples: [1]
polynomialfitwc function/************************************************************************* Weighted fitting by polynomials in barycentric form, with constraints on function values or first derivatives. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO: PolynomialFit() COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points, N>0. * if given, only leading N elements of X/Y/W are used * if not given, automatically determined from sizes of X/Y/W XC - points where polynomial values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that P(XC[i])=YC[i] * DC[i]=1 means that P'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints, 0<=K<M. K=0 means no constraints (XC/YC/DC are not used in such cases) M - number of basis functions (= polynomial_degree + 1), M>=1 OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints P - interpolant in barycentric form. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTES: you can convert P from barycentric form to the power or Chebyshev basis with PolynomialBar2Pow() or PolynomialBar2Cheb() functions from POLINT subpackage. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * even simple constraints can be inconsistent, see Wikipedia article on this subject: http://en.wikipedia.org/wiki/Birkhoff_interpolation * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints is NOT GUARANTEED. * in the one special cases, however, we can guarantee consistency. This case is: M>1 and constraints on the function values (NOT DERIVATIVES) Our final recommendation is to use constraints WHEN AND ONLY when you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 10.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::polynomialfitwc( real_1d_array x, real_1d_array y, real_1d_array w, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep); void alglib::polynomialfitwc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep); void alglib::smp_polynomialfitwc( real_1d_array x, real_1d_array y, real_1d_array w, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep); void alglib::smp_polynomialfitwc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, barycentricinterpolant& p, polynomialfitreport& rep);
Examples: [1]
spline1dfitcubic function/************************************************************************* Least squares fitting by cubic spline. This subroutine is "lightweight" alternative for more complex and feature- rich Spline1DFitCubicWC(). See Spline1DFitCubicWC() for more information about subroutine parameters (we don't duplicate it here because of length) COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::spline1dfitcubic( real_1d_array x, real_1d_array y, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::spline1dfitcubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitcubic( real_1d_array x, real_1d_array y, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitcubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep);
spline1dfitcubicwc function/************************************************************************* Weighted fitting by cubic spline, with constraints on function values or derivatives. Equidistant grid with M-2 nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are cubic splines with continuous second derivatives and non-fixed first derivatives at interval ends. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO Spline1DFitHermiteWC() - fitting by Hermite splines (more flexible, less smooth) Spline1DFitCubic() - "lightweight" fitting by cubic splines, without invididual weights and constraints COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes XC - points where spline values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints (optional): * 0<=K<M. * K=0 means no constraints (XC/YC/DC are not used) * if given, only first K elements of XC/YC/DC are used * if not given, automatically determined from XC/YC/DC M - number of basis functions ( = number_of_nodes+2), M>=4. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints S - spline interpolant. Rep - report, same format as in LSFitLinearWC() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * excessive constraints can be inconsistent. Splines are piecewise cubic functions, and it is easy to create an example, where large number of constraints concentrated in small area will result in inconsistency. Just because spline is not flexible enough to satisfy all of them. And same constraints spread across the [min(x),max(x)] will be perfectly consistent. * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints IS NOT GUARANTEED. * in the several special cases, however, we CAN guarantee consistency. * one of this cases is constraints on the function values AND/OR its derivatives at the interval boundaries. * another special case is ONE constraint on the function value (OR, but not AND, derivative) anywhere in the interval Our final recommendation is to use constraints WHEN AND ONLY WHEN you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::spline1dfitcubicwc( real_1d_array x, real_1d_array y, real_1d_array w, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::spline1dfitcubicwc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitcubicwc( real_1d_array x, real_1d_array y, real_1d_array w, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitcubicwc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep);
spline1dfithermite function/************************************************************************* Least squares fitting by Hermite spline. This subroutine is "lightweight" alternative for more complex and feature- rich Spline1DFitHermiteWC(). See Spline1DFitHermiteWC() description for more information about subroutine parameters (we don't duplicate it here because of length). COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::spline1dfithermite( real_1d_array x, real_1d_array y, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::spline1dfithermite( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfithermite( real_1d_array x, real_1d_array y, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfithermite( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep);
spline1dfithermitewc function/************************************************************************* Weighted fitting by Hermite spline, with constraints on function values or first derivatives. Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are Hermite splines. Small regularizing term is used when solving constrained tasks (to improve stability). Task is linear, so linear least squares solver is used. Complexity of this computational scheme is O(N*M^2), mostly dominated by least squares solver SEE ALSO Spline1DFitCubicWC() - fitting by Cubic splines (less flexible, more smooth) Spline1DFitHermite() - "lightweight" Hermite fitting, without invididual weights and constraints COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted task. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes XC - points where spline values/derivatives are constrained, array[0..K-1]. YC - values of constraints, array[0..K-1] DC - array[0..K-1], types of constraints: * DC[i]=0 means that S(XC[i])=YC[i] * DC[i]=1 means that S'(XC[i])=YC[i] SEE BELOW FOR IMPORTANT INFORMATION ON CONSTRAINTS K - number of constraints (optional): * 0<=K<M. * K=0 means no constraints (XC/YC/DC are not used) * if given, only first K elements of XC/YC/DC are used * if not given, automatically determined from XC/YC/DC M - number of basis functions (= 2 * number of nodes), M>=4, M IS EVEN! OUTPUT PARAMETERS: Info- same format as in LSFitLinearW() subroutine: * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD -3 means inconsistent constraints -2 means odd M was passed (which is not supported) -1 means another errors in parameters passed (N<=0, for example) S - spline interpolant. Rep - report, same format as in LSFitLinearW() subroutine. Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. IMPORTANT: this subroitine supports only even M's ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING CONSTRAINTS - DANGERS AND OPPORTUNITIES: Setting constraints can lead to undesired results, like ill-conditioned behavior, or inconsistency being detected. From the other side, it allows us to improve quality of the fit. Here we summarize our experience with constrained regression splines: * excessive constraints can be inconsistent. Splines are piecewise cubic functions, and it is easy to create an example, where large number of constraints concentrated in small area will result in inconsistency. Just because spline is not flexible enough to satisfy all of them. And same constraints spread across the [min(x),max(x)] will be perfectly consistent. * the more evenly constraints are spread across [min(x),max(x)], the more chances that they will be consistent * the greater is M (given fixed constraints), the more chances that constraints will be consistent * in the general case, consistency of constraints is NOT GUARANTEED. * in the several special cases, however, we can guarantee consistency. * one of this cases is M>=4 and constraints on the function value (AND/OR its derivative) at the interval boundaries. * another special case is M>=4 and ONE constraint on the function value (OR, BUT NOT AND, derivative) anywhere in [min(x),max(x)] Our final recommendation is to use constraints WHEN AND ONLY when you can't solve your task without them. Anything beyond special cases given above is not guaranteed and may result in inconsistency. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::spline1dfithermitewc( real_1d_array x, real_1d_array y, real_1d_array w, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::spline1dfithermitewc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfithermitewc( real_1d_array x, real_1d_array y, real_1d_array w, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfithermitewc( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, real_1d_array xc, real_1d_array yc, integer_1d_array dc, ae_int_t k, ae_int_t m, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep);
spline1dfitpenalized function/************************************************************************* Fitting by penalized cubic spline. Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are cubic splines with natural boundary conditions. Problem is regularized by adding non-linearity penalty to the usual least squares penalty function: S(x) = arg min { LS + P }, where LS = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty P = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty rho - tunable constant given by user C - automatically determined scale parameter, makes penalty invariant with respect to scaling of X, Y, W. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. N - number of points (optional): * N>0 * if given, only first N elements of X/Y are processed * if not given, automatically determined from X/Y sizes M - number of basis functions ( = number_of_nodes), M>=4. Rho - regularization constant passed by user. It penalizes nonlinearity in the regression spline. It is logarithmically scaled, i.e. actual value of regularization constant is calculated as 10^Rho. It is automatically scaled so that: * Rho=2.0 corresponds to moderate amount of nonlinearity * generally, it should be somewhere in the [-8.0,+8.0] If you do not want to penalize nonlineary, pass small Rho. Values as low as -15 should work. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD or Cholesky decomposition; problem may be too ill-conditioned (very rare) S - spline interpolant. Rep - Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTE 1: additional nodes are added to the spline outside of the fitting interval to force linearity when x<min(x,xc) or x>max(x,xc). It is done for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so it is natural to force linearity outside of this interval. NOTE 2: function automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 18.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::spline1dfitpenalized( real_1d_array x, real_1d_array y, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::spline1dfitpenalized( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitpenalized( real_1d_array x, real_1d_array y, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitpenalized( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep);
Examples: [1]
spline1dfitpenalizedw function/************************************************************************* Weighted fitting by penalized cubic spline. Equidistant grid with M nodes on [min(x,xc),max(x,xc)] is used to build basis functions. Basis functions are cubic splines with natural boundary conditions. Problem is regularized by adding non-linearity penalty to the usual least squares penalty function: S(x) = arg min { LS + P }, where LS = SUM { w[i]^2*(y[i] - S(x[i]))^2 } - least squares penalty P = C*10^rho*integral{ S''(x)^2*dx } - non-linearity penalty rho - tunable constant given by user C - automatically determined scale parameter, makes penalty invariant with respect to scaling of X, Y, W. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multithreading support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Speed-up provided by multithreading greatly depends on problem size ! - only large problems (number of coefficients is more than 500) can be ! efficiently multithreaded. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - points, array[0..N-1]. Y - function values, array[0..N-1]. W - weights, array[0..N-1] Each summand in square sum of approximation deviations from given values is multiplied by the square of corresponding weight. Fill it by 1's if you don't want to solve weighted problem. N - number of points (optional): * N>0 * if given, only first N elements of X/Y/W are processed * if not given, automatically determined from X/Y/W sizes M - number of basis functions ( = number_of_nodes), M>=4. Rho - regularization constant passed by user. It penalizes nonlinearity in the regression spline. It is logarithmically scaled, i.e. actual value of regularization constant is calculated as 10^Rho. It is automatically scaled so that: * Rho=2.0 corresponds to moderate amount of nonlinearity * generally, it should be somewhere in the [-8.0,+8.0] If you do not want to penalize nonlineary, pass small Rho. Values as low as -15 should work. OUTPUT PARAMETERS: Info- same format as in LSFitLinearWC() subroutine. * Info>0 task is solved * Info<=0 an error occured: -4 means inconvergence of internal SVD or Cholesky decomposition; problem may be too ill-conditioned (very rare) S - spline interpolant. Rep - Following fields are set: * RMSError rms error on the (X,Y). * AvgError average error on the (X,Y). * AvgRelError average relative error on the non-zero Y * MaxError maximum error NON-WEIGHTED ERRORS ARE CALCULATED IMPORTANT: this subroitine doesn't calculate task's condition number for K<>0. NOTE 1: additional nodes are added to the spline outside of the fitting interval to force linearity when x<min(x,xc) or x>max(x,xc). It is done for consistency - we penalize non-linearity at [min(x,xc),max(x,xc)], so it is natural to force linearity outside of this interval. NOTE 2: function automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 19.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::spline1dfitpenalizedw( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::spline1dfitpenalizedw( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitpenalizedw( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep); void alglib::smp_spline1dfitpenalizedw( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, ae_int_t m, double rho, ae_int_t& info, spline1dinterpolant& s, spline1dfitreport& rep);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// In this example we demonstrate linear fitting by f(x|a) = a*exp(0.5*x).
//
// We have:
// * y - vector of experimental data
// * fmatrix - matrix of basis functions calculated at sample points
// Actually, we have only one basis function F0 = exp(0.5*x).
//
real_2d_array fmatrix = "[[0.606531],[0.670320],[0.740818],[0.818731],[0.904837],[1.000000],[1.105171],[1.221403],[1.349859],[1.491825],[1.648721]]";
real_1d_array y = "[1.133719, 1.306522, 1.504604, 1.554663, 1.884638, 2.072436, 2.257285, 2.534068, 2.622017, 2.897713, 3.219371]";
ae_int_t info;
real_1d_array c;
lsfitreport rep;
//
// Linear fitting without weights
//
lsfitlinear(y, fmatrix, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.98650]
//
// Linear fitting with individual weights.
// Slightly different result is returned.
//
real_1d_array w = "[1.414213, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]";
lsfitlinearw(y, w, fmatrix, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [1.983354]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// In this example we demonstrate linear fitting by f(x|a,b) = a*x+b
// with simple constraint f(0)=0.
//
// We have:
// * y - vector of experimental data
// * fmatrix - matrix of basis functions sampled at [0,1] with step 0.2:
// [ 1.0 0.0 ]
// [ 1.0 0.2 ]
// [ 1.0 0.4 ]
// [ 1.0 0.6 ]
// [ 1.0 0.8 ]
// [ 1.0 1.0 ]
// first column contains value of first basis function (constant term)
// second column contains second basis function (linear term)
// * cmatrix - matrix of linear constraints:
// [ 1.0 0.0 0.0 ]
// first two columns contain coefficients before basis functions,
// last column contains desired value of their sum.
// So [1,0,0] means "1*constant_term + 0*linear_term = 0"
//
real_1d_array y = "[0.072436,0.246944,0.491263,0.522300,0.714064,0.921929]";
real_2d_array fmatrix = "[[1,0.0],[1,0.2],[1,0.4],[1,0.6],[1,0.8],[1,1.0]]";
real_2d_array cmatrix = "[[1,0,0]]";
ae_int_t info;
real_1d_array c;
lsfitreport rep;
//
// Constrained fitting without weights
//
lsfitlinearc(y, fmatrix, cmatrix, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [0,0.932933]
//
// Constrained fitting with individual weights
//
real_1d_array w = "[1, 1.414213, 1, 1, 1, 1]";
lsfitlinearwc(y, w, fmatrix, cmatrix, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", c.tostring(3).c_str()); // EXPECTED: [0,0.938322]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0))
// where x is a position on X-axis and c is adjustable parameter
func = exp(-c[0]*pow(x[0],2));
}
int main(int argc, char **argv)
{
//
// In this example we demonstrate exponential fitting
// by f(x) = exp(-c*x^2)
// using function value only.
//
// Gradient is estimated using combination of numerical differences
// and secant updates. diffstep variable stores differentiation step
// (we have to tell algorithm what step to use).
//
real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
real_1d_array c = "[0.3]";
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t info;
lsfitstate state;
lsfitreport rep;
double diffstep = 0.0001;
//
// Fitting without weights
//
lsfitcreatef(x, y, c, diffstep, state);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
//
// Fitting with weights
// (you can change weights and see how it changes result)
//
real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
lsfitcreatewf(x, y, w, c, diffstep, state);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0))
// where x is a position on X-axis and c is adjustable parameter
func = exp(-c[0]*pow(x[0],2));
}
int main(int argc, char **argv)
{
//
// In this example we demonstrate exponential fitting by
// f(x) = exp(-c*x^2)
// subject to bound constraints
// 0.0 <= c <= 1.0
// using function value only.
//
// Gradient is estimated using combination of numerical differences
// and secant updates. diffstep variable stores differentiation step
// (we have to tell algorithm what step to use).
//
// Unconstrained solution is c=1.5, but because of constraints we should
// get c=1.0 (at the boundary).
//
real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
real_1d_array c = "[0.3]";
real_1d_array bndl = "[0.0]";
real_1d_array bndu = "[1.0]";
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t info;
lsfitstate state;
lsfitreport rep;
double diffstep = 0.0001;
lsfitcreatef(x, y, c, diffstep, state);
lsfitsetbc(state, bndl, bndu);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func);
lsfitresults(state, info, c, rep);
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.0]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0))
// where x is a position on X-axis and c is adjustable parameter
func = exp(-c[0]*pow(x[0],2));
}
void function_cx_1_grad(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
// where x is a position on X-axis and c is adjustable parameter.
// IMPORTANT: gradient is calculated with respect to C, not to X
func = exp(-c[0]*pow(x[0],2));
grad[0] = -pow(x[0],2)*func;
}
int main(int argc, char **argv)
{
//
// In this example we demonstrate exponential fitting
// by f(x) = exp(-c*x^2)
// using function value and gradient (with respect to c).
//
real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
real_1d_array c = "[0.3]";
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t info;
lsfitstate state;
lsfitreport rep;
//
// Fitting without weights
//
lsfitcreatefg(x, y, c, true, state);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
//
// Fitting with weights
// (you can change weights and see how it changes result)
//
real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
lsfitcreatewfg(x, y, w, c, true, state);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
void function_cx_1_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0))
// where x is a position on X-axis and c is adjustable parameter
func = exp(-c[0]*pow(x[0],2));
}
void function_cx_1_grad(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0)) and gradient G={df/dc[i]}
// where x is a position on X-axis and c is adjustable parameter.
// IMPORTANT: gradient is calculated with respect to C, not to X
func = exp(-c[0]*pow(x[0],2));
grad[0] = -pow(x[0],2)*func;
}
void function_cx_1_hess(const real_1d_array &c, const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr)
{
// this callback calculates f(c,x)=exp(-c0*sqr(x0)), gradient G={df/dc[i]} and Hessian H={d2f/(dc[i]*dc[j])}
// where x is a position on X-axis and c is adjustable parameter.
// IMPORTANT: gradient/Hessian are calculated with respect to C, not to X
func = exp(-c[0]*pow(x[0],2));
grad[0] = -pow(x[0],2)*func;
hess[0][0] = pow(x[0],4)*func;
}
int main(int argc, char **argv)
{
//
// In this example we demonstrate exponential fitting
// by f(x) = exp(-c*x^2)
// using function value, gradient and Hessian (with respect to c)
//
real_2d_array x = "[[-1],[-0.8],[-0.6],[-0.4],[-0.2],[0],[0.2],[0.4],[0.6],[0.8],[1.0]]";
real_1d_array y = "[0.223130, 0.382893, 0.582748, 0.786628, 0.941765, 1.000000, 0.941765, 0.786628, 0.582748, 0.382893, 0.223130]";
real_1d_array c = "[0.3]";
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t info;
lsfitstate state;
lsfitreport rep;
//
// Fitting without weights
//
lsfitcreatefgh(x, y, c, state);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad, function_cx_1_hess);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
//
// Fitting with weights
// (you can change weights and see how it changes result)
//
real_1d_array w = "[1,1,1,1,1,1,1,1,1,1,1]";
lsfitcreatewfgh(x, y, w, c, state);
lsfitsetcond(state, epsx, maxits);
alglib::lsfitfit(state, function_cx_1_func, function_cx_1_grad, function_cx_1_hess);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(1).c_str()); // EXPECTED: [1.5]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
void function_debt_func(const real_1d_array &c, const real_1d_array &x, double &func, void *ptr)
{
//
// this callback calculates f(c,x)=c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1))
//
func = c[0]*(1+c[1]*(pow(x[0]-1999,c[2])-1));
}
int main(int argc, char **argv)
{
//
// In this example we demonstrate fitting by
// f(x) = c[0]*(1+c[1]*((x-1999)^c[2]-1))
// subject to bound constraints
// -INF < c[0] < +INF
// -10 <= c[1] <= +10
// 0.1 <= c[2] <= 2.0
// Data we want to fit are time series of Japan national debt
// collected from 2000 to 2008 measured in USD (dollars, not
// millions of dollars).
//
// Our variables are:
// c[0] - debt value at initial moment (2000),
// c[1] - direction coefficient (growth or decrease),
// c[2] - curvature coefficient.
// You may see that our variables are badly scaled - first one
// is order of 10^12, and next two are somewhere about 1 in
// magnitude. Such problem is difficult to solve without some
// kind of scaling.
// That is exactly where lsfitsetscale() function can be used.
// We set scale of our variables to [1.0E12, 1, 1], which allows
// us to easily solve this problem.
//
// You can try commenting out lsfitsetscale() call - and you will
// see that algorithm will fail to converge.
//
real_2d_array x = "[[2000],[2001],[2002],[2003],[2004],[2005],[2006],[2007],[2008]]";
real_1d_array y = "[4323239600000.0, 4560913100000.0, 5564091500000.0, 6743189300000.0, 7284064600000.0, 7050129600000.0, 7092221500000.0, 8483907600000.0, 8625804400000.0]";
real_1d_array c = "[1.0e+13, 1, 1]";
double epsx = 1.0e-5;
real_1d_array bndl = "[-inf, -10, 0.1]";
real_1d_array bndu = "[+inf, +10, 2.0]";
real_1d_array s = "[1.0e+12, 1, 1]";
ae_int_t maxits = 0;
ae_int_t info;
lsfitstate state;
lsfitreport rep;
double diffstep = 1.0e-5;
lsfitcreatef(x, y, c, diffstep, state);
lsfitsetcond(state, epsx, maxits);
lsfitsetbc(state, bndl, bndu);
lsfitsetscale(state, s);
alglib::lsfitfit(state, function_debt_func);
lsfitresults(state, info, c, rep);
printf("%d\n", int(info)); // EXPECTED: 2
printf("%s\n", c.tostring(-2).c_str()); // EXPECTED: [4.142560E+12, 0.434240, 0.565376]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates polynomial fitting.
//
// Fitting is done by two (M=2) functions from polynomial basis:
// f0 = 1
// f1 = x
// Basically, it just a linear fit; more complex polynomials may be used
// (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
// us to demonstrate polynomialfit() function in action.
//
// We have:
// * x set of abscissas
// * y experimental data
//
// Additionally we demonstrate weighted fitting, where second point has
// more weight than other ones.
//
real_1d_array x = "[0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0]";
real_1d_array y = "[0.00,0.05,0.26,0.32,0.33,0.43,0.60,0.60,0.77,0.98,1.02]";
ae_int_t m = 2;
double t = 2;
ae_int_t info;
barycentricinterpolant p;
polynomialfitreport rep;
double v;
//
// Fitting without individual weights
//
// NOTE: result is returned as barycentricinterpolant structure.
// if you want to get representation in the power basis,
// you can use barycentricbar2pow() function to convert
// from barycentric to power representation (see docs for
// POLINT subpackage for more info).
//
polynomialfit(x, y, m, info, p, rep);
v = barycentriccalc(p, t);
printf("%.2f\n", double(v)); // EXPECTED: 2.011
//
// Fitting with individual weights
//
// NOTE: slightly different result is returned
//
real_1d_array w = "[1,1.414213562,1,1,1,1,1,1,1,1,1]";
real_1d_array xc = "[]";
real_1d_array yc = "[]";
integer_1d_array dc = "[]";
polynomialfitwc(x, y, w, xc, yc, dc, m, info, p, rep);
v = barycentriccalc(p, t);
printf("%.2f\n", double(v)); // EXPECTED: 2.023
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates polynomial fitting.
//
// Fitting is done by two (M=2) functions from polynomial basis:
// f0 = 1
// f1 = x
// with simple constraint on function value
// f(0) = 0
// Basically, it just a linear fit; more complex polynomials may be used
// (e.g. parabolas with M=3, cubic with M=4), but even such simple fit allows
// us to demonstrate polynomialfit() function in action.
//
// We have:
// * x set of abscissas
// * y experimental data
// * xc points where constraints are placed
// * yc constraints on derivatives
// * dc derivative indices
// (0 means function itself, 1 means first derivative)
//
real_1d_array x = "[1.0,1.0]";
real_1d_array y = "[0.9,1.1]";
real_1d_array w = "[1,1]";
real_1d_array xc = "[0]";
real_1d_array yc = "[0]";
integer_1d_array dc = "[0]";
double t = 2;
ae_int_t m = 2;
ae_int_t info;
barycentricinterpolant p;
polynomialfitreport rep;
double v;
polynomialfitwc(x, y, w, xc, yc, dc, m, info, p, rep);
v = barycentriccalc(p, t);
printf("%.2f\n", double(v)); // EXPECTED: 2.000
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// In this example we demonstrate penalized spline fitting of noisy data
//
// We have:
// * x - abscissas
// * y - vector of experimental data, straight line with small noise
//
real_1d_array x = "[0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90]";
real_1d_array y = "[0.10,0.00,0.30,0.40,0.30,0.40,0.62,0.68,0.75,0.95]";
ae_int_t info;
double v;
spline1dinterpolant s;
spline1dfitreport rep;
double rho;
//
// Fit with VERY small amount of smoothing (rho = -5.0)
// and large number of basis functions (M=50).
//
// With such small regularization penalized spline almost fully reproduces function values
//
rho = -5.0;
spline1dfitpenalized(x, y, 50, rho, info, s, rep);
printf("%d\n", int(info)); // EXPECTED: 1
v = spline1dcalc(s, 0.0);
printf("%.1f\n", double(v)); // EXPECTED: 0.10
//
// Fit with VERY large amount of smoothing (rho = 10.0)
// and large number of basis functions (M=50).
//
// With such regularization our spline should become close to the straight line fit.
// We will compare its value in x=1.0 with results obtained from such fit.
//
rho = +10.0;
spline1dfitpenalized(x, y, 50, rho, info, s, rep);
printf("%d\n", int(info)); // EXPECTED: 1
v = spline1dcalc(s, 1.0);
printf("%.2f\n", double(v)); // EXPECTED: 0.969
//
// In real life applications you may need some moderate degree of fitting,
// so we try to fit once more with rho=3.0.
//
rho = +3.0;
spline1dfitpenalized(x, y, 50, rho, info, s, rep);
printf("%d\n", int(info)); // EXPECTED: 1
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_1d_array x = "[1,2,3,4,5,6,7,8]";
real_1d_array y = "[0.06313223,0.44552624,0.61838364,0.71385108,0.77345838,0.81383140,0.84280033,0.86449822]";
ae_int_t n = 8;
double a;
double b;
double c;
double d;
lsfitreport rep;
//
// Test logisticfit4() on carefully designed data with a priori known answer.
//
logisticfit4(x, y, n, a, b, c, d, rep);
printf("%.1f\n", double(a)); // EXPECTED: -1.000
printf("%.1f\n", double(b)); // EXPECTED: 1.200
printf("%.1f\n", double(c)); // EXPECTED: 0.900
printf("%.1f\n", double(d)); // EXPECTED: 1.000
//
// Evaluate model at point x=0.5
//
double v;
v = logisticcalc4(0.5, a, b, c, d);
printf("%.2f\n", double(v)); // EXPECTED: -0.33874308
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_1d_array x = "[1,2,3,4,5,6,7,8]";
real_1d_array y = "[0.1949776139,0.5710060208,0.726002637,0.8060434158,0.8534547965,0.8842071579,0.9054773317,0.9209088299]";
ae_int_t n = 8;
double a;
double b;
double c;
double d;
double g;
lsfitreport rep;
//
// Test logisticfit5() on carefully designed data with a priori known answer.
//
logisticfit5(x, y, n, a, b, c, d, g, rep);
printf("%.1f\n", double(a)); // EXPECTED: -1.000
printf("%.1f\n", double(b)); // EXPECTED: 1.200
printf("%.1f\n", double(c)); // EXPECTED: 0.900
printf("%.1f\n", double(d)); // EXPECTED: 1.000
printf("%.1f\n", double(g)); // EXPECTED: 1.200
//
// Evaluate model at point x=0.5
//
double v;
v = logisticcalc5(0.5, a, b, c, d, g);
printf("%.2f\n", double(v)); // EXPECTED: -0.2354656824
return 0;
}
mannwhitneyu subpackagemannwhitneyutest function/************************************************************************* Mann-Whitney U-test This test checks hypotheses about whether X and Y are samples of two continuous distributions of the same shape and same median or whether their medians are different. The following tests are performed: * two-tailed test (null hypothesis - the medians are equal) * left-tailed test (null hypothesis - the median of the first sample is greater than or equal to the median of the second sample) * right-tailed test (null hypothesis - the median of the first sample is less than or equal to the median of the second sample). Requirements: * the samples are independent * X and Y are continuous distributions (or discrete distributions well- approximating continuous distributions) * distributions of X and Y have the same shape. The only possible difference is their position (i.e. the value of the median) * the number of elements in each sample is not less than 5 * the scale of measurement should be ordinal, interval or ratio (i.e. the test could not be applied to nominal variables). The test is non-parametric and doesn't require distributions to be normal. Input parameters: X - sample 1. Array whose index goes from 0 to N-1. N - size of the sample. N>=5 Y - sample 2. Array whose index goes from 0 to M-1. M - size of the sample. M>=5 Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. To calculate p-values, special approximation is used. This method lets us calculate p-values with satisfactory accuracy in interval [0.0001, 1]. There is no approximation outside the [0.0001, 1] interval. Therefore, if the significance level outlies this interval, the test returns 0.0001. Relative precision of approximation of p-value: N M Max.err. Rms.err. 5..10 N..10 1.4e-02 6.0e-04 5..10 N..100 2.2e-02 5.3e-06 10..15 N..15 1.0e-02 3.2e-04 10..15 N..100 1.0e-02 2.2e-05 15..100 N..100 6.1e-03 2.7e-06 For N,M>100 accuracy checks weren't put into practice, but taking into account characteristics of asymptotic approximation used, precision should not be sharply different from the values for interval [5, 100]. NOTE: P-value approximation was optimized for 0.0001<=p<=0.2500. Thus, P's outside of this interval are enforced to these bounds. Say, you may quite often get P equal to exactly 0.25 or 0.0001. -- ALGLIB -- Copyright 09.04.2007 by Bochkanov Sergey *************************************************************************/void alglib::mannwhitneyutest( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, double& bothtails, double& lefttail, double& righttail);
matdet subpackage| matdet_d_1 | Determinant calculation, real matrix, short form | |
| matdet_d_2 | Determinant calculation, real matrix, full form | |
| matdet_d_3 | Determinant calculation, complex matrix, short form | |
| matdet_d_4 | Determinant calculation, complex matrix, full form | |
| matdet_d_5 | Determinant calculation, complex matrix with zero imaginary part, short form |
cmatrixdet function/************************************************************************* Calculation of the determinant of a general matrix Input parameters: A - matrix, array[0..N-1, 0..N-1] N - (optional) size of matrix A: * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, automatically determined from matrix size (A must be square matrix) Result: determinant of matrix A. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/alglib::complex alglib::cmatrixdet(complex_2d_array a); alglib::complex alglib::cmatrixdet(complex_2d_array a, ae_int_t n);
cmatrixludet function/************************************************************************* Determinant calculation of the matrix given by its LU decomposition. Input parameters: A - LU decomposition of the matrix (output of RMatrixLU subroutine). Pivots - table of permutations which were made during the LU decomposition. Output of RMatrixLU subroutine. N - (optional) size of matrix A: * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, automatically determined from matrix size (A must be square matrix) Result: matrix determinant. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/alglib::complex alglib::cmatrixludet( complex_2d_array a, integer_1d_array pivots); alglib::complex alglib::cmatrixludet( complex_2d_array a, integer_1d_array pivots, ae_int_t n);
rmatrixdet function/************************************************************************* Calculation of the determinant of a general matrix Input parameters: A - matrix, array[0..N-1, 0..N-1] N - (optional) size of matrix A: * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, automatically determined from matrix size (A must be square matrix) Result: determinant of matrix A. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/double alglib::rmatrixdet(real_2d_array a); double alglib::rmatrixdet(real_2d_array a, ae_int_t n);
rmatrixludet function/************************************************************************* Determinant calculation of the matrix given by its LU decomposition. Input parameters: A - LU decomposition of the matrix (output of RMatrixLU subroutine). Pivots - table of permutations which were made during the LU decomposition. Output of RMatrixLU subroutine. N - (optional) size of matrix A: * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, automatically determined from matrix size (A must be square matrix) Result: matrix determinant. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/double alglib::rmatrixludet(real_2d_array a, integer_1d_array pivots); double alglib::rmatrixludet( real_2d_array a, integer_1d_array pivots, ae_int_t n);
spdmatrixcholeskydet function/************************************************************************* Determinant calculation of the matrix given by the Cholesky decomposition. Input parameters: A - Cholesky decomposition, output of SMatrixCholesky subroutine. N - (optional) size of matrix A: * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, automatically determined from matrix size (A must be square matrix) As the determinant is equal to the product of squares of diagonal elements, it's not necessary to specify which triangle - lower or upper - the matrix is stored in. Result: matrix determinant. -- ALGLIB -- Copyright 2005-2008 by Bochkanov Sergey *************************************************************************/double alglib::spdmatrixcholeskydet(real_2d_array a); double alglib::spdmatrixcholeskydet(real_2d_array a, ae_int_t n);
spdmatrixdet function/************************************************************************* Determinant calculation of the symmetric positive definite matrix. Input parameters: A - matrix. Array with elements [0..N-1, 0..N-1]. N - (optional) size of matrix A: * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, automatically determined from matrix size (A must be square matrix) IsUpper - (optional) storage type: * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used/changed by function * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used/changed by function * if not given, both lower and upper triangles must be filled. Result: determinant of matrix A. If matrix A is not positive definite, exception is thrown. -- ALGLIB -- Copyright 2005-2008 by Bochkanov Sergey *************************************************************************/double alglib::spdmatrixdet(real_2d_array a); double alglib::spdmatrixdet(real_2d_array a, ae_int_t n, bool isupper);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array b = "[[1,2],[2,1]]";
double a;
a = rmatrixdet(b);
printf("%.3f\n", double(a)); // EXPECTED: -3
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array b = "[[5,4],[4,5]]";
double a;
a = rmatrixdet(b, 2);
printf("%.3f\n", double(a)); // EXPECTED: 9
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
complex_2d_array b = "[[1+1i,2],[2,1-1i]]";
alglib::complex a;
a = cmatrixdet(b);
printf("%s\n", a.tostring(3).c_str()); // EXPECTED: -2
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
alglib::complex a;
complex_2d_array b = "[[5i,4],[4i,5]]";
a = cmatrixdet(b, 2);
printf("%s\n", a.tostring(3).c_str()); // EXPECTED: 9i
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
alglib::complex a;
complex_2d_array b = "[[9,1],[2,1]]";
a = cmatrixdet(b);
printf("%s\n", a.tostring(3).c_str()); // EXPECTED: 7
return 0;
}
matgen subpackagecmatrixrndcond function/************************************************************************* Generation of random NxN complex matrix with given condition number C and norm2(A)=1 INPUT PARAMETERS: N - matrix size C - condition number (in 2-norm) OUTPUT PARAMETERS: A - random matrix with norm2(A)=1 and cond(A)=C -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixrndcond(ae_int_t n, double c, complex_2d_array& a);
cmatrixrndorthogonal function/************************************************************************* Generation of a random Haar distributed orthogonal complex matrix INPUT PARAMETERS: N - matrix size, N>=1 OUTPUT PARAMETERS: A - orthogonal NxN matrix, array[0..N-1,0..N-1] NOTE: this function uses algorithm described in Stewart, G. W. (1980), "The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators". Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it: * takes an NxN one * takes uniformly distributed unit vector of dimension N+1. * constructs a Householder reflection from the vector, then applies it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner). -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixrndorthogonal(ae_int_t n, complex_2d_array& a);
cmatrixrndorthogonalfromtheleft function/************************************************************************* Multiplication of MxN complex matrix by MxM random Haar distributed complex orthogonal matrix INPUT PARAMETERS: A - matrix, array[0..M-1, 0..N-1] M, N- matrix size OUTPUT PARAMETERS: A - Q*A, where Q is random MxM orthogonal matrix -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixrndorthogonalfromtheleft( complex_2d_array& a, ae_int_t m, ae_int_t n);
cmatrixrndorthogonalfromtheright function/************************************************************************* Multiplication of MxN complex matrix by NxN random Haar distributed complex orthogonal matrix INPUT PARAMETERS: A - matrix, array[0..M-1, 0..N-1] M, N- matrix size OUTPUT PARAMETERS: A - A*Q, where Q is random NxN orthogonal matrix -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::cmatrixrndorthogonalfromtheright( complex_2d_array& a, ae_int_t m, ae_int_t n);
hmatrixrndcond function/************************************************************************* Generation of random NxN Hermitian matrix with given condition number and norm2(A)=1 INPUT PARAMETERS: N - matrix size C - condition number (in 2-norm) OUTPUT PARAMETERS: A - random matrix with norm2(A)=1 and cond(A)=C -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::hmatrixrndcond(ae_int_t n, double c, complex_2d_array& a);
hmatrixrndmultiply function/************************************************************************* Hermitian multiplication of NxN matrix by random Haar distributed complex orthogonal matrix INPUT PARAMETERS: A - matrix, array[0..N-1, 0..N-1] N - matrix size OUTPUT PARAMETERS: A - Q^H*A*Q, where Q is random NxN orthogonal matrix -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::hmatrixrndmultiply(complex_2d_array& a, ae_int_t n);
hpdmatrixrndcond function/************************************************************************* Generation of random NxN Hermitian positive definite matrix with given condition number and norm2(A)=1 INPUT PARAMETERS: N - matrix size C - condition number (in 2-norm) OUTPUT PARAMETERS: A - random HPD matrix with norm2(A)=1 and cond(A)=C -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixrndcond(ae_int_t n, double c, complex_2d_array& a);
rmatrixrndcond function/************************************************************************* Generation of random NxN matrix with given condition number and norm2(A)=1 INPUT PARAMETERS: N - matrix size C - condition number (in 2-norm) OUTPUT PARAMETERS: A - random matrix with norm2(A)=1 and cond(A)=C -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixrndcond(ae_int_t n, double c, real_2d_array& a);
rmatrixrndorthogonal function/************************************************************************* Generation of a random uniformly distributed (Haar) orthogonal matrix INPUT PARAMETERS: N - matrix size, N>=1 OUTPUT PARAMETERS: A - orthogonal NxN matrix, array[0..N-1,0..N-1] NOTE: this function uses algorithm described in Stewart, G. W. (1980), "The Efficient Generation of Random Orthogonal Matrices with an Application to Condition Estimators". Speaking short, to generate an (N+1)x(N+1) orthogonal matrix, it: * takes an NxN one * takes uniformly distributed unit vector of dimension N+1. * constructs a Householder reflection from the vector, then applies it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner). -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixrndorthogonal(ae_int_t n, real_2d_array& a);
rmatrixrndorthogonalfromtheleft function/************************************************************************* Multiplication of MxN matrix by MxM random Haar distributed orthogonal matrix INPUT PARAMETERS: A - matrix, array[0..M-1, 0..N-1] M, N- matrix size OUTPUT PARAMETERS: A - Q*A, where Q is random MxM orthogonal matrix -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixrndorthogonalfromtheleft( real_2d_array& a, ae_int_t m, ae_int_t n);
rmatrixrndorthogonalfromtheright function/************************************************************************* Multiplication of MxN matrix by NxN random Haar distributed orthogonal matrix INPUT PARAMETERS: A - matrix, array[0..M-1, 0..N-1] M, N- matrix size OUTPUT PARAMETERS: A - A*Q, where Q is random NxN orthogonal matrix -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::rmatrixrndorthogonalfromtheright( real_2d_array& a, ae_int_t m, ae_int_t n);
smatrixrndcond function/************************************************************************* Generation of random NxN symmetric matrix with given condition number and norm2(A)=1 INPUT PARAMETERS: N - matrix size C - condition number (in 2-norm) OUTPUT PARAMETERS: A - random matrix with norm2(A)=1 and cond(A)=C -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::smatrixrndcond(ae_int_t n, double c, real_2d_array& a);
smatrixrndmultiply function/************************************************************************* Symmetric multiplication of NxN matrix by random Haar distributed orthogonal matrix INPUT PARAMETERS: A - matrix, array[0..N-1, 0..N-1] N - matrix size OUTPUT PARAMETERS: A - Q'*A*Q, where Q is random NxN orthogonal matrix -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::smatrixrndmultiply(real_2d_array& a, ae_int_t n);
spdmatrixrndcond function/************************************************************************* Generation of random NxN symmetric positive definite matrix with given condition number and norm2(A)=1 INPUT PARAMETERS: N - matrix size C - condition number (in 2-norm) OUTPUT PARAMETERS: A - random SPD matrix with norm2(A)=1 and cond(A)=C -- ALGLIB routine -- 04.12.2009 Bochkanov Sergey *************************************************************************/void alglib::spdmatrixrndcond(ae_int_t n, double c, real_2d_array& a);
matinv subpackage| matinv_d_c1 | Complex matrix inverse | |
| matinv_d_hpd1 | HPD matrix inverse | |
| matinv_d_r1 | Real matrix inverse | |
| matinv_d_spd1 | SPD matrix inverse |
matinvreport class/************************************************************************* Matrix inverse report: * R1 reciprocal of condition number in 1-norm * RInf reciprocal of condition number in inf-norm *************************************************************************/class matinvreport { double r1; double rinf; };
cmatrixinverse function/************************************************************************* Inversion of a general matrix. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that matrix inversion is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) Output parameters: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixinverse( complex_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::cmatrixinverse( complex_2d_array& a, ae_int_t n, ae_int_t& info, matinvreport& rep); void alglib::smp_cmatrixinverse( complex_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::smp_cmatrixinverse( complex_2d_array& a, ae_int_t n, ae_int_t& info, matinvreport& rep);
Examples: [1]
cmatrixluinverse function/************************************************************************* Inversion of a matrix given by its LU decomposition. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that matrix inversion is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - LU decomposition of the matrix (output of CMatrixLU subroutine). Pivots - table of permutations (the output of CMatrixLU subroutine). N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) OUTPUT PARAMETERS: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse -- ALGLIB routine -- 05.02.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixluinverse( complex_2d_array& a, integer_1d_array pivots, ae_int_t& info, matinvreport& rep); void alglib::cmatrixluinverse( complex_2d_array& a, integer_1d_array pivots, ae_int_t n, ae_int_t& info, matinvreport& rep); void alglib::smp_cmatrixluinverse( complex_2d_array& a, integer_1d_array pivots, ae_int_t& info, matinvreport& rep); void alglib::smp_cmatrixluinverse( complex_2d_array& a, integer_1d_array pivots, ae_int_t n, ae_int_t& info, matinvreport& rep);
cmatrixtrinverse function/************************************************************************* Triangular matrix inverse (complex) The subroutine inverts the following types of matrices: * upper triangular * upper triangular with unit diagonal * lower triangular * lower triangular with unit diagonal In case of an upper (lower) triangular matrix, the inverse matrix will also be upper (lower) triangular, and after the end of the algorithm, the inverse matrix replaces the source matrix. The elements below (above) the main diagonal are not changed by the algorithm. If the matrix has a unit diagonal, the inverse matrix also has a unit diagonal, and the diagonal elements are not passed to the algorithm. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that triangular inverse is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix, array[0..N-1, 0..N-1]. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) IsUpper - True, if the matrix is upper triangular. IsUnit - diagonal type (optional): * if True, matrix has unit diagonal (a[i,i] are NOT used) * if False, matrix diagonal is arbitrary * if not given, False is assumed Output parameters: Info - same as for RMatrixLUInverse Rep - same as for RMatrixLUInverse A - same as for RMatrixLUInverse. -- ALGLIB -- Copyright 05.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::cmatrixtrinverse( complex_2d_array& a, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::cmatrixtrinverse( complex_2d_array& a, ae_int_t n, bool isupper, bool isunit, ae_int_t& info, matinvreport& rep); void alglib::smp_cmatrixtrinverse( complex_2d_array& a, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::smp_cmatrixtrinverse( complex_2d_array& a, ae_int_t n, bool isupper, bool isunit, ae_int_t& info, matinvreport& rep);
hpdmatrixcholeskyinverse function/************************************************************************* Inversion of a Hermitian positive definite matrix which is given by Cholesky decomposition. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. However, Cholesky inversion is a "difficult" ! algorithm - it has lots of internal synchronization points which ! prevents efficient parallelization of algorithm. Only very large ! problems (N=thousands) can be efficiently parallelized. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - Cholesky decomposition of the matrix to be inverted: A=U'*U or A = L*L'. Output of HPDMatrixCholesky subroutine. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) IsUpper - storage type (optional): * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used/changed by function * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used/changed by function * if not given, lower half is used. Output parameters: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse -- ALGLIB routine -- 10.02.2010 Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixcholeskyinverse( complex_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::hpdmatrixcholeskyinverse( complex_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::smp_hpdmatrixcholeskyinverse( complex_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::smp_hpdmatrixcholeskyinverse( complex_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep);
hpdmatrixinverse function/************************************************************************* Inversion of a Hermitian positive definite matrix. Given an upper or lower triangle of a Hermitian positive definite matrix, the algorithm generates matrix A^-1 and saves the upper or lower triangle depending on the input. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. However, Cholesky inversion is a "difficult" ! algorithm - it has lots of internal synchronization points which ! prevents efficient parallelization of algorithm. Only very large ! problems (N=thousands) can be efficiently parallelized. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix to be inverted (upper or lower triangle). Array with elements [0..N-1,0..N-1]. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) IsUpper - storage type (optional): * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used/changed by function * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used/changed by function * if not given, both lower and upper triangles must be filled. Output parameters: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse -- ALGLIB routine -- 10.02.2010 Bochkanov Sergey *************************************************************************/void alglib::hpdmatrixinverse( complex_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::hpdmatrixinverse( complex_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::smp_hpdmatrixinverse( complex_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::smp_hpdmatrixinverse( complex_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep);
Examples: [1]
rmatrixinverse function/************************************************************************* Inversion of a general matrix. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that matrix inversion is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) Output parameters: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse Result: True, if the matrix is not singular. False, if the matrix is singular. -- ALGLIB -- Copyright 2005-2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixinverse( real_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::rmatrixinverse( real_2d_array& a, ae_int_t n, ae_int_t& info, matinvreport& rep); void alglib::smp_rmatrixinverse( real_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::smp_rmatrixinverse( real_2d_array& a, ae_int_t n, ae_int_t& info, matinvreport& rep);
Examples: [1]
rmatrixluinverse function/************************************************************************* Inversion of a matrix given by its LU decomposition. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that matrix inversion is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - LU decomposition of the matrix (output of RMatrixLU subroutine). Pivots - table of permutations (the output of RMatrixLU subroutine). N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) OUTPUT PARAMETERS: Info - return code: * -3 A is singular, or VERY close to singular. it is filled by zeros in such cases. * 1 task is solved (but matrix A may be ill-conditioned, check R1/RInf parameters for condition numbers). Rep - solver report, see below for more info A - inverse of matrix A. Array whose indexes range within [0..N-1, 0..N-1]. SOLVER REPORT Subroutine sets following fields of the Rep structure: * R1 reciprocal of condition number: 1/cond(A), 1-norm. * RInf reciprocal of condition number: 1/cond(A), inf-norm. -- ALGLIB routine -- 05.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixluinverse( real_2d_array& a, integer_1d_array pivots, ae_int_t& info, matinvreport& rep); void alglib::rmatrixluinverse( real_2d_array& a, integer_1d_array pivots, ae_int_t n, ae_int_t& info, matinvreport& rep); void alglib::smp_rmatrixluinverse( real_2d_array& a, integer_1d_array pivots, ae_int_t& info, matinvreport& rep); void alglib::smp_rmatrixluinverse( real_2d_array& a, integer_1d_array pivots, ae_int_t n, ae_int_t& info, matinvreport& rep);
rmatrixtrinverse function/************************************************************************* Triangular matrix inverse (real) The subroutine inverts the following types of matrices: * upper triangular * upper triangular with unit diagonal * lower triangular * lower triangular with unit diagonal In case of an upper (lower) triangular matrix, the inverse matrix will also be upper (lower) triangular, and after the end of the algorithm, the inverse matrix replaces the source matrix. The elements below (above) the main diagonal are not changed by the algorithm. If the matrix has a unit diagonal, the inverse matrix also has a unit diagonal, and the diagonal elements are not passed to the algorithm. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that triangular inverse is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix, array[0..N-1, 0..N-1]. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) IsUpper - True, if the matrix is upper triangular. IsUnit - diagonal type (optional): * if True, matrix has unit diagonal (a[i,i] are NOT used) * if False, matrix diagonal is arbitrary * if not given, False is assumed Output parameters: Info - same as for RMatrixLUInverse Rep - same as for RMatrixLUInverse A - same as for RMatrixLUInverse. -- ALGLIB -- Copyright 05.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::rmatrixtrinverse( real_2d_array& a, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::rmatrixtrinverse( real_2d_array& a, ae_int_t n, bool isupper, bool isunit, ae_int_t& info, matinvreport& rep); void alglib::smp_rmatrixtrinverse( real_2d_array& a, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::smp_rmatrixtrinverse( real_2d_array& a, ae_int_t n, bool isupper, bool isunit, ae_int_t& info, matinvreport& rep);
spdmatrixcholeskyinverse function/************************************************************************* Inversion of a symmetric positive definite matrix which is given by Cholesky decomposition. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. However, Cholesky inversion is a "difficult" ! algorithm - it has lots of internal synchronization points which ! prevents efficient parallelization of algorithm. Only very large ! problems (N=thousands) can be efficiently parallelized. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - Cholesky decomposition of the matrix to be inverted: A=U'*U or A = L*L'. Output of SPDMatrixCholesky subroutine. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) IsUpper - storage type (optional): * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used/changed by function * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used/changed by function * if not given, lower half is used. Output parameters: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse -- ALGLIB routine -- 10.02.2010 Bochkanov Sergey *************************************************************************/void alglib::spdmatrixcholeskyinverse( real_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::spdmatrixcholeskyinverse( real_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::smp_spdmatrixcholeskyinverse( real_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::smp_spdmatrixcholeskyinverse( real_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep);
spdmatrixinverse function/************************************************************************* Inversion of a symmetric positive definite matrix. Given an upper or lower triangle of a symmetric positive definite matrix, the algorithm generates matrix A^-1 and saves the upper or lower triangle depending on the input. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. However, Cholesky inversion is a "difficult" ! algorithm - it has lots of internal synchronization points which ! prevents efficient parallelization of algorithm. Only very large ! problems (N=thousands) can be efficiently parallelized. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix to be inverted (upper or lower triangle). Array with elements [0..N-1,0..N-1]. N - size of matrix A (optional) : * if given, only principal NxN submatrix is processed and overwritten. other elements are unchanged. * if not given, size is automatically determined from matrix size (A must be square matrix) IsUpper - storage type (optional): * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used/changed by function * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used/changed by function * if not given, both lower and upper triangles must be filled. Output parameters: Info - return code, same as in RMatrixLUInverse Rep - solver report, same as in RMatrixLUInverse A - inverse of matrix A, same as in RMatrixLUInverse -- ALGLIB routine -- 10.02.2010 Bochkanov Sergey *************************************************************************/void alglib::spdmatrixinverse( real_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::spdmatrixinverse( real_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep); void alglib::smp_spdmatrixinverse( real_2d_array& a, ae_int_t& info, matinvreport& rep); void alglib::smp_spdmatrixinverse( real_2d_array& a, ae_int_t n, bool isupper, ae_int_t& info, matinvreport& rep);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
complex_2d_array a = "[[1i,-1],[1i,1]]";
ae_int_t info;
matinvreport rep;
cmatrixinverse(a, info, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[-0.5i,-0.5i],[-0.5,0.5]]
printf("%.4f\n", double(rep.r1)); // EXPECTED: 0.5
printf("%.4f\n", double(rep.rinf)); // EXPECTED: 0.5
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
complex_2d_array a = "[[2,1],[1,2]]";
ae_int_t info;
matinvreport rep;
hpdmatrixinverse(a, info, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.666666,-0.333333],[-0.333333,0.666666]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array a = "[[1,-1],[1,1]]";
ae_int_t info;
matinvreport rep;
rmatrixinverse(a, info, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.5,0.5],[-0.5,0.5]]
printf("%.4f\n", double(rep.r1)); // EXPECTED: 0.5
printf("%.4f\n", double(rep.rinf)); // EXPECTED: 0.5
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array a = "[[2,1],[1,2]]";
ae_int_t info;
matinvreport rep;
spdmatrixinverse(a, info, rep);
printf("%d\n", int(info)); // EXPECTED: 1
printf("%s\n", a.tostring(4).c_str()); // EXPECTED: [[0.666666,-0.333333],[-0.333333,0.666666]]
return 0;
}
mcpd subpackage| mcpd_simple1 | Simple unconstrained MCPD model (no entry/exit states) | |
| mcpd_simple2 | Simple MCPD model (no entry/exit states) with equality constraints |
mcpdreport class/************************************************************************* This structure is a MCPD training report: InnerIterationsCount - number of inner iterations of the underlying optimization algorithm OuterIterationsCount - number of outer iterations of the underlying optimization algorithm NFEV - number of merit function evaluations TerminationType - termination type (same as for MinBLEIC optimizer, positive values denote success, negative ones - failure) -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/class mcpdreport { ae_int_t inneriterationscount; ae_int_t outeriterationscount; ae_int_t nfev; ae_int_t terminationtype; };
mcpdstate class/************************************************************************* This structure is a MCPD (Markov Chains for Population Data) solver. You should use ALGLIB functions in order to work with this object. -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/class mcpdstate { };
mcpdaddbc function/************************************************************************* This function is used to add bound constraints on the elements of the transition matrix P. MCPD solver has four types of constraints which can be placed on P: * user-specified equality constraints (optional) * user-specified bound constraints (optional) * user-specified general linear constraints (optional) * basic constraints (always present): * non-negativity: P[i,j]>=0 * consistency: every column of P sums to 1.0 Final constraints which are passed to the underlying optimizer are calculated as intersection of all present constraints. For example, you may specify boundary constraint on P[0,0] and equality one: 0.1<=P[0,0]<=0.9 P[0,0]=0.5 Such combination of constraints will be silently reduced to their intersection, which is P[0,0]=0.5. This function can be used to ADD bound constraint for one element of P without changing constraints for other elements. You can also use MCPDSetBC() function which allows to place bound constraints on arbitrary subset of elements of P. Set of constraints is specified by BndL/BndU matrices, which may contain arbitrary combination of finite numbers or infinities (like -INF<x<=0.5 or 0.1<=x<+INF). These functions (MCPDSetBC and MCPDAddBC) interact as follows: * there is internal matrix of bound constraints which is stored in the MCPD solver * MCPDSetBC() replaces this matrix by another one (SET) * MCPDAddBC() modifies one element of this matrix and leaves other ones unchanged (ADD) * thus MCPDAddBC() call preserves all modifications done by previous calls, while MCPDSetBC() completely discards all changes done to the equality constraints. INPUT PARAMETERS: S - solver I - row index of element being constrained J - column index of element being constrained BndL - lower bound BndU - upper bound -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdaddbc( mcpdstate s, ae_int_t i, ae_int_t j, double bndl, double bndu);
mcpdaddec function/************************************************************************* This function is used to add equality constraints on the elements of the transition matrix P. MCPD solver has four types of constraints which can be placed on P: * user-specified equality constraints (optional) * user-specified bound constraints (optional) * user-specified general linear constraints (optional) * basic constraints (always present): * non-negativity: P[i,j]>=0 * consistency: every column of P sums to 1.0 Final constraints which are passed to the underlying optimizer are calculated as intersection of all present constraints. For example, you may specify boundary constraint on P[0,0] and equality one: 0.1<=P[0,0]<=0.9 P[0,0]=0.5 Such combination of constraints will be silently reduced to their intersection, which is P[0,0]=0.5. This function can be used to ADD equality constraint for one element of P without changing constraints for other elements. You can also use MCPDSetEC() function which allows you to specify arbitrary set of equality constraints in one call. These functions (MCPDSetEC and MCPDAddEC) interact as follows: * there is internal matrix of equality constraints which is stored in the MCPD solver * MCPDSetEC() replaces this matrix by another one (SET) * MCPDAddEC() modifies one element of this matrix and leaves other ones unchanged (ADD) * thus MCPDAddEC() call preserves all modifications done by previous calls, while MCPDSetEC() completely discards all changes done to the equality constraints. INPUT PARAMETERS: S - solver I - row index of element being constrained J - column index of element being constrained C - value (constraint for P[I,J]). Can be either NAN (no constraint) or finite value from [0,1]. NOTES: 1. infinite values of C will lead to exception being thrown. Values less than 0.0 or greater than 1.0 will lead to error code being returned after call to MCPDSolve(). -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdaddec(mcpdstate s, ae_int_t i, ae_int_t j, double c);
Examples: [1]
mcpdaddtrack function/************************************************************************* This function is used to add a track - sequence of system states at the different moments of its evolution. You may add one or several tracks to the MCPD solver. In case you have several tracks, they won't overwrite each other. For example, if you pass two tracks, A1-A2-A3 (system at t=A+1, t=A+2 and t=A+3) and B1-B2-B3, then solver will try to model transitions from t=A+1 to t=A+2, t=A+2 to t=A+3, t=B+1 to t=B+2, t=B+2 to t=B+3. But it WONT mix these two tracks - i.e. it wont try to model transition from t=A+3 to t=B+1. INPUT PARAMETERS: S - solver XY - track, array[K,N]: * I-th row is a state at t=I * elements of XY must be non-negative (exception will be thrown on negative elements) K - number of points in a track * if given, only leading K rows of XY are used * if not given, automatically determined from size of XY NOTES: 1. Track may contain either proportional or population data: * with proportional data all rows of XY must sum to 1.0, i.e. we have proportions instead of absolute population values * with population data rows of XY contain population counts and generally do not sum to 1.0 (although they still must be non-negative) -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdaddtrack(mcpdstate s, real_2d_array xy); void alglib::mcpdaddtrack(mcpdstate s, real_2d_array xy, ae_int_t k);
mcpdcreate function/************************************************************************* DESCRIPTION: This function creates MCPD (Markov Chains for Population Data) solver. This solver can be used to find transition matrix P for N-dimensional prediction problem where transition from X[i] to X[i+1] is modelled as X[i+1] = P*X[i] where X[i] and X[i+1] are N-dimensional population vectors (components of each X are non-negative), and P is a N*N transition matrix (elements of P are non-negative, each column sums to 1.0). Such models arise when when: * there is some population of individuals * individuals can have different states * individuals can transit from one state to another * population size is constant, i.e. there is no new individuals and no one leaves population * you want to model transitions of individuals from one state into another USAGE: Here we give very brief outline of the MCPD. We strongly recommend you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide on data analysis which is available at http://www.alglib.net/dataanalysis/ 1. User initializes algorithm state with MCPDCreate() call 2. User adds one or more tracks - sequences of states which describe evolution of a system being modelled from different starting conditions 3. User may add optional boundary, equality and/or linear constraints on the coefficients of P by calling one of the following functions: * MCPDSetEC() to set equality constraints * MCPDSetBC() to set bound constraints * MCPDSetLC() to set linear constraints 4. Optionally, user may set custom weights for prediction errors (by default, algorithm assigns non-equal, automatically chosen weights for errors in the prediction of different components of X). It can be done with a call of MCPDSetPredictionWeights() function. 5. User calls MCPDSolve() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 6. User calls MCPDResults() to get solution INPUT PARAMETERS: N - problem dimension, N>=1 OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdcreate(ae_int_t n, mcpdstate& s);
mcpdcreateentry function/************************************************************************* DESCRIPTION: This function is a specialized version of MCPDCreate() function, and we recommend you to read comments for this function for general information about MCPD solver. This function creates MCPD (Markov Chains for Population Data) solver for "Entry-state" model, i.e. model where transition from X[i] to X[i+1] is modelled as X[i+1] = P*X[i] where X[i] and X[i+1] are N-dimensional state vectors P is a N*N transition matrix and one selected component of X[] is called "entry" state and is treated in a special way: system state always transits from "entry" state to some another state system state can not transit from any state into "entry" state Such conditions basically mean that row of P which corresponds to "entry" state is zero. Such models arise when: * there is some population of individuals * individuals can have different states * individuals can transit from one state to another * population size is NOT constant - at every moment of time there is some (unpredictable) amount of "new" individuals, which can transit into one of the states at the next turn, but still no one leaves population * you want to model transitions of individuals from one state into another * but you do NOT want to predict amount of "new" individuals because it does not depends on individuals already present (hence system can not transit INTO entry state - it can only transit FROM it). This model is discussed in more details in the ALGLIB User Guide (see http://www.alglib.net/dataanalysis/ for more data). INPUT PARAMETERS: N - problem dimension, N>=2 EntryState- index of entry state, in 0..N-1 OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdcreateentry( ae_int_t n, ae_int_t entrystate, mcpdstate& s);
mcpdcreateentryexit function/************************************************************************* DESCRIPTION: This function is a specialized version of MCPDCreate() function, and we recommend you to read comments for this function for general information about MCPD solver. This function creates MCPD (Markov Chains for Population Data) solver for "Entry-Exit-states" model, i.e. model where transition from X[i] to X[i+1] is modelled as X[i+1] = P*X[i] where X[i] and X[i+1] are N-dimensional state vectors P is a N*N transition matrix one selected component of X[] is called "entry" state and is treated in a special way: system state always transits from "entry" state to some another state system state can not transit from any state into "entry" state and another one component of X[] is called "exit" state and is treated in a special way too: system state can transit from any state into "exit" state system state can not transit from "exit" state into any other state transition operator discards "exit" state (makes it zero at each turn) Such conditions basically mean that: row of P which corresponds to "entry" state is zero column of P which corresponds to "exit" state is zero Multiplication by such P may decrease sum of vector components. Such models arise when: * there is some population of individuals * individuals can have different states * individuals can transit from one state to another * population size is NOT constant * at every moment of time there is some (unpredictable) amount of "new" individuals, which can transit into one of the states at the next turn * some individuals can move (predictably) into "exit" state and leave population at the next turn * you want to model transitions of individuals from one state into another, including transitions from the "entry" state and into the "exit" state. * but you do NOT want to predict amount of "new" individuals because it does not depends on individuals already present (hence system can not transit INTO entry state - it can only transit FROM it). This model is discussed in more details in the ALGLIB User Guide (see http://www.alglib.net/dataanalysis/ for more data). INPUT PARAMETERS: N - problem dimension, N>=2 EntryState- index of entry state, in 0..N-1 ExitState- index of exit state, in 0..N-1 OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdcreateentryexit( ae_int_t n, ae_int_t entrystate, ae_int_t exitstate, mcpdstate& s);
mcpdcreateexit function/************************************************************************* DESCRIPTION: This function is a specialized version of MCPDCreate() function, and we recommend you to read comments for this function for general information about MCPD solver. This function creates MCPD (Markov Chains for Population Data) solver for "Exit-state" model, i.e. model where transition from X[i] to X[i+1] is modelled as X[i+1] = P*X[i] where X[i] and X[i+1] are N-dimensional state vectors P is a N*N transition matrix and one selected component of X[] is called "exit" state and is treated in a special way: system state can transit from any state into "exit" state system state can not transit from "exit" state into any other state transition operator discards "exit" state (makes it zero at each turn) Such conditions basically mean that column of P which corresponds to "exit" state is zero. Multiplication by such P may decrease sum of vector components. Such models arise when: * there is some population of individuals * individuals can have different states * individuals can transit from one state to another * population size is NOT constant - individuals can move into "exit" state and leave population at the next turn, but there are no new individuals * amount of individuals which leave population can be predicted * you want to model transitions of individuals from one state into another (including transitions into the "exit" state) This model is discussed in more details in the ALGLIB User Guide (see http://www.alglib.net/dataanalysis/ for more data). INPUT PARAMETERS: N - problem dimension, N>=2 ExitState- index of exit state, in 0..N-1 OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdcreateexit(ae_int_t n, ae_int_t exitstate, mcpdstate& s);
mcpdresults function/************************************************************************* MCPD results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: P - array[N,N], transition matrix Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one. Speaking short, positive values denote success, negative ones are failures. More information about fields of this structure can be found in the comments on MCPDReport datatype. -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdresults(mcpdstate s, real_2d_array& p, mcpdreport& rep);
mcpdsetbc function/************************************************************************* This function is used to add bound constraints on the elements of the transition matrix P. MCPD solver has four types of constraints which can be placed on P: * user-specified equality constraints (optional) * user-specified bound constraints (optional) * user-specified general linear constraints (optional) * basic constraints (always present): * non-negativity: P[i,j]>=0 * consistency: every column of P sums to 1.0 Final constraints which are passed to the underlying optimizer are calculated as intersection of all present constraints. For example, you may specify boundary constraint on P[0,0] and equality one: 0.1<=P[0,0]<=0.9 P[0,0]=0.5 Such combination of constraints will be silently reduced to their intersection, which is P[0,0]=0.5. This function can be used to place bound constraints on arbitrary subset of elements of P. Set of constraints is specified by BndL/BndU matrices, which may contain arbitrary combination of finite numbers or infinities (like -INF<x<=0.5 or 0.1<=x<+INF). You can also use MCPDAddBC() function which allows to ADD bound constraint for one element of P without changing constraints for other elements. These functions (MCPDSetBC and MCPDAddBC) interact as follows: * there is internal matrix of bound constraints which is stored in the MCPD solver * MCPDSetBC() replaces this matrix by another one (SET) * MCPDAddBC() modifies one element of this matrix and leaves other ones unchanged (ADD) * thus MCPDAddBC() call preserves all modifications done by previous calls, while MCPDSetBC() completely discards all changes done to the equality constraints. INPUT PARAMETERS: S - solver BndL - lower bounds constraints, array[N,N]. Elements of BndL can be finite numbers or -INF. BndU - upper bounds constraints, array[N,N]. Elements of BndU can be finite numbers or +INF. -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsetbc( mcpdstate s, real_2d_array bndl, real_2d_array bndu);
mcpdsetec function/************************************************************************* This function is used to add equality constraints on the elements of the transition matrix P. MCPD solver has four types of constraints which can be placed on P: * user-specified equality constraints (optional) * user-specified bound constraints (optional) * user-specified general linear constraints (optional) * basic constraints (always present): * non-negativity: P[i,j]>=0 * consistency: every column of P sums to 1.0 Final constraints which are passed to the underlying optimizer are calculated as intersection of all present constraints. For example, you may specify boundary constraint on P[0,0] and equality one: 0.1<=P[0,0]<=0.9 P[0,0]=0.5 Such combination of constraints will be silently reduced to their intersection, which is P[0,0]=0.5. This function can be used to place equality constraints on arbitrary subset of elements of P. Set of constraints is specified by EC, which may contain either NAN's or finite numbers from [0,1]. NAN denotes absence of constraint, finite number denotes equality constraint on specific element of P. You can also use MCPDAddEC() function which allows to ADD equality constraint for one element of P without changing constraints for other elements. These functions (MCPDSetEC and MCPDAddEC) interact as follows: * there is internal matrix of equality constraints which is stored in the MCPD solver * MCPDSetEC() replaces this matrix by another one (SET) * MCPDAddEC() modifies one element of this matrix and leaves other ones unchanged (ADD) * thus MCPDAddEC() call preserves all modifications done by previous calls, while MCPDSetEC() completely discards all changes done to the equality constraints. INPUT PARAMETERS: S - solver EC - equality constraints, array[N,N]. Elements of EC can be either NAN's or finite numbers from [0,1]. NAN denotes absence of constraints, while finite value denotes equality constraint on the corresponding element of P. NOTES: 1. infinite values of EC will lead to exception being thrown. Values less than 0.0 or greater than 1.0 will lead to error code being returned after call to MCPDSolve(). -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsetec(mcpdstate s, real_2d_array ec);
mcpdsetlc function/************************************************************************* This function is used to set linear equality/inequality constraints on the elements of the transition matrix P. This function can be used to set one or several general linear constraints on the elements of P. Two types of constraints are supported: * equality constraints * inequality constraints (both less-or-equal and greater-or-equal) Coefficients of constraints are specified by matrix C (one of the parameters). One row of C corresponds to one constraint. Because transition matrix P has N*N elements, we need N*N columns to store all coefficients (they are stored row by row), and one more column to store right part - hence C has N*N+1 columns. Constraint kind is stored in the CT array. Thus, I-th linear constraint is P[0,0]*C[I,0] + P[0,1]*C[I,1] + .. + P[0,N-1]*C[I,N-1] + + P[1,0]*C[I,N] + P[1,1]*C[I,N+1] + ... + + P[N-1,N-1]*C[I,N*N-1] ?=? C[I,N*N] where ?=? can be either "=" (CT[i]=0), "<=" (CT[i]<0) or ">=" (CT[i]>0). Your constraint may involve only some subset of P (less than N*N elements). For example it can be something like P[0,0] + P[0,1] = 0.5 In this case you still should pass matrix with N*N+1 columns, but all its elements (except for C[0,0], C[0,1] and C[0,N*N-1]) will be zero. INPUT PARAMETERS: S - solver C - array[K,N*N+1] - coefficients of constraints (see above for complete description) CT - array[K] - constraint types (see above for complete description) K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsetlc(mcpdstate s, real_2d_array c, integer_1d_array ct); void alglib::mcpdsetlc( mcpdstate s, real_2d_array c, integer_1d_array ct, ae_int_t k);
mcpdsetpredictionweights function/************************************************************************* This function is used to change prediction weights MCPD solver scales prediction errors as follows Error(P) = ||W*(y-P*x)||^2 where x is a system state at time t y is a system state at time t+1 P is a transition matrix W is a diagonal scaling matrix By default, weights are chosen in order to minimize relative prediction error instead of absolute one. For example, if one component of state is about 0.5 in magnitude and another one is about 0.05, then algorithm will make corresponding weights equal to 2.0 and 20.0. INPUT PARAMETERS: S - solver PW - array[N], weights: * must be non-negative values (exception will be thrown otherwise) * zero values will be replaced by automatically chosen values -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsetpredictionweights(mcpdstate s, real_1d_array pw);
mcpdsetprior function/************************************************************************* This function allows to set prior values used for regularization of your problem. By default, regularizing term is equal to r*||P-prior_P||^2, where r is a small non-zero value, P is transition matrix, prior_P is identity matrix, ||X||^2 is a sum of squared elements of X. This function allows you to change prior values prior_P. You can also change r with MCPDSetTikhonovRegularizer() function. INPUT PARAMETERS: S - solver PP - array[N,N], matrix of prior values: 1. elements must be real numbers from [0,1] 2. columns must sum to 1.0. First property is checked (exception is thrown otherwise), while second one is not checked/enforced. -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsetprior(mcpdstate s, real_2d_array pp);
mcpdsettikhonovregularizer function/************************************************************************* This function allows to tune amount of Tikhonov regularization being applied to your problem. By default, regularizing term is equal to r*||P-prior_P||^2, where r is a small non-zero value, P is transition matrix, prior_P is identity matrix, ||X||^2 is a sum of squared elements of X. This function allows you to change coefficient r. You can also change prior values with MCPDSetPrior() function. INPUT PARAMETERS: S - solver V - regularization coefficient, finite non-negative value. It is not recommended to specify zero value unless you are pretty sure that you want it. -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsettikhonovregularizer(mcpdstate s, double v);
mcpdsolve function/************************************************************************* This function is used to start solution of the MCPD problem. After return from this function, you can use MCPDResults() to get solution and completion code. -- ALGLIB -- Copyright 23.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::mcpdsolve(mcpdstate s);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// The very simple MCPD example
//
// We have a loan portfolio. Our loans can be in one of two states:
// * normal loans ("good" ones)
// * past due loans ("bad" ones)
//
// We assume that:
// * loans can transition from any state to any other state. In
// particular, past due loan can become "good" one at any moment
// with same (fixed) probability. Not realistic, but it is toy example :)
// * portfolio size does not change over time
//
// Thus, we have following model
// state_new = P*state_old
// where
// ( p00 p01 )
// P = ( )
// ( p10 p11 )
//
// We want to model transitions between these two states using MCPD
// approach (Markov Chains for Proportional/Population Data), i.e.
// to restore hidden transition matrix P using actual portfolio data.
// We have:
// * poportional data, i.e. proportion of loans in the normal and past
// due states (not portfolio size measured in some currency, although
// it is possible to work with population data too)
// * two tracks, i.e. two sequences which describe portfolio
// evolution from two different starting states: [1,0] (all loans
// are "good") and [0.8,0.2] (only 80% of portfolio is in the "good"
// state)
//
mcpdstate s;
mcpdreport rep;
real_2d_array p;
real_2d_array track0 = "[[1.00000,0.00000],[0.95000,0.05000],[0.92750,0.07250],[0.91738,0.08263],[0.91282,0.08718]]";
real_2d_array track1 = "[[0.80000,0.20000],[0.86000,0.14000],[0.88700,0.11300],[0.89915,0.10085]]";
mcpdcreate(2, s);
mcpdaddtrack(s, track0);
mcpdaddtrack(s, track1);
mcpdsolve(s);
mcpdresults(s, p, rep);
//
// Hidden matrix P is equal to
// ( 0.95 0.50 )
// ( )
// ( 0.05 0.50 )
// which means that "good" loans can become "bad" with 5% probability,
// while "bad" loans will return to good state with 50% probability.
//
printf("%s\n", p.tostring(2).c_str()); // EXPECTED: [[0.95,0.50],[0.05,0.50]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Simple MCPD example
//
// We have a loan portfolio. Our loans can be in one of three states:
// * normal loans
// * past due loans
// * charged off loans
//
// We assume that:
// * normal loan can stay normal or become past due (but not charged off)
// * past due loan can stay past due, become normal or charged off
// * charged off loan will stay charged off for the rest of eternity
// * portfolio size does not change over time
// Not realistic, but it is toy example :)
//
// Thus, we have following model
// state_new = P*state_old
// where
// ( p00 p01 )
// P = ( p10 p11 )
// ( p21 1 )
// i.e. four elements of P are known a priori.
//
// Although it is possible (given enough data) to In order to enforce
// this property we set equality constraints on these elements.
//
// We want to model transitions between these two states using MCPD
// approach (Markov Chains for Proportional/Population Data), i.e.
// to restore hidden transition matrix P using actual portfolio data.
// We have:
// * poportional data, i.e. proportion of loans in the current and past
// due states (not portfolio size measured in some currency, although
// it is possible to work with population data too)
// * two tracks, i.e. two sequences which describe portfolio
// evolution from two different starting states: [1,0,0] (all loans
// are "good") and [0.8,0.2,0.0] (only 80% of portfolio is in the "good"
// state)
//
mcpdstate s;
mcpdreport rep;
real_2d_array p;
real_2d_array track0 = "[[1.000000,0.000000,0.000000],[0.950000,0.050000,0.000000],[0.927500,0.060000,0.012500],[0.911125,0.061375,0.027500],[0.896256,0.060900,0.042844]]";
real_2d_array track1 = "[[0.800000,0.200000,0.000000],[0.860000,0.090000,0.050000],[0.862000,0.065500,0.072500],[0.851650,0.059475,0.088875],[0.838805,0.057451,0.103744]]";
mcpdcreate(3, s);
mcpdaddtrack(s, track0);
mcpdaddtrack(s, track1);
mcpdaddec(s, 0, 2, 0.0);
mcpdaddec(s, 1, 2, 0.0);
mcpdaddec(s, 2, 2, 1.0);
mcpdaddec(s, 2, 0, 0.0);
mcpdsolve(s);
mcpdresults(s, p, rep);
//
// Hidden matrix P is equal to
// ( 0.95 0.50 )
// ( 0.05 0.25 )
// ( 0.25 1.00 )
// which means that "good" loans can become past due with 5% probability,
// while past due loans will become charged off with 25% probability or
// return back to normal state with 50% probability.
//
printf("%s\n", p.tostring(2).c_str()); // EXPECTED: [[0.95,0.50,0.00],[0.05,0.25,0.00],[0.00,0.25,1.00]]
return 0;
}
minbc subpackage| minbc_d_1 | Nonlinear optimization with box constraints | |
| minbc_numdiff | Nonlinear optimization with bound constraints and numerical differentiation |
minbcreport class/************************************************************************* This structure stores optimization report: * IterationsCount number of iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. -7 gradient verification failed. See MinBCSetGradientCheck() for more information. -3 inconsistent constraints. 1 relative function improvement is no more than EpsF. 2 relative step is no more than EpsX. 4 gradient norm is no more than EpsG 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. 8 terminated by user who called minbcrequesttermination(). X contains point which was "current accepted" when termination request was submitted. ADDITIONAL FIELDS There are additional fields which can be used for debugging: * DebugEqErr error in the equality constraints (2-norm) * DebugFS f, calculated at projection of initial point to the feasible set * DebugFF f, calculated at the final point * DebugDX |X_start-X_final| *************************************************************************/class minbcreport { ae_int_t iterationscount; ae_int_t nfev; ae_int_t varidx; ae_int_t terminationtype; };
minbcstate class/************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinBC subpackage to work with this object *************************************************************************/class minbcstate { };
minbccreate function/************************************************************************* BOX CONSTRAINED OPTIMIZATION WITH FAST ACTIVATION OF MULTIPLE BOX CONSTRAINTS DESCRIPTION: The subroutine minimizes function F(x) of N arguments subject to box constraints (with some of box constraints actually being equality ones). This optimizer uses algorithm similar to that of MinBLEIC (optimizer with general linear constraints), but presence of box-only constraints allows us to use faster constraint activation strategies. On large-scale problems, with multiple constraints active at the solution, this optimizer can be several times faster than BLEIC. REQUIREMENTS: * user must provide function value and gradient * starting point X0 must be feasible or not too far away from the feasible set * grad(f) must be Lipschitz continuous on a level set: L = { x : f(x)<=f(x0) } * function must be defined everywhere on the feasible set F USAGE: Constrained optimization if far more complex than the unconstrained one. Here we give very brief outline of the BC optimizer. We strongly recommend you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide on optimization, which is available at http://www.alglib.net/optimization/ 1. User initializes algorithm state with MinBCCreate() call 2. USer adds box constraints by calling MinBCSetBC() function. 3. User sets stopping conditions with MinBCSetCond(). 4. User calls MinBCOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 5. User calls MinBCResults() to get solution 6. Optionally user may call MinBCRestartFrom() to solve another problem with same N but another starting point. MinBCRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size ofX X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbccreate(real_1d_array x, minbcstate& state); void alglib::minbccreate(ae_int_t n, real_1d_array x, minbcstate& state);
Examples: [1]
minbccreatef function/************************************************************************* The subroutine is finite difference variant of MinBCCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinBCCreate() in order to get more information about creation of BC optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinBCSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. CG needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/void alglib::minbccreatef( real_1d_array x, double diffstep, minbcstate& state); void alglib::minbccreatef( ae_int_t n, real_1d_array x, double diffstep, minbcstate& state);
Examples: [1]
minbcoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinBCCreate() for analytical gradient or MinBCCreateF() for numerical differentiation) you should choose appropriate variant of MinBCOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinBCOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinBCOptimize() and specific function used to create optimizer. | USER PASSED TO MinBCOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinBCCreateF() | works FAILS MinBCCreate() | FAILS works Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinBCOptimize() version. Attemps to use such combination (for example, to create optimizer with MinBCCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void minbcoptimize(minbcstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minbcoptimize(minbcstate &state, void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minbcrequesttermination function/************************************************************************* This subroutine submits request for termination of running optimizer. It should be called from user-supplied callback when user decides that it is time to "smoothly" terminate optimization process. As result, optimizer stops at point which was "current accepted" when termination request was submitted and returns error code 8 (successful termination). INPUT PARAMETERS: State - optimizer structure NOTE: after request for termination optimizer may perform several additional calls to user-supplied callbacks. It does NOT guarantee to stop immediately - it just guarantees that these additional calls will be discarded later. NOTE: calling this function on optimizer which is NOT running will have no effect. NOTE: multiple calls to this function are possible. First call is counted, subsequent calls are silently ignored. -- ALGLIB -- Copyright 08.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::minbcrequesttermination(minbcstate state);
minbcrestartfrom function/************************************************************************* This subroutine restarts algorithm from new point. All optimization parameters (including constraints) are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure previously allocated with MinBCCreate call. X - new starting point. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcrestartfrom(minbcstate state, real_1d_array x);
minbcresults function/************************************************************************* BC results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one: * -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. * -7 gradient verification failed. See MinBCSetGradientCheck() for more information. * -3 inconsistent constraints. * 1 relative function improvement is no more than EpsF. * 2 scaled step is no more than EpsX. * 4 scaled gradient norm is no more than EpsG. * 5 MaxIts steps was taken * 8 terminated by user who called minbcrequesttermination(). X contains point which was "current accepted" when termination request was submitted. More information about fields of this structure can be found in the comments on MinBCReport datatype. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcresults( minbcstate state, real_1d_array& x, minbcreport& rep);
minbcresultsbuf function/************************************************************************* BC results Buffered implementation of MinBCResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcresultsbuf( minbcstate state, real_1d_array& x, minbcreport& rep);
minbcsetbc function/************************************************************************* This function sets boundary constraints for BC optimizer. Boundary constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinBCRestartFrom(). INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF. BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF. NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints, even when numerical differentiation is used (algorithm adjusts nodes according to boundary constraints) -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetbc( minbcstate state, real_1d_array bndl, real_1d_array bndu);
minbcsetcond function/************************************************************************* This function sets stopping conditions for the optimizer. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinBCSetScale() EpsF - >=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - step vector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinBCSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection. NOTE: when SetCond() called with non-zero MaxIts, BC solver may perform slightly more than MaxIts iterations. I.e., MaxIts sets non-strict limit on iterations count. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetcond( minbcstate state, double epsg, double epsf, double epsx, ae_int_t maxits);
minbcsetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before optimization begins * MinBCOptimize() is called * prior to actual optimization, for each component of parameters being optimized X[i] algorithm performs following steps: * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i], where X[i] is i-th component of the initial point and S[i] is a scale of i-th parameter * if needed, steps are bounded with respect to constraints on X[] * F(X) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N (parameters count) gradient evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with MinBCSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 15.06.2012 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetgradientcheck(minbcstate state, double teststep);
minbcsetprecdefault function/************************************************************************* Modification of the preconditioner: preconditioning is turned off. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetprecdefault(minbcstate state);
minbcsetprecdiag function/************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE 1: D[i] should be positive. Exception will be thrown otherwise. NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetprecdiag(minbcstate state, real_1d_array d);
minbcsetprecscale function/************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinBCSetScale() call (before or after MinBCSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetprecscale(minbcstate state);
minbcsetscale function/************************************************************************* This function sets scaling coefficients for BC optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the BC too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinBCSetPrec...() functions. There is a special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetscale(minbcstate state, real_1d_array s);
minbcsetstpmax function/************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which lead to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetstpmax(minbcstate state, double stpmax);
minbcsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinBCOptimize(). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbcsetxrep(minbcstate state, bool needxrep);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// subject to bound constraints -1<=x<=+1, -1<=y<=+1, using MinBC optimizer.
//
real_1d_array x = "[0,0]";
real_1d_array bndl = "[-1,-1]";
real_1d_array bndu = "[+1,+1]";
minbcstate state;
minbcreport rep;
//
// These variables define stopping conditions for the optimizer.
//
// We use very simple condition - |g|<=epsg
//
double epsg = 0.000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
//
// Now we are ready to actually optimize something:
// * first we create optimizer
// * we add boundary constraints
// * we tune stopping conditions
// * and, finally, optimize and obtain results...
//
minbccreate(x, state);
minbcsetbc(state, bndl, bndu);
minbcsetcond(state, epsg, epsf, epsx, maxits);
alglib::minbcoptimize(state, function1_grad);
minbcresults(state, x, rep);
//
// ...and evaluate these results
//
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// subject to bound constraints -1<=x<=+1, -1<=y<=+1, using MinBC optimizer.
//
real_1d_array x = "[0,0]";
real_1d_array bndl = "[-1,-1]";
real_1d_array bndu = "[+1,+1]";
minbcstate state;
minbcreport rep;
//
// These variables define stopping conditions for the optimizer.
//
// We use very simple condition - |g|<=epsg
//
double epsg = 0.000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
//
// This variable contains differentiation step
//
double diffstep = 1.0e-6;
//
// Now we are ready to actually optimize something:
// * first we create optimizer
// * we add boundary constraints
// * we tune stopping conditions
// * and, finally, optimize and obtain results...
//
minbccreatef(x, diffstep, state);
minbcsetbc(state, bndl, bndu);
minbcsetcond(state, epsg, epsf, epsx, maxits);
alglib::minbcoptimize(state, function1_func);
minbcresults(state, x, rep);
//
// ...and evaluate these results
//
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
return 0;
}
minbleic subpackage| minbleic_d_1 | Nonlinear optimization with bound constraints | |
| minbleic_d_2 | Nonlinear optimization with linear inequality constraints | |
| minbleic_ftrim | Nonlinear optimization by BLEIC, function with singularities | |
| minbleic_numdiff | Nonlinear optimization with bound constraints and numerical differentiation |
minbleicreport class/************************************************************************* This structure stores optimization report: * IterationsCount number of iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. -7 gradient verification failed. See MinBLEICSetGradientCheck() for more information. -3 inconsistent constraints. Feasible point is either nonexistent or too hard to find. Try to restart optimizer with better initial approximation 1 relative function improvement is no more than EpsF. 2 relative step is no more than EpsX. 4 gradient norm is no more than EpsG 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. 8 terminated by user who called minbleicrequesttermination(). X contains point which was "current accepted" when termination request was submitted. ADDITIONAL FIELDS There are additional fields which can be used for debugging: * DebugEqErr error in the equality constraints (2-norm) * DebugFS f, calculated at projection of initial point to the feasible set * DebugFF f, calculated at the final point * DebugDX |X_start-X_final| *************************************************************************/class minbleicreport { ae_int_t iterationscount; ae_int_t nfev; ae_int_t varidx; ae_int_t terminationtype; double debugeqerr; double debugfs; double debugff; double debugdx; ae_int_t debugfeasqpits; ae_int_t debugfeasgpaits; ae_int_t inneriterationscount; ae_int_t outeriterationscount; };
minbleicstate class/************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinBLEIC subpackage to work with this object *************************************************************************/class minbleicstate { };
minbleiccreate function/************************************************************************* BOUND CONSTRAINED OPTIMIZATION WITH ADDITIONAL LINEAR EQUALITY AND INEQUALITY CONSTRAINTS DESCRIPTION: The subroutine minimizes function F(x) of N arguments subject to any combination of: * bound constraints * linear inequality constraints * linear equality constraints REQUIREMENTS: * user must provide function value and gradient * starting point X0 must be feasible or not too far away from the feasible set * grad(f) must be Lipschitz continuous on a level set: L = { x : f(x)<=f(x0) } * function must be defined everywhere on the feasible set F USAGE: Constrained optimization if far more complex than the unconstrained one. Here we give very brief outline of the BLEIC optimizer. We strongly recommend you to read examples in the ALGLIB Reference Manual and to read ALGLIB User Guide on optimization, which is available at http://www.alglib.net/optimization/ 1. User initializes algorithm state with MinBLEICCreate() call 2. USer adds boundary and/or linear constraints by calling MinBLEICSetBC() and MinBLEICSetLC() functions. 3. User sets stopping conditions with MinBLEICSetCond(). 4. User calls MinBLEICOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 5. User calls MinBLEICResults() to get solution 6. Optionally user may call MinBLEICRestartFrom() to solve another problem with same N but another starting point. MinBLEICRestartFrom() allows to reuse already initialized structure. NOTE: if you have box-only constraints (no general linear constraints), then MinBC optimizer can be better option. It uses special, faster constraint activation method, which performs better on problems with multiple constraints active at the solution. On small-scale problems performance of MinBC is similar to that of MinBLEIC, but on large-scale ones (hundreds and thousands of active constraints) it can be several times faster than MinBLEIC. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size ofX X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleiccreate(real_1d_array x, minbleicstate& state); void alglib::minbleiccreate( ae_int_t n, real_1d_array x, minbleicstate& state);
minbleiccreatef function/************************************************************************* The subroutine is finite difference variant of MinBLEICCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinBLEICCreate() in order to get more information about creation of BLEIC optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinBLEICSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. CG needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/void alglib::minbleiccreatef( real_1d_array x, double diffstep, minbleicstate& state); void alglib::minbleiccreatef( ae_int_t n, real_1d_array x, double diffstep, minbleicstate& state);
Examples: [1]
minbleicoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinBLEICCreate() for analytical gradient or MinBLEICCreateF() for numerical differentiation) you should choose appropriate variant of MinBLEICOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinBLEICOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinBLEICOptimize() and specific function used to create optimizer. | USER PASSED TO MinBLEICOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinBLEICCreateF() | work FAIL MinBLEICCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinBLEICOptimize() version. Attemps to use such combination (for example, to create optimizer with MinBLEICCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void minbleicoptimize(minbleicstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minbleicoptimize(minbleicstate &state, void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minbleicrequesttermination function/************************************************************************* This subroutine submits request for termination of running optimizer. It should be called from user-supplied callback when user decides that it is time to "smoothly" terminate optimization process. As result, optimizer stops at point which was "current accepted" when termination request was submitted and returns error code 8 (successful termination). INPUT PARAMETERS: State - optimizer structure NOTE: after request for termination optimizer may perform several additional calls to user-supplied callbacks. It does NOT guarantee to stop immediately - it just guarantees that these additional calls will be discarded later. NOTE: calling this function on optimizer which is NOT running will have no effect. NOTE: multiple calls to this function are possible. First call is counted, subsequent calls are silently ignored. -- ALGLIB -- Copyright 08.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::minbleicrequesttermination(minbleicstate state);
minbleicrestartfrom function/************************************************************************* This subroutine restarts algorithm from new point. All optimization parameters (including constraints) are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure previously allocated with MinBLEICCreate call. X - new starting point. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicrestartfrom(minbleicstate state, real_1d_array x);
minbleicresults function/************************************************************************* BLEIC results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one: * -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. * -7 gradient verification failed. See MinBLEICSetGradientCheck() for more information. * -3 inconsistent constraints. Feasible point is either nonexistent or too hard to find. Try to restart optimizer with better initial approximation * 1 relative function improvement is no more than EpsF. * 2 scaled step is no more than EpsX. * 4 scaled gradient norm is no more than EpsG. * 5 MaxIts steps was taken * 8 terminated by user who called minbleicrequesttermination(). X contains point which was "current accepted" when termination request was submitted. More information about fields of this structure can be found in the comments on MinBLEICReport datatype. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicresults( minbleicstate state, real_1d_array& x, minbleicreport& rep);
minbleicresultsbuf function/************************************************************************* BLEIC results Buffered implementation of MinBLEICResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicresultsbuf( minbleicstate state, real_1d_array& x, minbleicreport& rep);
minbleicsetbc function/************************************************************************* This function sets boundary constraints for BLEIC optimizer. Boundary constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinBLEICRestartFrom(). NOTE: if you have box-only constraints (no general linear constraints), then MinBC optimizer can be better option. It uses special, faster constraint activation method, which performs better on problems with multiple constraints active at the solution. On small-scale problems performance of MinBC is similar to that of MinBLEIC, but on large-scale ones (hundreds and thousands of active constraints) it can be several times faster than MinBLEIC. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF. BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF. NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints, even when numerical differentiation is used (algorithm adjusts nodes according to boundary constraints) -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetbc( minbleicstate state, real_1d_array bndl, real_1d_array bndu);
minbleicsetcond function/************************************************************************* This function sets stopping conditions for the optimizer. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinBLEICSetScale() EpsF - >=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - step vector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinBLEICSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection. NOTE: when SetCond() called with non-zero MaxIts, BLEIC solver may perform slightly more than MaxIts iterations. I.e., MaxIts sets non-strict limit on iterations count. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetcond( minbleicstate state, double epsg, double epsf, double epsx, ae_int_t maxits);
minbleicsetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before optimization begins * MinBLEICOptimize() is called * prior to actual optimization, for each component of parameters being optimized X[i] algorithm performs following steps: * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i], where X[i] is i-th component of the initial point and S[i] is a scale of i-th parameter * if needed, steps are bounded with respect to constraints on X[] * F(X) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N (parameters count) gradient evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with MinBLEICSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 15.06.2012 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetgradientcheck( minbleicstate state, double teststep);
minbleicsetlc function/************************************************************************* This function sets linear constraints for BLEIC optimizer. Linear constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinBLEICRestartFrom(). INPUT PARAMETERS: State - structure previously allocated with MinBLEICCreate call. C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT NOTE 1: linear (non-bound) constraints are satisfied only approximately: * there always exists some minor violation (about Epsilon in magnitude) due to rounding errors * numerical differentiation, if used, may lead to function evaluations outside of the feasible area, because algorithm does NOT change numerical differentiation formula according to linear constraints. If you want constraints to be satisfied exactly, try to reformulate your problem in such manner that all constraints will become boundary ones (this kind of constraints is always satisfied exactly, both in the final solution and in all intermediate points). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetlc( minbleicstate state, real_2d_array c, integer_1d_array ct); void alglib::minbleicsetlc( minbleicstate state, real_2d_array c, integer_1d_array ct, ae_int_t k);
Examples: [1]
minbleicsetprecdefault function/************************************************************************* Modification of the preconditioner: preconditioning is turned off. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetprecdefault(minbleicstate state);
minbleicsetprecdiag function/************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE 1: D[i] should be positive. Exception will be thrown otherwise. NOTE 2: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetprecdiag(minbleicstate state, real_1d_array d);
minbleicsetprecscale function/************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinBLEICSetScale() call (before or after MinBLEICSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetprecscale(minbleicstate state);
minbleicsetscale function/************************************************************************* This function sets scaling coefficients for BLEIC optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the BLEIC too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinBLEICSetPrec...() functions. There is a special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetscale(minbleicstate state, real_1d_array s);
minbleicsetstpmax function/************************************************************************* This function sets maximum step length IMPORTANT: this feature is hard to combine with preconditioning. You can't set upper limit on step length, when you solve optimization problem with linear (non-boundary) constraints AND preconditioner turned on. When non-boundary constraints are present, you have to either a) use preconditioner, or b) use upper limit on step length. YOU CAN'T USE BOTH! In this case algorithm will terminate with appropriate error code. INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which lead to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetstpmax(minbleicstate state, double stpmax);
minbleicsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinBLEICOptimize(). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetxrep(minbleicstate state, bool needxrep);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// subject to bound constraints -1<=x<=+1, -1<=y<=+1, using BLEIC optimizer.
//
real_1d_array x = "[0,0]";
real_1d_array bndl = "[-1,-1]";
real_1d_array bndu = "[+1,+1]";
minbleicstate state;
minbleicreport rep;
//
// These variables define stopping conditions for the optimizer.
//
// We use very simple condition - |g|<=epsg
//
double epsg = 0.000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
//
// Now we are ready to actually optimize something:
// * first we create optimizer
// * we add boundary constraints
// * we tune stopping conditions
// * and, finally, optimize and obtain results...
//
minbleiccreate(x, state);
minbleicsetbc(state, bndl, bndu);
minbleicsetcond(state, epsg, epsf, epsx, maxits);
alglib::minbleicoptimize(state, function1_grad);
minbleicresults(state, x, rep);
//
// ...and evaluate these results
//
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// subject to inequality constraints:
// * x>=2 (posed as general linear constraint),
// * x+y>=6
// using BLEIC optimizer.
//
real_1d_array x = "[5,5]";
real_2d_array c = "[[1,0,2],[1,1,6]]";
integer_1d_array ct = "[1,1]";
minbleicstate state;
minbleicreport rep;
//
// These variables define stopping conditions for the optimizer.
//
// We use very simple condition - |g|<=epsg
//
double epsg = 0.000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
//
// Now we are ready to actually optimize something:
// * first we create optimizer
// * we add linear constraints
// * we tune stopping conditions
// * and, finally, optimize and obtain results...
//
minbleiccreate(x, state);
minbleicsetlc(state, c, ct);
minbleicsetcond(state, epsg, epsf, epsx, maxits);
alglib::minbleicoptimize(state, function1_grad);
minbleicresults(state, x, rep);
//
// ...and evaluate these results
//
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,4]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void s1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
//
// function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
// Note that we do NOT calculate gradient in this case.
//
if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
{
func = 1.0E+300;
return;
}
func = pow(1+x[0],-0.2) + pow(1-x[0],-0.3) + 1000*x[0];
grad[0] = -0.2*pow(1+x[0],-1.2) +0.3*pow(1-x[0],-1.3) + 1000;
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
//
// This function is undefined outside of (-1,+1) and has singularities at x=-1 and x=+1.
// Special technique called "function trimming" allows us to solve this optimization problem
// - without using boundary constraints!
//
// See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
// on this subject.
//
real_1d_array x = "[0]";
double epsg = 1.0e-6;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
minbleicstate state;
minbleicreport rep;
minbleiccreate(x, state);
minbleicsetcond(state, epsg, epsf, epsx, maxits);
alglib::minbleicoptimize(state, s1_grad);
minbleicresults(state, x, rep);
printf("%s\n", x.tostring(5).c_str()); // EXPECTED: [-0.99917305]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// subject to bound constraints -1<=x<=+1, -1<=y<=+1, using BLEIC optimizer.
//
real_1d_array x = "[0,0]";
real_1d_array bndl = "[-1,-1]";
real_1d_array bndu = "[+1,+1]";
minbleicstate state;
minbleicreport rep;
//
// These variables define stopping conditions for the optimizer.
//
// We use very simple condition - |g|<=epsg
//
double epsg = 0.000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
//
// This variable contains differentiation step
//
double diffstep = 1.0e-6;
//
// Now we are ready to actually optimize something:
// * first we create optimizer
// * we add boundary constraints
// * we tune stopping conditions
// * and, finally, optimize and obtain results...
//
minbleiccreatef(x, diffstep, state);
minbleicsetbc(state, bndl, bndu);
minbleicsetcond(state, epsg, epsf, epsx, maxits);
alglib::minbleicoptimize(state, function1_func);
minbleicresults(state, x, rep);
//
// ...and evaluate these results
//
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,1]
return 0;
}
mincg subpackage| mincg_d_1 | Nonlinear optimization by CG | |
| mincg_d_2 | Nonlinear optimization with additional settings and restarts | |
| mincg_ftrim | Nonlinear optimization by CG, function with singularities | |
| mincg_numdiff | Nonlinear optimization by CG with numerical differentiation |
mincgreport class/************************************************************************* This structure stores optimization report: * IterationsCount total number of inner iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. -7 gradient verification failed. See MinCGSetGradientCheck() for more information. 1 relative function improvement is no more than EpsF. 2 relative step is no more than EpsX. 4 gradient norm is no more than EpsG 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. 8 terminated by user who called mincgrequesttermination(). X contains point which was "current accepted" when termination request was submitted. Other fields of this structure are not documented and should not be used! *************************************************************************/class mincgreport { ae_int_t iterationscount; ae_int_t nfev; ae_int_t varidx; ae_int_t terminationtype; };
mincgstate class/************************************************************************* This object stores state of the nonlinear CG optimizer. You should use ALGLIB functions to work with this object. *************************************************************************/class mincgstate { };
mincgcreate function/************************************************************************* NONLINEAR CONJUGATE GRADIENT METHOD DESCRIPTION: The subroutine minimizes function F(x) of N arguments by using one of the nonlinear conjugate gradient methods. These CG methods are globally convergent (even on non-convex functions) as long as grad(f) is Lipschitz continuous in a some neighborhood of the L = { x : f(x)<=f(x0) }. REQUIREMENTS: Algorithm will request following information during its operation: * function value F and its gradient G (simultaneously) at given point X USAGE: 1. User initializes algorithm state with MinCGCreate() call 2. User tunes solver parameters with MinCGSetCond(), MinCGSetStpMax() and other functions 3. User calls MinCGOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 4. User calls MinCGResults() to get solution 5. Optionally, user may call MinCGRestartFrom() to solve another problem with same N but another starting point and/or another function. MinCGRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. OUTPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 25.03.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgcreate(real_1d_array x, mincgstate& state); void alglib::mincgcreate(ae_int_t n, real_1d_array x, mincgstate& state);
mincgcreatef function/************************************************************************* The subroutine is finite difference variant of MinCGCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinCGCreate() in order to get more information about creation of CG optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinCGSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. L-BFGS needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/void alglib::mincgcreatef( real_1d_array x, double diffstep, mincgstate& state); void alglib::mincgcreatef( ae_int_t n, real_1d_array x, double diffstep, mincgstate& state);
Examples: [1]
mincgoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinCGCreate() for analytical gradient or MinCGCreateF() for numerical differentiation) you should choose appropriate variant of MinCGOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinCGOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinCGOptimize() and specific function used to create optimizer. | USER PASSED TO MinCGOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinCGCreateF() | work FAIL MinCGCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinCGOptimize() version. Attemps to use such combination (for example, to create optimizer with MinCGCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/void mincgoptimize(mincgstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void mincgoptimize(mincgstate &state, void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
mincgrequesttermination function/************************************************************************* This subroutine submits request for termination of running optimizer. It should be called from user-supplied callback when user decides that it is time to "smoothly" terminate optimization process. As result, optimizer stops at point which was "current accepted" when termination request was submitted and returns error code 8 (successful termination). INPUT PARAMETERS: State - optimizer structure NOTE: after request for termination optimizer may perform several additional calls to user-supplied callbacks. It does NOT guarantee to stop immediately - it just guarantees that these additional calls will be discarded later. NOTE: calling this function on optimizer which is NOT running will have no effect. NOTE: multiple calls to this function are possible. First call is counted, subsequent calls are silently ignored. -- ALGLIB -- Copyright 08.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::mincgrequesttermination(mincgstate state);
mincgrestartfrom function/************************************************************************* This subroutine restarts CG algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used to store algorithm state. X - new starting point. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgrestartfrom(mincgstate state, real_1d_array x);
Examples: [1]
mincgresults function/************************************************************************* Conjugate gradient results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. * -7 gradient verification failed. See MinCGSetGradientCheck() for more information. * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible, we return best X found so far * 8 terminated by user * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/void alglib::mincgresults( mincgstate state, real_1d_array& x, mincgreport& rep);
mincgresultsbuf function/************************************************************************* Conjugate gradient results Buffered implementation of MinCGResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.04.2009 by Bochkanov Sergey *************************************************************************/void alglib::mincgresultsbuf( mincgstate state, real_1d_array& x, mincgreport& rep);
mincgsetcgtype function/************************************************************************* This function sets CG algorithm. INPUT PARAMETERS: State - structure which stores algorithm state CGType - algorithm type: * -1 automatic selection of the best algorithm * 0 DY (Dai and Yuan) algorithm * 1 Hybrid DY-HS algorithm -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetcgtype(mincgstate state, ae_int_t cgtype);
mincgsetcond function/************************************************************************* This function sets stopping conditions for CG optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinCGSetScale() EpsF - >=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinCGSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetcond( mincgstate state, double epsg, double epsf, double epsx, ae_int_t maxits);
mincgsetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before optimization begins * MinCGOptimize() is called * prior to actual optimization, for each component of parameters being optimized X[i] algorithm performs following steps: * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i], where X[i] is i-th component of the initial point and S[i] is a scale of i-th parameter * F(X) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N (parameters count) gradient evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with MinCGSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 31.05.2012 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetgradientcheck(mincgstate state, double teststep);
mincgsetprecdefault function/************************************************************************* Modification of the preconditioner: preconditioning is turned off. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetprecdefault(mincgstate state);
mincgsetprecdiag function/************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: D[i] should be positive. Exception will be thrown otherwise. NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetprecdiag(mincgstate state, real_1d_array d);
mincgsetprecscale function/************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinCGSetScale() call (before or after MinCGSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetprecscale(mincgstate state);
mincgsetscale function/************************************************************************* This function sets scaling coefficients for CG optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of CG optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the CG too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinCGSetPrec...() functions. There is special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetscale(mincgstate state, real_1d_array s);
mincgsetstpmax function/************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetstpmax(mincgstate state, double stpmax);
mincgsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinCGOptimize(). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsetxrep(mincgstate state, bool needxrep);
mincgsuggeststep function/************************************************************************* This function allows to suggest initial step length to the CG algorithm. Suggested step length is used as starting point for the line search. It can be useful when you have badly scaled problem, i.e. when ||grad|| (which is used as initial estimate for the first step) is many orders of magnitude different from the desired step. Line search may fail on such problems without good estimate of initial step length. Imagine, for example, problem with ||grad||=10^50 and desired step equal to 0.1 Line search function will use 10^50 as initial step, then it will decrease step length by 2 (up to 20 attempts) and will get 10^44, which is still too large. This function allows us to tell than line search should be started from some moderate step length, like 1.0, so algorithm will be able to detect desired step length in a several searches. Default behavior (when no step is suggested) is to use preconditioner, if it is available, to generate initial estimate of step length. This function influences only first iteration of algorithm. It should be called between MinCGCreate/MinCGRestartFrom() call and MinCGOptimize call. Suggested step is ignored if you have preconditioner. INPUT PARAMETERS: State - structure used to store algorithm state. Stp - initial estimate of the step length. Can be zero (no estimate). -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/void alglib::mincgsuggeststep(mincgstate state, double stp);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// with nonlinear conjugate gradient method.
//
real_1d_array x = "[0,0]";
double epsg = 0.0000000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
mincgstate state;
mincgreport rep;
mincgcreate(x, state);
mincgsetcond(state, epsg, epsf, epsx, maxits);
alglib::mincgoptimize(state, function1_grad);
mincgresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// with nonlinear conjugate gradient method.
//
// Several advanced techniques are demonstrated:
// * upper limit on step size
// * restart from new point
//
real_1d_array x = "[0,0]";
double epsg = 0.0000000001;
double epsf = 0;
double epsx = 0;
double stpmax = 0.1;
ae_int_t maxits = 0;
mincgstate state;
mincgreport rep;
// first run
mincgcreate(x, state);
mincgsetcond(state, epsg, epsf, epsx, maxits);
mincgsetstpmax(state, stpmax);
alglib::mincgoptimize(state, function1_grad);
mincgresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
// second run - algorithm is restarted with mincgrestartfrom()
x = "[10,10]";
mincgrestartfrom(state, x);
alglib::mincgoptimize(state, function1_grad);
mincgresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void s1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
//
// function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
// Note that we do NOT calculate gradient in this case.
//
if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
{
func = 1.0E+300;
return;
}
func = pow(1+x[0],-0.2) + pow(1-x[0],-0.3) + 1000*x[0];
grad[0] = -0.2*pow(1+x[0],-1.2) +0.3*pow(1-x[0],-1.3) + 1000;
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
// This function has singularities at the boundary of the [-1,+1], but technique called
// "function trimming" allows us to solve this optimization problem.
//
// See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
// on this subject.
//
real_1d_array x = "[0]";
double epsg = 1.0e-6;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
mincgstate state;
mincgreport rep;
mincgcreate(x, state);
mincgsetcond(state, epsg, epsf, epsx, maxits);
alglib::mincgoptimize(state, s1_grad);
mincgresults(state, x, rep);
printf("%s\n", x.tostring(5).c_str()); // EXPECTED: [-0.99917305]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// using numerical differentiation to calculate gradient.
//
real_1d_array x = "[0,0]";
double epsg = 0.0000000001;
double epsf = 0;
double epsx = 0;
double diffstep = 1.0e-6;
ae_int_t maxits = 0;
mincgstate state;
mincgreport rep;
mincgcreatef(x, diffstep, state);
mincgsetcond(state, epsg, epsf, epsx, maxits);
alglib::mincgoptimize(state, function1_func);
mincgresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
return 0;
}
mincomp subpackageminasareport class/************************************************************************* *************************************************************************/class minasareport { ae_int_t iterationscount; ae_int_t nfev; ae_int_t terminationtype; ae_int_t activeconstraints; };
minasastate class/************************************************************************* *************************************************************************/class minasastate { };
minasacreate function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 25.03.2010 by Bochkanov Sergey *************************************************************************/void alglib::minasacreate( real_1d_array x, real_1d_array bndl, real_1d_array bndu, minasastate& state); void alglib::minasacreate( ae_int_t n, real_1d_array x, real_1d_array bndl, real_1d_array bndu, minasastate& state);
minasaoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/void minasaoptimize(minasastate &state, void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minasarestartfrom function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/void alglib::minasarestartfrom( minasastate state, real_1d_array x, real_1d_array bndl, real_1d_array bndu);
minasaresults function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minasaresults( minasastate state, real_1d_array& x, minasareport& rep);
minasaresultsbuf function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minasaresultsbuf( minasastate state, real_1d_array& x, minasareport& rep);
minasasetalgorithm function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minasasetalgorithm(minasastate state, ae_int_t algotype);
minasasetcond function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minasasetcond( minasastate state, double epsg, double epsf, double epsx, ae_int_t maxits);
minasasetstpmax function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minasasetstpmax(minasastate state, double stpmax);
minasasetxrep function/************************************************************************* Obsolete optimization algorithm. Was replaced by MinBLEIC subpackage. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minasasetxrep(minasastate state, bool needxrep);
minbleicsetbarrierdecay function/************************************************************************* This is obsolete function which was used by previous version of the BLEIC optimizer. It does nothing in the current version of BLEIC. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetbarrierdecay(minbleicstate state, double mudecay);
minbleicsetbarrierwidth function/************************************************************************* This is obsolete function which was used by previous version of the BLEIC optimizer. It does nothing in the current version of BLEIC. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minbleicsetbarrierwidth(minbleicstate state, double mu);
minlbfgssetcholeskypreconditioner function/************************************************************************* Obsolete function, use MinLBFGSSetCholeskyPreconditioner() instead. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetcholeskypreconditioner( minlbfgsstate state, real_2d_array p, bool isupper);
minlbfgssetdefaultpreconditioner function/************************************************************************* Obsolete function, use MinLBFGSSetPrecDefault() instead. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetdefaultpreconditioner(minlbfgsstate state);
minlbfgs subpackage| minlbfgs_d_1 | Nonlinear optimization by L-BFGS | |
| minlbfgs_d_2 | Nonlinear optimization with additional settings and restarts | |
| minlbfgs_ftrim | Nonlinear optimization by LBFGS, function with singularities | |
| minlbfgs_numdiff | Nonlinear optimization by L-BFGS with numerical differentiation |
minlbfgsreport class/************************************************************************* This structure stores optimization report: * IterationsCount total number of inner iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. -7 gradient verification failed. See MinLBFGSSetGradientCheck() for more information. 1 relative function improvement is no more than EpsF. 2 relative step is no more than EpsX. 4 gradient norm is no more than EpsG 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. 8 terminated by user who called minlbfgsrequesttermination(). X contains point which was "current accepted" when termination request was submitted. Other fields of this structure are not documented and should not be used! *************************************************************************/class minlbfgsreport { ae_int_t iterationscount; ae_int_t nfev; ae_int_t varidx; ae_int_t terminationtype; };
minlbfgsstate class/************************************************************************* *************************************************************************/class minlbfgsstate { };
minlbfgscreate function/************************************************************************* LIMITED MEMORY BFGS METHOD FOR LARGE SCALE OPTIMIZATION DESCRIPTION: The subroutine minimizes function F(x) of N arguments by using a quasi- Newton method (LBFGS scheme) which is optimized to use a minimum amount of memory. The subroutine generates the approximation of an inverse Hessian matrix by using information about the last M steps of the algorithm (instead of N). It lessens a required amount of memory from a value of order N^2 to a value of order 2*N*M. REQUIREMENTS: Algorithm will request following information during its operation: * function value F and its gradient G (simultaneously) at given point X USAGE: 1. User initializes algorithm state with MinLBFGSCreate() call 2. User tunes solver parameters with MinLBFGSSetCond() MinLBFGSSetStpMax() and other functions 3. User calls MinLBFGSOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G. 4. User calls MinLBFGSResults() to get solution 5. Optionally user may call MinLBFGSRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLBFGSRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension. N>0 M - number of corrections in the BFGS scheme of Hessian approximation update. Recommended value: 3<=M<=7. The smaller value causes worse convergence, the bigger will not cause a considerably better convergence, but will cause a fall in the performance. M<=N. X - initial solution approximation, array[0..N-1]. OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLBFGSSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLBFGSSetStpMax() function to bound algorithm's steps. However, L-BFGS rarely needs such a tuning. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgscreate( ae_int_t m, real_1d_array x, minlbfgsstate& state); void alglib::minlbfgscreate( ae_int_t n, ae_int_t m, real_1d_array x, minlbfgsstate& state);
minlbfgscreatef function/************************************************************************* The subroutine is finite difference variant of MinLBFGSCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinLBFGSCreate() in order to get more information about creation of LBFGS optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of corrections in the BFGS scheme of Hessian approximation update. Recommended value: 3<=M<=7. The smaller value causes worse convergence, the bigger will not cause a considerably better convergence, but will cause a fall in the performance. M<=N. X - starting point, array[0..N-1]. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinLBFGSSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large truncation errors, while too small step will result in too large numerical errors. 1.0E-6 can be good value to start with. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. LBFGS needs exact gradient values. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 16.05.2011 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgscreatef( ae_int_t m, real_1d_array x, double diffstep, minlbfgsstate& state); void alglib::minlbfgscreatef( ae_int_t n, ae_int_t m, real_1d_array x, double diffstep, minlbfgsstate& state);
Examples: [1]
minlbfgsoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied gradient, and one which uses function value only and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object (either MinLBFGSCreate() for analytical gradient or MinLBFGSCreateF() for numerical differentiation) you should choose appropriate variant of MinLBFGSOptimize() - one which accepts function AND gradient or one which accepts function ONLY. Be careful to choose variant of MinLBFGSOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinLBFGSOptimize() and specific function used to create optimizer. | USER PASSED TO MinLBFGSOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinLBFGSCreateF() | work FAIL MinLBFGSCreate() | FAIL work Here "FAIL" denotes inappropriate combinations of optimizer creation function and MinLBFGSOptimize() version. Attemps to use such combination (for example, to create optimizer with MinLBFGSCreateF() and to pass gradient information to MinCGOptimize()) will lead to exception being thrown. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/void minlbfgsoptimize(minlbfgsstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minlbfgsoptimize(minlbfgsstate &state, void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minlbfgsrequesttermination function/************************************************************************* This subroutine submits request for termination of running optimizer. It should be called from user-supplied callback when user decides that it is time to "smoothly" terminate optimization process. As result, optimizer stops at point which was "current accepted" when termination request was submitted and returns error code 8 (successful termination). INPUT PARAMETERS: State - optimizer structure NOTE: after request for termination optimizer may perform several additional calls to user-supplied callbacks. It does NOT guarantee to stop immediately - it just guarantees that these additional calls will be discarded later. NOTE: calling this function on optimizer which is NOT running will have no effect. NOTE: multiple calls to this function are possible. First call is counted, subsequent calls are silently ignored. -- ALGLIB -- Copyright 08.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgsrequesttermination(minlbfgsstate state);
minlbfgsrestartfrom function/************************************************************************* This subroutine restarts LBFGS algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used to store algorithm state X - new starting point. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgsrestartfrom(minlbfgsstate state, real_1d_array x);
Examples: [1]
minlbfgsresults function/************************************************************************* L-BFGS algorithm results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. * -7 gradient verification failed. See MinLBFGSSetGradientCheck() for more information. * -2 rounding errors prevent further improvement. X contains best point found. * -1 incorrect parameters were specified * 1 relative function improvement is no more than EpsF. * 2 relative step is no more than EpsX. * 4 gradient norm is no more than EpsG * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * 8 terminated by user who called minlbfgsrequesttermination(). X contains point which was "current accepted" when termination request was submitted. * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgsresults( minlbfgsstate state, real_1d_array& x, minlbfgsreport& rep);
minlbfgsresultsbuf function/************************************************************************* L-BFGS algorithm results Buffered implementation of MinLBFGSResults which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.08.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgsresultsbuf( minlbfgsstate state, real_1d_array& x, minlbfgsreport& rep);
minlbfgssetcond function/************************************************************************* This function sets stopping conditions for L-BFGS optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinLBFGSSetScale() EpsF - >=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinLBFGSSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0, EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetcond( minlbfgsstate state, double epsg, double epsf, double epsx, ae_int_t maxits);
minlbfgssetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before optimization begins * MinLBFGSOptimize() is called * prior to actual optimization, for each component of parameters being optimized X[i] algorithm performs following steps: * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i], where X[i] is i-th component of the initial point and S[i] is a scale of i-th parameter * if needed, steps are bounded with respect to constraints on X[] * F(X) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N (parameters count) gradient evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with MinLBFGSSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 24.05.2012 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetgradientcheck( minlbfgsstate state, double teststep);
minlbfgssetpreccholesky function/************************************************************************* Modification of the preconditioner: Cholesky factorization of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state P - triangular preconditioner, Cholesky factorization of the approximate Hessian. array[0..N-1,0..N-1], (if larger, only leading N elements are used). IsUpper - whether upper or lower triangle of P is given (other triangle is not referenced) After call to this function preconditioner is changed to P (P is copied into the internal buffer). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: P should be nonsingular. Exception will be thrown otherwise. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetpreccholesky( minlbfgsstate state, real_2d_array p, bool isupper);
minlbfgssetprecdefault function/************************************************************************* Modification of the preconditioner: default preconditioner (simple scaling, same for all elements of X) is used. INPUT PARAMETERS: State - structure which stores algorithm state NOTE: you can change preconditioner "on the fly", during algorithm iterations. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetprecdefault(minlbfgsstate state);
minlbfgssetprecdiag function/************************************************************************* Modification of the preconditioner: diagonal of approximate Hessian is used. INPUT PARAMETERS: State - structure which stores algorithm state D - diagonal of the approximate Hessian, array[0..N-1], (if larger, only leading N elements are used). NOTE: you can change preconditioner "on the fly", during algorithm iterations. NOTE 2: D[i] should be positive. Exception will be thrown otherwise. NOTE 3: you should pass diagonal of approximate Hessian - NOT ITS INVERSE. -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetprecdiag(minlbfgsstate state, real_1d_array d);
minlbfgssetprecscale function/************************************************************************* Modification of the preconditioner: scale-based diagonal preconditioning. This preconditioning mode can be useful when you don't have approximate diagonal of Hessian, but you know that your variables are badly scaled (for example, one variable is in [1,10], and another in [1000,100000]), and most part of the ill-conditioning comes from different scales of vars. In this case simple scale-based preconditioner, with H[i] = 1/(s[i]^2), can greatly improve convergence. IMPRTANT: you should set scale of your variables with MinLBFGSSetScale() call (before or after MinLBFGSSetPrecScale() call). Without knowledge of the scale of your variables scale-based preconditioner will be just unit matrix. INPUT PARAMETERS: State - structure which stores algorithm state -- ALGLIB -- Copyright 13.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetprecscale(minlbfgsstate state);
minlbfgssetscale function/************************************************************************* This function sets scaling coefficients for LBFGS optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. In most optimizers (and in the LBFGS too) scaling is NOT a form of preconditioning. It just affects stopping conditions. You should set preconditioner by separate call to one of the MinLBFGSSetPrec...() functions. There is special preconditioning mode, however, which uses scaling coefficients to form diagonal preconditioning matrix. You can turn this mode on, if you want. But you should understand that scaling is not the same thing as preconditioning - these are two different, although related forms of tuning solver. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetscale(minlbfgsstate state, real_1d_array s);
minlbfgssetstpmax function/************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0 (default), if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetstpmax(minlbfgsstate state, double stpmax);
minlbfgssetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinLBFGSOptimize(). -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlbfgssetxrep(minlbfgsstate state, bool needxrep);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// using LBFGS method.
//
real_1d_array x = "[0,0]";
double epsg = 0.0000000001;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
minlbfgsstate state;
minlbfgsreport rep;
minlbfgscreate(1, x, state);
minlbfgssetcond(state, epsg, epsf, epsx, maxits);
alglib::minlbfgsoptimize(state, function1_grad);
minlbfgsresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// using LBFGS method.
//
// Several advanced techniques are demonstrated:
// * upper limit on step size
// * restart from new point
//
real_1d_array x = "[0,0]";
double epsg = 0.0000000001;
double epsf = 0;
double epsx = 0;
double stpmax = 0.1;
ae_int_t maxits = 0;
minlbfgsstate state;
minlbfgsreport rep;
// first run
minlbfgscreate(1, x, state);
minlbfgssetcond(state, epsg, epsf, epsx, maxits);
minlbfgssetstpmax(state, stpmax);
alglib::minlbfgsoptimize(state, function1_grad);
minlbfgsresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
// second run - algorithm is restarted
x = "[10,10]";
minlbfgsrestartfrom(state, x);
alglib::minlbfgsoptimize(state, function1_grad);
minlbfgsresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void s1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x and its gradient.
//
// function is trimmed when we calculate it near the singular points or outside of the [-1,+1].
// Note that we do NOT calculate gradient in this case.
//
if( (x[0]<=-0.999999999999) || (x[0]>=+0.999999999999) )
{
func = 1.0E+300;
return;
}
func = pow(1+x[0],-0.2) + pow(1-x[0],-0.3) + 1000*x[0];
grad[0] = -0.2*pow(1+x[0],-1.2) +0.3*pow(1-x[0],-1.3) + 1000;
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x) = (1+x)^(-0.2) + (1-x)^(-0.3) + 1000*x.
// This function has singularities at the boundary of the [-1,+1], but technique called
// "function trimming" allows us to solve this optimization problem.
//
// See http://www.alglib.net/optimization/tipsandtricks.php#ftrimming for more information
// on this subject.
//
real_1d_array x = "[0]";
double epsg = 1.0e-6;
double epsf = 0;
double epsx = 0;
ae_int_t maxits = 0;
minlbfgsstate state;
minlbfgsreport rep;
minlbfgscreate(1, x, state);
minlbfgssetcond(state, epsg, epsf, epsx, maxits);
alglib::minlbfgsoptimize(state, s1_grad);
minlbfgsresults(state, x, rep);
printf("%s\n", x.tostring(5).c_str()); // EXPECTED: [-0.99917305]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of f(x,y) = 100*(x+3)^4+(y-3)^4
// using numerical differentiation to calculate gradient.
//
real_1d_array x = "[0,0]";
double epsg = 0.0000000001;
double epsf = 0;
double epsx = 0;
double diffstep = 1.0e-6;
ae_int_t maxits = 0;
minlbfgsstate state;
minlbfgsreport rep;
minlbfgscreatef(1, x, diffstep, state);
minlbfgssetcond(state, epsg, epsf, epsx, maxits);
alglib::minlbfgsoptimize(state, function1_func);
minlbfgsresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,3]
return 0;
}
minlm subpackage| minlm_d_fgh | Nonlinear Hessian-based optimization for general functions | |
| minlm_d_restarts | Efficient restarts of LM optimizer | |
| minlm_d_v | Nonlinear least squares optimization using function vector only | |
| minlm_d_vb | Bound constrained nonlinear least squares optimization | |
| minlm_d_vj | Nonlinear least squares optimization using function vector and Jacobian |
minlmreport class/************************************************************************* Optimization report, filled by MinLMResults() function FIELDS: * TerminationType, completetion code: * -8 optimizer detected NAN/INF values either in the function itself, or in its Jacobian * -7 derivative correctness check failed; see rep.funcidx, rep.varidx for more information. * -5 inappropriate solver was used: * solver created with minlmcreatefgh() used on problem with general linear constraints (set with minlmsetlc() call). * -3 constraints are inconsistent * 2 relative step is no more than EpsX. * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * 8 terminated by user who called MinLMRequestTermination(). X contains point which was "current accepted" when termination request was submitted. * IterationsCount, contains iterations count * NFunc, number of function calculations * NJac, number of Jacobi matrix calculations * NGrad, number of gradient calculations * NHess, number of Hessian calculations * NCholesky, number of Cholesky decomposition calculations *************************************************************************/class minlmreport { ae_int_t iterationscount; ae_int_t terminationtype; ae_int_t funcidx; ae_int_t varidx; ae_int_t nfunc; ae_int_t njac; ae_int_t ngrad; ae_int_t nhess; ae_int_t ncholesky; };
minlmstate class/************************************************************************* Levenberg-Marquardt optimizer. This structure should be created using one of the MinLMCreate???() functions. You should not access its fields directly; use ALGLIB functions to work with it. *************************************************************************/class minlmstate { };
minlmcreatefgh function/************************************************************************* LEVENBERG-MARQUARDT-LIKE METHOD FOR NON-LINEAR OPTIMIZATION DESCRIPTION: This function is used to find minimum of general form (not "sum-of- -squares") function F = F(x[0], ..., x[n-1]) using its gradient and Hessian. Levenberg-Marquardt modification with L-BFGS pre-optimization and internal pre-conditioned L-BFGS optimization after each Levenberg-Marquardt step is used. REQUIREMENTS: This algorithm will request following information during its operation: * function value F at given point X * F and gradient G (simultaneously) at given point X * F, G and Hessian H (simultaneously) at given point X There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts func(), grad() and hess() function pointers. First pointer is used to calculate F at given point, second one calculates F(x) and grad F(x), third one calculates F(x), grad F(x), hess F(x). You can try to initialize MinLMState structure with FGH-function and then use incorrect version of MinLMOptimize() (for example, version which does not provide Hessian matrix), but it will lead to exception being thrown after first attempt to calculate Hessian. USAGE: 1. User initializes algorithm state with MinLMCreateFGH() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and pointers (delegates, etc.) to callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - initial solution, array[0..N-1] OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmcreatefgh(real_1d_array x, minlmstate& state); void alglib::minlmcreatefgh( ae_int_t n, real_1d_array x, minlmstate& state);
Examples: [1]
minlmcreatefgj function/************************************************************************* This is obsolete function. Since ALGLIB 3.3 it is equivalent to MinLMCreateFJ(). -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmcreatefgj( ae_int_t m, real_1d_array x, minlmstate& state); void alglib::minlmcreatefgj( ae_int_t n, ae_int_t m, real_1d_array x, minlmstate& state);
minlmcreatefj function/************************************************************************* This function is considered obsolete since ALGLIB 3.1.0 and is present for backward compatibility only. We recommend to use MinLMCreateVJ, which provides similar, but more consistent and feature-rich interface. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmcreatefj( ae_int_t m, real_1d_array x, minlmstate& state); void alglib::minlmcreatefj( ae_int_t n, ae_int_t m, real_1d_array x, minlmstate& state);
minlmcreatev function/************************************************************************* IMPROVED LEVENBERG-MARQUARDT METHOD FOR NON-LINEAR LEAST SQUARES OPTIMIZATION DESCRIPTION: This function is used to find minimum of function which is represented as sum of squares: F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1]) using value of function vector f[] only. Finite differences are used to calculate Jacobian. REQUIREMENTS: This algorithm will request following information during its operation: * function vector f[] at given point X There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts fvec() callback. You can try to initialize MinLMState structure with VJ function and then use incorrect version of MinLMOptimize() (for example, version which works with general form function and does not accept function vector), but it will lead to exception being thrown after first attempt to calculate Jacobian. USAGE: 1. User initializes algorithm state with MinLMCreateV() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of functions f[i] X - initial solution, array[0..N-1] DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure which stores algorithm state See also MinLMIteration, MinLMResults. NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmcreatev( ae_int_t m, real_1d_array x, double diffstep, minlmstate& state); void alglib::minlmcreatev( ae_int_t n, ae_int_t m, real_1d_array x, double diffstep, minlmstate& state);
minlmcreatevgj function/************************************************************************* This is obsolete function. Since ALGLIB 3.3 it is equivalent to MinLMCreateVJ(). -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmcreatevgj( ae_int_t m, real_1d_array x, minlmstate& state); void alglib::minlmcreatevgj( ae_int_t n, ae_int_t m, real_1d_array x, minlmstate& state);
minlmcreatevj function/************************************************************************* IMPROVED LEVENBERG-MARQUARDT METHOD FOR NON-LINEAR LEAST SQUARES OPTIMIZATION DESCRIPTION: This function is used to find minimum of function which is represented as sum of squares: F(x) = f[0]^2(x[0],...,x[n-1]) + ... + f[m-1]^2(x[0],...,x[n-1]) using value of function vector f[] and Jacobian of f[]. REQUIREMENTS: This algorithm will request following information during its operation: * function vector f[] at given point X * function vector f[] and Jacobian of f[] (simultaneously) at given point There are several overloaded versions of MinLMOptimize() function which correspond to different LM-like optimization algorithms provided by this unit. You should choose version which accepts fvec() and jac() callbacks. First one is used to calculate f[] at given point, second one calculates f[] and Jacobian df[i]/dx[j]. You can try to initialize MinLMState structure with VJ function and then use incorrect version of MinLMOptimize() (for example, version which works with general form function and does not provide Jacobian), but it will lead to exception being thrown after first attempt to calculate Jacobian. USAGE: 1. User initializes algorithm state with MinLMCreateVJ() call 2. User tunes solver parameters with MinLMSetCond(), MinLMSetStpMax() and other functions 3. User calls MinLMOptimize() function which takes algorithm state and callback functions. 4. User calls MinLMResults() to get solution 5. Optionally, user may call MinLMRestartFrom() to solve another problem with same N/M but another starting point and/or another function. MinLMRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - dimension, N>1 * if given, only leading N elements of X are used * if not given, automatically determined from size of X M - number of functions f[i] X - initial solution, array[0..N-1] OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with MinLMSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use MinLMSetStpMax() function to bound algorithm's steps. -- ALGLIB -- Copyright 30.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmcreatevj( ae_int_t m, real_1d_array x, minlmstate& state); void alglib::minlmcreatevj( ae_int_t n, ae_int_t m, real_1d_array x, minlmstate& state);
Examples: [1]
minlmoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x grad - callback which calculates function (or merit function) value func and gradient grad at given point x hess - callback which calculates function (or merit function) value func, gradient grad and Hessian hess at given point x fvec - callback which calculates function vector fi[] at given point x jac - callback which calculates function vector fi[] and Jacobian jac at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. Depending on function used to create state structure, this algorithm may accept Jacobian and/or Hessian and/or gradient. According to the said above, there ase several versions of this function, which accept different sets of callbacks. This flexibility opens way to subtle errors - you may create state with MinLMCreateFGH() (optimization using Hessian), but call function which does not accept Hessian. So when algorithm will request Hessian, there will be no callback to call. In this case exception will be thrown. Be careful to avoid such errors because there is no way to find them at compile time - you can see them at runtime only. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/void minlmoptimize(minlmstate &state, void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minlmoptimize(minlmstate &state, void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr), void (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minlmoptimize(minlmstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*hess)(const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minlmoptimize(minlmstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minlmoptimize(minlmstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*grad)(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr), void (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minlmrequesttermination function/************************************************************************* This subroutine submits request for termination of running optimizer. It should be called from user-supplied callback when user decides that it is time to "smoothly" terminate optimization process. As result, optimizer stops at point which was "current accepted" when termination request was submitted and returns error code 8 (successful termination). INPUT PARAMETERS: State - optimizer structure NOTE: after request for termination optimizer may perform several additional calls to user-supplied callbacks. It does NOT guarantee to stop immediately - it just guarantees that these additional calls will be discarded later. NOTE: calling this function on optimizer which is NOT running will have no effect. NOTE: multiple calls to this function are possible. First call is counted, subsequent calls are silently ignored. -- ALGLIB -- Copyright 08.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::minlmrequesttermination(minlmstate state);
minlmrestartfrom function/************************************************************************* This subroutine restarts LM algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used for reverse communication previously allocated with MinLMCreateXXX call. X - new starting point. -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlmrestartfrom(minlmstate state, real_1d_array x);
Examples: [1]
minlmresults function/************************************************************************* Levenberg-Marquardt algorithm results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report; includes termination codes and additional information. Termination codes are listed below, see comments for this structure for more info. Termination code is stored in rep.terminationtype field: * -8 optimizer detected NAN/INF values either in the function itself, or in its Jacobian * -7 derivative correctness check failed; see rep.funcidx, rep.varidx for more information. * -3 constraints are inconsistent * 2 relative step is no more than EpsX. * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * 8 terminated by user who called minlmrequesttermination(). X contains point which was "current accepted" when termination request was submitted. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmresults( minlmstate state, real_1d_array& x, minlmreport& rep);
minlmresultsbuf function/************************************************************************* Levenberg-Marquardt algorithm results Buffered implementation of MinLMResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::minlmresultsbuf( minlmstate state, real_1d_array& x, minlmreport& rep);
minlmsetacctype function/************************************************************************* This function is used to change acceleration settings You can choose between three acceleration strategies: * AccType=0, no acceleration. * AccType=1, secant updates are used to update quadratic model after each iteration. After fixed number of iterations (or after model breakdown) we recalculate quadratic model using analytic Jacobian or finite differences. Number of secant-based iterations depends on optimization settings: about 3 iterations - when we have analytic Jacobian, up to 2*N iterations - when we use finite differences to calculate Jacobian. AccType=1 is recommended when Jacobian calculation cost is prohibitively high (several Mx1 function vector calculations followed by several NxN Cholesky factorizations are faster than calculation of one M*N Jacobian). It should also be used when we have no Jacobian, because finite difference approximation takes too much time to compute. Table below list optimization protocols (XYZ protocol corresponds to MinLMCreateXYZ) and acceleration types they support (and use by default). ACCELERATION TYPES SUPPORTED BY OPTIMIZATION PROTOCOLS: protocol 0 1 comment V + + VJ + + FGH + DEFAULT VALUES: protocol 0 1 comment V x without acceleration it is so slooooooooow VJ x FGH x NOTE: this function should be called before optimization. Attempt to call it during algorithm iterations may result in unexpected behavior. NOTE: attempt to call this function with unsupported protocol/acceleration combination will result in exception being thrown. -- ALGLIB -- Copyright 14.10.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetacctype(minlmstate state, ae_int_t acctype);
minlmsetbc function/************************************************************************* This function sets boundary constraints for LM optimizer Boundary constraints are inactive by default (after initial creation). They are preserved until explicitly turned off with another SetBC() call. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF (latter is recommended because it will allow solver to use better algorithm). BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF (latter is recommended because it will allow solver to use better algorithm). NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: this solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints or at its boundary -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetbc( minlmstate state, real_1d_array bndl, real_1d_array bndu);
minlmsetcond function/************************************************************************* This function sets stopping conditions for Levenberg-Marquardt optimization algorithm. INPUT PARAMETERS: State - structure which stores algorithm state EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - ste pvector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinLMSetScale() Recommended values: 1E-9 ... 1E-12. MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Only Levenberg-Marquardt iterations are counted (L-BFGS/CG iterations are NOT counted because their cost is very low compared to that of LM). Passing EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (small EpsX). NOTE: it is not recommended to set large EpsX (say, 0.001). Because LM is a second-order method, it performs very precise steps anyway. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetcond(minlmstate state, double epsx, ae_int_t maxits);
minlmsetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before optimization begins * MinLMOptimize() is called * prior to actual optimization, for each function Fi and each component of parameters being optimized X[j] algorithm performs following steps: * two trial steps are made to X[j]-TestStep*S[j] and X[j]+TestStep*S[j], where X[j] is j-th parameter and S[j] is a scale of j-th parameter * if needed, steps are bounded with respect to constraints on X[] * Fi(X) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative, Rep.FuncIdx is set to index of the function. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N (parameters count) Jacobian evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with MinLMSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 15.06.2012 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetgradientcheck(minlmstate state, double teststep);
minlmsetlc function/************************************************************************* This function sets general linear constraints for LM optimizer Linear constraints are inactive by default (after initial creation). They are preserved until explicitly turned off with another minlmsetlc() call. INPUT PARAMETERS: State - structure stores algorithm state C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT IMPORTANT: if you have linear constraints, it is strongly recommended to set scale of variables with minlmsetscale(). QP solver which is used to calculate linearly constrained steps heavily relies on good scaling of input problems. IMPORTANT: solvers created with minlmcreatefgh() do not support linear constraints. NOTE: linear (non-bound) constraints are satisfied only approximately - there always exists some violation due to numerical errors and algorithmic limitations. NOTE: general linear constraints add significant overhead to solution process. Although solver performs roughly same amount of iterations (when compared with similar box-only constrained problem), each iteration now involves solution of linearly constrained QP subproblem, which requires ~3-5 times more Cholesky decompositions. Thus, if you can reformulate your problem in such way this it has only box constraints, it may be beneficial to do so. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetlc( minlmstate state, real_2d_array c, integer_1d_array ct); void alglib::minlmsetlc( minlmstate state, real_2d_array c, integer_1d_array ct, ae_int_t k);
minlmsetscale function/************************************************************************* This function sets scaling coefficients for LM optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Generally, scale is NOT considered to be a form of preconditioner. But LM optimizer is unique in that it uses scaling matrix both in the stopping condition tests and as Marquardt damping factor. Proper scaling is very important for the algorithm performance. It is less important for the quality of results, but still has some influence (it is easier to converge when variables are properly scaled, so premature stopping is possible when very badly scalled variables are combined with relaxed stopping conditions). INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetscale(minlmstate state, real_1d_array s);
minlmsetstpmax function/************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. NOTE: non-zero StpMax leads to moderate performance degradation because intermediate step of preconditioned L-BFGS optimization is incompatible with limits on step size. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetstpmax(minlmstate state, double stpmax);
minlmsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinLMOptimize(). Both Levenberg-Marquardt and internal L-BFGS iterations are reported. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minlmsetxrep(minlmstate state, bool needxrep);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_func(const real_1d_array &x, double &func, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
}
void function1_grad(const real_1d_array &x, double &func, real_1d_array &grad, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// and its derivatives df/d0 and df/dx1
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
}
void function1_hess(const real_1d_array &x, double &func, real_1d_array &grad, real_2d_array &hess, void *ptr)
{
//
// this callback calculates f(x0,x1) = 100*(x0+3)^4 + (x1-3)^4
// its derivatives df/d0 and df/dx1
// and its Hessian.
//
func = 100*pow(x[0]+3,4) + pow(x[1]-3,4);
grad[0] = 400*pow(x[0]+3,3);
grad[1] = 4*pow(x[1]-3,3);
hess[0][0] = 1200*pow(x[0]+3,2);
hess[0][1] = 0;
hess[1][0] = 0;
hess[1][1] = 12*pow(x[1]-3,2);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = 100*(x0+3)^4+(x1-3)^4
// using "FGH" mode of the Levenberg-Marquardt optimizer.
//
// F is treated like a monolitic function without internal structure,
// i.e. we do NOT represent it as a sum of squares.
//
// Optimization algorithm uses:
// * function value F(x0,x1)
// * gradient G={dF/dxi}
// * Hessian H={d2F/(dxi*dxj)}
//
real_1d_array x = "[0,0]";
double epsx = 0.0000000001;
ae_int_t maxits = 0;
minlmstate state;
minlmreport rep;
minlmcreatefgh(x, state);
minlmsetcond(state, epsx, maxits);
alglib::minlmoptimize(state, function1_func, function1_grad, function1_hess);
minlmresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
//
// this callback calculates
// f0(x0,x1) = 100*(x0+3)^4,
// f1(x0,x1) = (x1-3)^4
//
fi[0] = 10*pow(x[0]+3,2);
fi[1] = pow(x[1]-3,2);
}
void function2_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
//
// this callback calculates
// f0(x0,x1) = x0^2+1
// f1(x0,x1) = x1-1
//
fi[0] = x[0]*x[0]+1;
fi[1] = x[1]-1;
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
//
// f0(x0,x1) = 10*(x0+3)^2
// f1(x0,x1) = (x1-3)^2
//
// using several starting points and efficient restarts.
//
real_1d_array x;
double epsx = 0.0000000001;
ae_int_t maxits = 0;
minlmstate state;
minlmreport rep;
//
// create optimizer using minlmcreatev()
//
x = "[10,10]";
minlmcreatev(2, x, 0.0001, state);
minlmsetcond(state, epsx, maxits);
alglib::minlmoptimize(state, function1_fvec);
minlmresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
//
// restart optimizer using minlmrestartfrom()
//
// we can use different starting point, different function,
// different stopping conditions, but problem size
// must remain unchanged.
//
x = "[4,4]";
minlmrestartfrom(state, x);
alglib::minlmoptimize(state, function2_fvec);
minlmresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [0,1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
//
// this callback calculates
// f0(x0,x1) = 100*(x0+3)^4,
// f1(x0,x1) = (x1-3)^4
//
fi[0] = 10*pow(x[0]+3,2);
fi[1] = pow(x[1]-3,2);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
//
// f0(x0,x1) = 10*(x0+3)^2
// f1(x0,x1) = (x1-3)^2
//
// using "V" mode of the Levenberg-Marquardt optimizer.
//
// Optimization algorithm uses:
// * function vector f[] = {f1,f2}
//
// No other information (Jacobian, gradient, etc.) is needed.
//
real_1d_array x = "[0,0]";
double epsx = 0.0000000001;
ae_int_t maxits = 0;
minlmstate state;
minlmreport rep;
minlmcreatev(2, x, 0.0001, state);
minlmsetcond(state, epsx, maxits);
alglib::minlmoptimize(state, function1_fvec);
minlmresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
//
// this callback calculates
// f0(x0,x1) = 100*(x0+3)^4,
// f1(x0,x1) = (x1-3)^4
//
fi[0] = 10*pow(x[0]+3,2);
fi[1] = pow(x[1]-3,2);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
//
// f0(x0,x1) = 10*(x0+3)^2
// f1(x0,x1) = (x1-3)^2
//
// with boundary constraints
//
// -1 <= x0 <= +1
// -1 <= x1 <= +1
//
// using "V" mode of the Levenberg-Marquardt optimizer.
//
// Optimization algorithm uses:
// * function vector f[] = {f1,f2}
//
// No other information (Jacobian, gradient, etc.) is needed.
//
real_1d_array x = "[0,0]";
real_1d_array bndl = "[-1,-1]";
real_1d_array bndu = "[+1,+1]";
double epsx = 0.0000000001;
ae_int_t maxits = 0;
minlmstate state;
minlmreport rep;
minlmcreatev(2, x, 0.0001, state);
minlmsetbc(state, bndl, bndu);
minlmsetcond(state, epsx, maxits);
alglib::minlmoptimize(state, function1_fvec);
minlmresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-1,+1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void function1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
//
// this callback calculates
// f0(x0,x1) = 100*(x0+3)^4,
// f1(x0,x1) = (x1-3)^4
//
fi[0] = 10*pow(x[0]+3,2);
fi[1] = pow(x[1]-3,2);
}
void function1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates
// f0(x0,x1) = 100*(x0+3)^4,
// f1(x0,x1) = (x1-3)^4
// and Jacobian matrix J = [dfi/dxj]
//
fi[0] = 10*pow(x[0]+3,2);
fi[1] = pow(x[1]-3,2);
jac[0][0] = 20*(x[0]+3);
jac[0][1] = 0;
jac[1][0] = 0;
jac[1][1] = 2*(x[1]-3);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = f0^2+f1^2, where
//
// f0(x0,x1) = 10*(x0+3)^2
// f1(x0,x1) = (x1-3)^2
//
// using "VJ" mode of the Levenberg-Marquardt optimizer.
//
// Optimization algorithm uses:
// * function vector f[] = {f1,f2}
// * Jacobian matrix J = {dfi/dxj}.
//
real_1d_array x = "[0,0]";
double epsx = 0.0000000001;
ae_int_t maxits = 0;
minlmstate state;
minlmreport rep;
minlmcreatevj(2, x, state);
minlmsetcond(state, epsx, maxits);
alglib::minlmoptimize(state, function1_fvec, function1_jac);
minlmresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [-3,+3]
return 0;
}
minnlc subpackage| minnlc_d_equality | Nonlinearly constrained optimization (equality constraints) | |
| minnlc_d_inequality | Nonlinearly constrained optimization (inequality constraints) | |
| minnlc_d_mixed | Nonlinearly constrained optimization with mixed equality/inequality constraints |
minnlcreport class/************************************************************************* This structure stores optimization report: * IterationsCount total number of inner iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) TERMINATION CODES TerminationType field contains completion code, which can be: -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. -7 gradient verification failed. See MinNLCSetGradientCheck() for more information. 1 relative function improvement is no more than EpsF. 2 relative step is no more than EpsX. 4 gradient norm is no more than EpsG 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. Other fields of this structure are not documented and should not be used! *************************************************************************/class minnlcreport { ae_int_t iterationscount; ae_int_t nfev; ae_int_t varidx; ae_int_t funcidx; ae_int_t terminationtype; ae_int_t dbgphase0its; };
minnlcstate class/************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinNLC subpackage to work with this object *************************************************************************/class minnlcstate { };
minnlccreate function/************************************************************************* NONLINEARLY CONSTRAINED OPTIMIZATION WITH PRECONDITIONED AUGMENTED LAGRANGIAN ALGORITHM DESCRIPTION: The subroutine minimizes function F(x) of N arguments subject to any combination of: * bound constraints * linear inequality constraints * linear equality constraints * nonlinear equality constraints Gi(x)=0 * nonlinear inequality constraints Hi(x)<=0 REQUIREMENTS: * user must provide function value and gradient for F(), H(), G() * starting point X0 must be feasible or not too far away from the feasible set * F(), G(), H() are twice continuously differentiable on the feasible set and its neighborhood * nonlinear constraints G() and H() must have non-zero gradient at G(x)=0 and at H(x)=0. Say, constraint like x^2>=1 is supported, but x^2>=0 is NOT supported. USAGE: Constrained optimization if far more complex than the unconstrained one. Nonlinearly constrained optimization is one of the most esoteric numerical procedures. Here we give very brief outline of the MinNLC optimizer. We strongly recommend you to study examples in the ALGLIB Reference Manual and to read ALGLIB User Guide on optimization, which is available at http://www.alglib.net/optimization/ 1. User initializes algorithm state with MinNLCCreate() call and chooses what NLC solver to use. There is some solver which is used by default, with default settings, but you should NOT rely on default choice. It may change in future releases of ALGLIB without notice, and no one can guarantee that new solver will be able to solve your problem with default settings. From the other side, if you choose solver explicitly, you can be pretty sure that it will work with new ALGLIB releases. In the current release following solvers can be used: * AUL solver (activated with MinNLCSetAlgoAUL() function) 2. User adds boundary and/or linear and/or nonlinear constraints by means of calling one of the following functions: a) MinNLCSetBC() for boundary constraints b) MinNLCSetLC() for linear constraints c) MinNLCSetNLC() for nonlinear constraints You may combine (a), (b) and (c) in one optimization problem. 3. User sets scale of the variables with MinNLCSetScale() function. It is VERY important to set scale of the variables, because nonlinearly constrained problems are hard to solve when variables are badly scaled. 4. User sets stopping conditions with MinNLCSetCond(). If NLC solver uses inner/outer iteration layout, this function sets stopping conditions for INNER iterations. 5. User chooses one of the preconditioning methods. Preconditioning is very important for efficient handling of boundary/linear/nonlinear constraints. Without preconditioning algorithm would require thousands of iterations even for simple problems. Several preconditioners can be used: a) inexact LBFGS-based, with O(N*K) evaluation time b) exact low rank one, with O(N*K^2) evaluation time c) exact robust one, with O(N^3+K*N^2) evaluation time where K is a total number of general linear and nonlinear constraints (box ones are not counted). Since version 3.11.0 ALGLIB uses exact robust preconditioner as default option, but in some cases exact low rank one may be better option. 6. Finally, user calls MinNLCOptimize() function which takes algorithm state and pointer (delegate, etc.) to callback function which calculates F/G/H. 7. User calls MinNLCResults() to get solution 8. Optionally user may call MinNLCRestartFrom() to solve another problem with same N but another starting point. MinNLCRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size ofX X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlccreate(real_1d_array x, minnlcstate& state); void alglib::minnlccreate( ae_int_t n, real_1d_array x, minnlcstate& state);
minnlccreatef function/************************************************************************* This subroutine is a finite difference variant of MinNLCCreate(). It uses finite differences in order to differentiate target function. Description below contains information which is specific to this function only. We recommend to read comments on MinNLCCreate() in order to get more information about creation of NLC optimizer. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size ofX X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. DiffStep- differentiation step, >0 OUTPUT PARAMETERS: State - structure stores algorithm state NOTES: 1. algorithm uses 4-point central formula for differentiation. 2. differentiation step along I-th axis is equal to DiffStep*S[I] where S[] is scaling vector which can be set by MinNLCSetScale() call. 3. we recommend you to use moderate values of differentiation step. Too large step will result in too large TRUNCATION errors, while too small step will result in too large NUMERICAL errors. 1.0E-4 can be good value to start from. 4. Numerical differentiation is very inefficient - one gradient calculation needs 4*N function evaluations. This function will work for any N - either small (1...10), moderate (10...100) or large (100...). However, performance penalty will be too severe for any N's except for small ones. We should also say that code which relies on numerical differentiation is less robust and precise. Imprecise gradient may slow down convergence, especially on highly nonlinear problems. Thus we recommend to use this function for fast prototyping on small- dimensional problems only, and to implement analytical gradient as soon as possible. -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlccreatef( real_1d_array x, double diffstep, minnlcstate& state); void alglib::minnlccreatef( ae_int_t n, real_1d_array x, double diffstep, minnlcstate& state);
minnlcoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state fvec - callback which calculates function vector fi[] at given point x jac - callback which calculates function vector fi[] and Jacobian jac at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied Jacobian, and one which uses only function vector and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object you should choose appropriate variant of MinNLCOptimize() - one which accepts function AND Jacobian or one which accepts ONLY function. Be careful to choose variant of MinNLCOptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to MinNLCOptimize() and specific function used to create optimizer. | USER PASSED TO MinNLCOptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ MinNLCCreateF() | works FAILS MinNLCCreate() | FAILS works Here "FAILS" denotes inappropriate combinations of optimizer creation function and MinNLCOptimize() version. Attemps to use such combination will lead to exception. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void minnlcoptimize(minnlcstate &state, void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minnlcoptimize(minnlcstate &state, void (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minnlcrestartfrom function/************************************************************************* This subroutine restarts algorithm from new point. All optimization parameters (including constraints) are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure previously allocated with MinNLCCreate call. X - new starting point. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minnlcrestartfrom(minnlcstate state, real_1d_array x);
minnlcresults function/************************************************************************* MinNLC results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one: * -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. * -7 gradient verification failed. See MinNLCSetGradientCheck() for more information. * 1 relative function improvement is no more than EpsF. * 2 scaled step is no more than EpsX. * 4 scaled gradient norm is no more than EpsG. * 5 MaxIts steps was taken More information about fields of this structure can be found in the comments on MinNLCReport datatype. -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcresults( minnlcstate state, real_1d_array& x, minnlcreport& rep);
minnlcresultsbuf function/************************************************************************* NLC results Buffered implementation of MinNLCResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minnlcresultsbuf( minnlcstate state, real_1d_array& x, minnlcreport& rep);
minnlcsetalgoaul function/************************************************************************* This function tells MinNLC unit to use Augmented Lagrangian algorithm for nonlinearly constrained optimization. This algorithm is a slight modification of one described in "A Modified Barrier-Augmented Lagrangian Method for Constrained Minimization (1999)" by D.GOLDFARB, R.POLYAK, K. SCHEINBERG, I.YUZEFOVICH. Augmented Lagrangian algorithm works by converting problem of minimizing F(x) subject to equality/inequality constraints to unconstrained problem of the form min[ f(x) + + Rho*PENALTY_EQ(x) + SHIFT_EQ(x,Nu1) + + Rho*PENALTY_INEQ(x) + SHIFT_INEQ(x,Nu2) ] where: * Rho is a fixed penalization coefficient * PENALTY_EQ(x) is a penalty term, which is used to APPROXIMATELY enforce equality constraints * SHIFT_EQ(x) is a special "shift" term which is used to "fine-tune" equality constraints, greatly increasing precision * PENALTY_INEQ(x) is a penalty term which is used to approximately enforce inequality constraints * SHIFT_INEQ(x) is a special "shift" term which is used to "fine-tune" inequality constraints, greatly increasing precision * Nu1/Nu2 are vectors of Lagrange coefficients which are fine-tuned during outer iterations of algorithm This version of AUL algorithm uses preconditioner, which greatly accelerates convergence. Because this algorithm is similar to penalty methods, it may perform steps into infeasible area. All kinds of constraints (boundary, linear and nonlinear ones) may be violated in intermediate points - and in the solution. However, properly configured AUL method is significantly better at handling constraints than barrier and/or penalty methods. The very basic outline of algorithm is given below: 1) first outer iteration is performed with "default" values of Lagrange multipliers Nu1/Nu2. Solution quality is low (candidate point can be too far away from true solution; large violation of constraints is possible) and is comparable with that of penalty methods. 2) subsequent outer iterations refine Lagrange multipliers and improve quality of the solution. INPUT PARAMETERS: State - structure which stores algorithm state Rho - penalty coefficient, Rho>0: * large enough that algorithm converges with desired precision. Minimum value is 10*max(S'*diag(H)*S), where S is a scale matrix (set by MinNLCSetScale) and H is a Hessian of the function being minimized. If you can not easily estimate Hessian norm, see our recommendations below. * not TOO large to prevent ill-conditioning * for unit-scale problems (variables and Hessian have unit magnitude), Rho=100 or Rho=1000 can be used. * it is important to note that Rho is internally multiplied by scaling matrix, i.e. optimum value of Rho depends on scale of variables specified by MinNLCSetScale(). ItsCnt - number of outer iterations: * ItsCnt=0 means that small number of outer iterations is automatically chosen (10 iterations in current version). * ItsCnt=1 means that AUL algorithm performs just as usual barrier method. * ItsCnt>1 means that AUL algorithm performs specified number of outer iterations HOW TO CHOOSE PARAMETERS Nonlinear optimization is a tricky area and Augmented Lagrangian algorithm is sometimes hard to tune. Good values of Rho and ItsCnt are problem- specific. In order to help you we prepared following set of recommendations: * for unit-scale problems (variables and Hessian have unit magnitude), Rho=100 or Rho=1000 can be used. * start from some small value of Rho and solve problem with just one outer iteration (ItcCnt=1). In this case algorithm behaves like penalty method. Increase Rho in 2x or 10x steps until you see that one outer iteration returns point which is "rough approximation to solution". It is very important to have Rho so large that penalty term becomes constraining i.e. modified function becomes highly convex in constrained directions. From the other side, too large Rho may prevent you from converging to the solution. You can diagnose it by studying number of inner iterations performed by algorithm: too few (5-10 on 1000-dimensional problem) or too many (orders of magnitude more than dimensionality) usually means that Rho is too large. * with just one outer iteration you usually have low-quality solution. Some constraints can be violated with very large margin, while other ones (which are NOT violated in the true solution) can push final point too far in the inner area of the feasible set. For example, if you have constraint x0>=0 and true solution x0=1, then merely a presence of "x0>=0" will introduce a bias towards larger values of x0. Say, algorithm may stop at x0=1.5 instead of 1.0. * after you found good Rho, you may increase number of outer iterations. ItsCnt=10 is a good value. Subsequent outer iteration will refine values of Lagrange multipliers. Constraints which were violated will be enforced, inactive constraints will be dropped (corresponding multipliers will be decreased). Ideally, you should see 10-1000x improvement in constraint handling (constraint violation is reduced). * if you see that algorithm converges to vicinity of solution, but additional outer iterations do not refine solution, it may mean that algorithm is unstable - it wanders around true solution, but can not approach it. Sometimes algorithm may be stabilized by increasing Rho one more time, making it 5x or 10x larger. SCALING OF CONSTRAINTS [IMPORTANT] AUL optimizer scales variables according to scale specified by MinNLCSetScale() function, so it can handle problems with badly scaled variables (as long as we KNOW their scales). However, because function being optimized is a mix of original function and constraint-dependent penalty functions, it is important to rescale both variables AND constraints. Say, if you minimize f(x)=x^2 subject to 1000000*x>=0, then you have constraint whose scale is different from that of target function (another example is 0.000001*x>=0). It is also possible to have constraints whose scales are misaligned: 1000000*x0>=0, 0.000001*x1<=0. Inappropriate scaling may ruin convergence because minimizing x^2 subject to x>=0 is NOT same as minimizing it subject to 1000000*x>=0. Because we know coefficients of boundary/linear constraints, we can automatically rescale and normalize them. However, there is no way to automatically rescale nonlinear constraints Gi(x) and Hi(x) - they are black boxes. It means that YOU are the one who is responsible for correct scaling of nonlinear constraints Gi(x) and Hi(x). We recommend you to rescale nonlinear constraints in such way that I-th component of dG/dX (or dH/dx) has magnitude approximately equal to 1/S[i] (where S is a scale set by MinNLCSetScale() function). WHAT IF IT DOES NOT CONVERGE? It is possible that AUL algorithm fails to converge to precise values of Lagrange multipliers. It stops somewhere around true solution, but candidate point is still too far from solution, and some constraints are violated. Such kind of failure is specific for Lagrangian algorithms - technically, they stop at some point, but this point is not constrained solution. There are exist several reasons why algorithm may fail to converge: a) too loose stopping criteria for inner iteration b) degenerate, redundant constraints c) target function has unconstrained extremum exactly at the boundary of some constraint d) numerical noise in the target function In all these cases algorithm is unstable - each outer iteration results in large and almost random step which improves handling of some constraints, but violates other ones (ideally outer iterations should form a sequence of progressively decreasing steps towards solution). First reason possible is that too loose stopping criteria for inner iteration were specified. Augmented Lagrangian algorithm solves a sequence of intermediate problems, and requries each of them to be solved with high precision. Insufficient precision results in incorrect update of Lagrange multipliers. Another reason is that you may have specified degenerate constraints: say, some constraint was repeated twice. In most cases AUL algorithm gracefully handles such situations, but sometimes it may spend too much time figuring out subtle degeneracies in constraint matrix. Third reason is tricky and hard to diagnose. Consider situation when you minimize f=x^2 subject to constraint x>=0. Unconstrained extremum is located exactly at the boundary of constrained area. In this case algorithm will tend to oscillate between negative and positive x. Each time it stops at x<0 it "reinforces" constraint x>=0, and each time it is bounced to x>0 it "relaxes" constraint (and is attracted to x<0). Such situation sometimes happens in problems with hidden symetries. Algorithm is got caught in a loop with Lagrange multipliers being continuously increased/decreased. Luckily, such loop forms after at least three iterations, so this problem can be solved by DECREASING number of outer iterations down to 1-2 and increasing penalty coefficient Rho as much as possible. Final reason is numerical noise. AUL algorithm is robust against moderate noise (more robust than, say, active set methods), but large noise may destabilize algorithm. -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetalgoaul( minnlcstate state, double rho, ae_int_t itscnt);
minnlcsetbc function/************************************************************************* This function sets boundary constraints for NLC optimizer. Boundary constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinNLCRestartFrom(). You may combine boundary constraints with general linear ones - and with nonlinear ones! Boundary constraints are handled more efficiently than other types. Thus, if your problem has mixed constraints, you may explicitly specify some of them as boundary and save some time/space. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF. BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF. NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: when you solve your problem with augmented Lagrangian solver, boundary constraints are satisfied only approximately! It is possible that algorithm will evaluate function outside of feasible area! -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetbc( minnlcstate state, real_1d_array bndl, real_1d_array bndu);
minnlcsetcond function/************************************************************************* This function sets stopping conditions for inner iterations of optimizer. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinNLCSetScale() EpsF - >=0 The subroutine finishes its work if on k+1-th iteration the condition |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} is satisfied. EpsX - >=0 The subroutine finishes its work if on k+1-th iteration the condition |v|<=EpsX is fulfilled, where: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - step vector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinNLCSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection. -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetcond( minnlcstate state, double epsg, double epsf, double epsx, ae_int_t maxits);
minnlcsetgradientcheck function/************************************************************************* This subroutine turns on verification of the user-supplied analytic gradient: * user calls this subroutine before optimization begins * MinNLCOptimize() is called * prior to actual optimization, for each component of parameters being optimized X[i] algorithm performs following steps: * two trial steps are made to X[i]-TestStep*S[i] and X[i]+TestStep*S[i], where X[i] is i-th component of the initial point and S[i] is a scale of i-th parameter * F(X) is evaluated at these trial points * we perform one more evaluation in the middle point of the interval * we build cubic model using function values and derivatives at trial points and we compare its prediction with actual value in the middle point * in case difference between prediction and actual value is higher than some predetermined threshold, algorithm stops with completion code -7; Rep.VarIdx is set to index of the parameter with incorrect derivative, and Rep.FuncIdx is set to index of the function. * after verification is over, algorithm proceeds to the actual optimization. NOTE 1: verification needs N (parameters count) gradient evaluations. It is very costly and you should use it only for low dimensional problems, when you want to be sure that you've correctly calculated analytic derivatives. You should not use it in the production code (unless you want to check derivatives provided by some third party). NOTE 2: you should carefully choose TestStep. Value which is too large (so large that function behaviour is significantly non-cubic) will lead to false alarms. You may use different step for different parameters by means of setting scale with MinNLCSetScale(). NOTE 3: this function may lead to false positives. In case it reports that I-th derivative was calculated incorrectly, you may decrease test step and try one more time - maybe your function changes too sharply and your step is too large for such rapidly chanding function. INPUT PARAMETERS: State - structure used to store algorithm state TestStep - verification step: * TestStep=0 turns verification off * TestStep>0 activates verification -- ALGLIB -- Copyright 15.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetgradientcheck(minnlcstate state, double teststep);
minnlcsetlc function/************************************************************************* This function sets linear constraints for MinNLC optimizer. Linear constraints are inactive by default (after initial creation). They are preserved after algorithm restart with MinNLCRestartFrom(). You may combine linear constraints with boundary ones - and with nonlinear ones! If your problem has mixed constraints, you may explicitly specify some of them as linear. It may help optimizer to handle them more efficiently. INPUT PARAMETERS: State - structure previously allocated with MinNLCCreate call. C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT NOTE 1: when you solve your problem with augmented Lagrangian solver, linear constraints are satisfied only approximately! It is possible that algorithm will evaluate function outside of feasible area! -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetlc( minnlcstate state, real_2d_array c, integer_1d_array ct); void alglib::minnlcsetlc( minnlcstate state, real_2d_array c, integer_1d_array ct, ae_int_t k);
minnlcsetnlc function/************************************************************************* This function sets nonlinear constraints for MinNLC optimizer. In fact, this function sets NUMBER of nonlinear constraints. Constraints itself (constraint functions) are passed to MinNLCOptimize() method. This method requires user-defined vector function F[] and its Jacobian J[], where: * first component of F[] and first row of Jacobian J[] corresponds to function being minimized * next NLEC components of F[] (and rows of J) correspond to nonlinear equality constraints G_i(x)=0 * next NLIC components of F[] (and rows of J) correspond to nonlinear inequality constraints H_i(x)<=0 NOTE: you may combine nonlinear constraints with linear/boundary ones. If your problem has mixed constraints, you may explicitly specify some of them as linear ones. It may help optimizer to handle them more efficiently. INPUT PARAMETERS: State - structure previously allocated with MinNLCCreate call. NLEC - number of Non-Linear Equality Constraints (NLEC), >=0 NLIC - number of Non-Linear Inquality Constraints (NLIC), >=0 NOTE 1: when you solve your problem with augmented Lagrangian solver, nonlinear constraints are satisfied only approximately! It is possible that algorithm will evaluate function outside of feasible area! NOTE 2: algorithm scales variables according to scale specified by MinNLCSetScale() function, so it can handle problems with badly scaled variables (as long as we KNOW their scales). However, there is no way to automatically scale nonlinear constraints Gi(x) and Hi(x). Inappropriate scaling of Gi/Hi may ruin convergence. Solving problem with constraint "1000*G0(x)=0" is NOT same as solving it with constraint "0.001*G0(x)=0". It means that YOU are the one who is responsible for correct scaling of nonlinear constraints Gi(x) and Hi(x). We recommend you to scale nonlinear constraints in such way that I-th component of dG/dX (or dH/dx) has approximately unit magnitude (for problems with unit scale) or has magnitude approximately equal to 1/S[i] (where S is a scale set by MinNLCSetScale() function). -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetnlc( minnlcstate state, ae_int_t nlec, ae_int_t nlic);
minnlcsetprecexactlowrank function/************************************************************************* This function sets preconditioner to "exact low rank" mode. Preconditioning is very important for convergence of Augmented Lagrangian algorithm because presence of penalty term makes problem ill-conditioned. Difference between performance of preconditioned and unpreconditioned methods can be as large as 100x! MinNLC optimizer may use following preconditioners, each with its own benefits and drawbacks: a) inexact LBFGS-based, with O(N*K) evaluation time b) exact low rank one, with O(N*K^2) evaluation time c) exact robust one, with O(N^3+K*N^2) evaluation time where K is a total number of general linear and nonlinear constraints (box ones are not counted). It also provides special unpreconditioned mode of operation which can be used for test purposes. Comments below discuss low rank preconditioner. Exact low-rank preconditioner uses Woodbury matrix identity to build quadratic model of the penalized function. It has following features: * no special assumptions about orthogonality of constraints * preconditioner evaluation is optimized for K<<N. Its cost is O(N*K^2), so it may become prohibitively slow for K>=N. * finally, stability of the process is guaranteed only for K<<N. Woodbury update often fail for K>=N due to degeneracy of intermediate matrices. That's why we recommend to use "exact robust" preconditioner for such cases. RECOMMENDATIONS We recommend to choose between "exact low rank" and "exact robust" preconditioners, with "low rank" version being chosen when you know in advance that total count of non-box constraints won't exceed N, and "robust" version being chosen when you need bulletproof solution. INPUT PARAMETERS: State - structure stores algorithm state UpdateFreq- update frequency. Preconditioner is rebuilt after every UpdateFreq iterations. Recommended value: 10 or higher. Zero value means that good default value will be used. -- ALGLIB -- Copyright 26.09.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetprecexactlowrank( minnlcstate state, ae_int_t updatefreq);
minnlcsetprecexactrobust function/************************************************************************* This function sets preconditioner to "exact robust" mode. Preconditioning is very important for convergence of Augmented Lagrangian algorithm because presence of penalty term makes problem ill-conditioned. Difference between performance of preconditioned and unpreconditioned methods can be as large as 100x! MinNLC optimizer may use following preconditioners, each with its own benefits and drawbacks: a) inexact LBFGS-based, with O(N*K) evaluation time b) exact low rank one, with O(N*K^2) evaluation time c) exact robust one, with O(N^3+K*N^2) evaluation time where K is a total number of general linear and nonlinear constraints (box ones are not counted). It also provides special unpreconditioned mode of operation which can be used for test purposes. Comments below discuss robust preconditioner. Exact robust preconditioner uses Cholesky decomposition to invert approximate Hessian matrix H=D+W'*C*W (where D stands for diagonal terms of Hessian, combined result of initial scaling matrix and penalty from box constraints; W stands for general linear constraints and linearization of nonlinear ones; C stands for diagonal matrix of penalty coefficients). This preconditioner has following features: * no special assumptions about constraint structure * preconditioner is optimized for stability; unlike "exact low rank" version which fails for K>=N, this one works well for any value of K. * the only drawback is that is takes O(N^3+K*N^2) time to build it. No economical Woodbury update is applied even when it makes sense, thus there are exist situations (K<<N) when "exact low rank" preconditioner outperforms this one. RECOMMENDATIONS We recommend to choose between "exact low rank" and "exact robust" preconditioners, with "low rank" version being chosen when you know in advance that total count of non-box constraints won't exceed N, and "robust" version being chosen when you need bulletproof solution. INPUT PARAMETERS: State - structure stores algorithm state UpdateFreq- update frequency. Preconditioner is rebuilt after every UpdateFreq iterations. Recommended value: 10 or higher. Zero value means that good default value will be used. -- ALGLIB -- Copyright 26.09.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetprecexactrobust( minnlcstate state, ae_int_t updatefreq);
minnlcsetprecinexact function/************************************************************************* This function sets preconditioner to "inexact LBFGS-based" mode. Preconditioning is very important for convergence of Augmented Lagrangian algorithm because presence of penalty term makes problem ill-conditioned. Difference between performance of preconditioned and unpreconditioned methods can be as large as 100x! MinNLC optimizer may use following preconditioners, each with its own benefits and drawbacks: a) inexact LBFGS-based, with O(N*K) evaluation time b) exact low rank one, with O(N*K^2) evaluation time c) exact robust one, with O(N^3+K*N^2) evaluation time where K is a total number of general linear and nonlinear constraints (box ones are not counted). Inexact LBFGS-based preconditioner uses L-BFGS formula combined with orthogonality assumption to perform very fast updates. For a N-dimensional problem with K general linear or nonlinear constraints (boundary ones are not counted) it has O(N*K) cost per iteration. This preconditioner has best quality (less iterations) when general linear and nonlinear constraints are orthogonal to each other (orthogonality with respect to boundary constraints is not required). Number of iterations increases when constraints are non-orthogonal, because algorithm assumes orthogonality, but still it is better than no preconditioner at all. INPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 26.09.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetprecinexact(minnlcstate state);
minnlcsetprecnone function/************************************************************************* This function sets preconditioner to "turned off" mode. Preconditioning is very important for convergence of Augmented Lagrangian algorithm because presence of penalty term makes problem ill-conditioned. Difference between performance of preconditioned and unpreconditioned methods can be as large as 100x! MinNLC optimizer may utilize two preconditioners, each with its own benefits and drawbacks: a) inexact LBFGS-based, and b) exact low rank one. It also provides special unpreconditioned mode of operation which can be used for test purposes. This function activates this test mode. Do not use it in production code to solve real-life problems. INPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 26.09.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetprecnone(minnlcstate state);
minnlcsetscale function/************************************************************************* This function sets scaling coefficients for NLC optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 06.06.2014 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetscale(minnlcstate state, real_1d_array s);
minnlcsetstpmax function/************************************************************************* This function sets maximum step length (after scaling of step vector with respect to variable scales specified by minnlcsetscale() call). INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0 (default), if you don't want to limit step length. Use this subroutine when you optimize target function which contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 02.04.2010 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetstpmax(minnlcstate state, double stpmax);
minnlcsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to MinNLCOptimize(). NOTE: algorithm passes two parameters to rep() callback - current point and penalized function value at current point. Important - function value which is returned is NOT function being minimized. It is sum of the value of the function being minimized - and penalty term. -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minnlcsetxrep(minnlcstate state, bool needxrep);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nlcfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates
//
// f0(x0,x1) = -x0+x1
// f1(x0,x1) = x0^2+x1^2-1
//
// and Jacobian matrix J = [dfi/dxj]
//
fi[0] = -x[0]+x[1];
fi[1] = x[0]*x[0] + x[1]*x[1] - 1.0;
jac[0][0] = -1.0;
jac[0][1] = +1.0;
jac[1][0] = 2*x[0];
jac[1][1] = 2*x[1];
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = -x0+x1
//
// subject to nonlinear equality constraint
//
// x0^2 + x1^2 - 1 = 0
//
real_1d_array x0 = "[0,0]";
real_1d_array s = "[1,1]";
double epsg = 0;
double epsf = 0;
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t outerits = 5;
ae_int_t updatefreq = 10;
double rho = 1000;
minnlcstate state;
minnlcreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AUL algorithm and tune its settings:
// * rho=1000 penalty coefficient
// * outerits=5 number of outer iterations to tune Lagrange coefficients
// * epsx=0.000001 stopping condition for inner iterations
// * s=[1,1] all variables have unit scale
// * exact low-rank preconditioner is used, updated after each 10 iterations
//
minnlccreate(2, x0, state);
minnlcsetalgoaul(state, rho, outerits);
minnlcsetcond(state, epsg, epsf, epsx, maxits);
minnlcsetscale(state, s);
minnlcsetprecexactlowrank(state, updatefreq);
//
// Set constraints:
//
// Nonlinear constraints are tricky - you can not "pack" general
// nonlinear function into double precision array. That's why
// minnlcsetnlc() does not accept constraints itself - only constraint
// counts are passed: first parameter is number of equality constraints,
// second one is number of inequality constraints.
//
// As for constraining functions - these functions are passed as part
// of problem Jacobian (see below).
//
// NOTE: MinNLC optimizer supports arbitrary combination of boundary, general
// linear and general nonlinear constraints. This example does not
// show how to work with general linear constraints, but you can
// easily find it in documentation on minnlcsetbc() and
// minnlcsetlc() functions.
//
minnlcsetnlc(state, 1, 0);
//
// Optimize and test results.
//
// Optimizer object accepts vector function and its Jacobian, with first
// component (Jacobian row) being target function, and next components
// (Jacobian rows) being nonlinear equality and inequality constraints.
//
// So, our vector function has form
//
// {f0,f1} = { -x0+x1 , x0^2+x1^2-1 }
//
// with Jacobian
//
// [ -1 +1 ]
// J = [ ]
// [ 2*x0 2*x1 ]
//
// with f0 being target function, f1 being constraining function. Number
// of equality/inequality constraints is specified by minnlcsetnlc(),
// with equality ones always being first, inequality ones being last.
//
alglib::minnlcoptimize(state, nlcfunc1_jac);
minnlcresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.70710,-0.70710]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nlcfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates
//
// f0(x0,x1) = -x0+x1
// f1(x0,x1) = x0^2+x1^2-1
//
// and Jacobian matrix J = [dfi/dxj]
//
fi[0] = -x[0]+x[1];
fi[1] = x[0]*x[0] + x[1]*x[1] - 1.0;
jac[0][0] = -1.0;
jac[0][1] = +1.0;
jac[1][0] = 2*x[0];
jac[1][1] = 2*x[1];
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = -x0+x1
//
// subject to boundary constraints
//
// x0>=0, x1>=0
//
// and nonlinear inequality constraint
//
// x0^2 + x1^2 - 1 <= 0
//
real_1d_array x0 = "[0,0]";
real_1d_array s = "[1,1]";
double epsg = 0;
double epsf = 0;
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t outerits = 5;
ae_int_t updatefreq = 10;
double rho = 1000;
real_1d_array bndl = "[0,0]";
real_1d_array bndu = "[+inf,+inf]";
minnlcstate state;
minnlcreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AUL algorithm and tune its settings:
// * rho=1000 penalty coefficient
// * outerits=5 number of outer iterations to tune Lagrange coefficients
// * epsx=0.000001 stopping condition for inner iterations
// * s=[1,1] all variables have unit scale
// * exact low-rank preconditioner is used, updated after each 10 iterations
//
minnlccreate(2, x0, state);
minnlcsetalgoaul(state, rho, outerits);
minnlcsetcond(state, epsg, epsf, epsx, maxits);
minnlcsetscale(state, s);
minnlcsetprecexactlowrank(state, updatefreq);
//
// Set constraints:
//
// 1. boundary constraints are passed with minnlcsetbc() call
//
// 2. nonlinear constraints are more tricky - you can not "pack" general
// nonlinear function into double precision array. That's why
// minnlcsetnlc() does not accept constraints itself - only constraint
// counts are passed: first parameter is number of equality constraints,
// second one is number of inequality constraints.
//
// As for constraining functions - these functions are passed as part
// of problem Jacobian (see below).
//
// NOTE: MinNLC optimizer supports arbitrary combination of boundary, general
// linear and general nonlinear constraints. This example does not
// show how to work with general linear constraints, but you can
// easily find it in documentation on minnlcsetlc() function.
//
minnlcsetbc(state, bndl, bndu);
minnlcsetnlc(state, 0, 1);
//
// Optimize and test results.
//
// Optimizer object accepts vector function and its Jacobian, with first
// component (Jacobian row) being target function, and next components
// (Jacobian rows) being nonlinear equality and inequality constraints.
//
// So, our vector function has form
//
// {f0,f1} = { -x0+x1 , x0^2+x1^2-1 }
//
// with Jacobian
//
// [ -1 +1 ]
// J = [ ]
// [ 2*x0 2*x1 ]
//
// with f0 being target function, f1 being constraining function. Number
// of equality/inequality constraints is specified by minnlcsetnlc(),
// with equality ones always being first, inequality ones being last.
//
alglib::minnlcoptimize(state, nlcfunc1_jac);
minnlcresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nlcfunc2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates
//
// f0(x0,x1,x2) = x0+x1
// f1(x0,x1,x2) = x2-exp(x0)
// f2(x0,x1,x2) = x0^2+x1^2-1
//
// and Jacobian matrix J = [dfi/dxj]
//
fi[0] = x[0]+x[1];
fi[1] = x[2]-exp(x[0]);
fi[2] = x[0]*x[0] + x[1]*x[1] - 1.0;
jac[0][0] = 1.0;
jac[0][1] = 1.0;
jac[0][2] = 0.0;
jac[1][0] = -exp(x[0]);
jac[1][1] = 0.0;
jac[1][2] = 1.0;
jac[2][0] = 2*x[0];
jac[2][1] = 2*x[1];
jac[2][2] = 0.0;
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = x0+x1
//
// subject to nonlinear inequality constraint
//
// x0^2 + x1^2 - 1 <= 0
//
// and nonlinear equality constraint
//
// x2-exp(x0) = 0
//
real_1d_array x0 = "[0,0,0]";
real_1d_array s = "[1,1,1]";
double epsg = 0;
double epsf = 0;
double epsx = 0.000001;
ae_int_t maxits = 0;
ae_int_t outerits = 5;
ae_int_t updatefreq = 10;
double rho = 1000;
minnlcstate state;
minnlcreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AUL algorithm and tune its settings:
// * rho=1000 penalty coefficient
// * outerits=5 number of outer iterations to tune Lagrange coefficients
// * epsx=0.000001 stopping condition for inner iterations
// * s=[1,1] all variables have unit scale
// * exact low-rank preconditioner is used, updated after each 10 iterations
// * upper limit on step length is specified (to avoid probing locations where exp() is large)
//
minnlccreate(3, x0, state);
minnlcsetalgoaul(state, rho, outerits);
minnlcsetcond(state, epsg, epsf, epsx, maxits);
minnlcsetscale(state, s);
minnlcsetprecexactlowrank(state, updatefreq);
minnlcsetstpmax(state, 10.0);
//
// Set constraints:
//
// Nonlinear constraints are tricky - you can not "pack" general
// nonlinear function into double precision array. That's why
// minnlcsetnlc() does not accept constraints itself - only constraint
// counts are passed: first parameter is number of equality constraints,
// second one is number of inequality constraints.
//
// As for constraining functions - these functions are passed as part
// of problem Jacobian (see below).
//
// NOTE: MinNLC optimizer supports arbitrary combination of boundary, general
// linear and general nonlinear constraints. This example does not
// show how to work with boundary or general linear constraints, but you
// can easily find it in documentation on minnlcsetbc() and
// minnlcsetlc() functions.
//
minnlcsetnlc(state, 1, 1);
//
// Optimize and test results.
//
// Optimizer object accepts vector function and its Jacobian, with first
// component (Jacobian row) being target function, and next components
// (Jacobian rows) being nonlinear equality and inequality constraints.
//
// So, our vector function has form
//
// {f0,f1,f2} = { x0+x1 , x2-exp(x0) , x0^2+x1^2-1 }
//
// with Jacobian
//
// [ +1 +1 0 ]
// J = [-exp(x0) 0 1 ]
// [ 2*x0 2*x1 0 ]
//
// with f0 being target function, f1 being equality constraint "f1=0",
// f2 being inequality constraint "f2<=0". Number of equality/inequality
// constraints is specified by minnlcsetnlc(), with equality ones always
// being first, inequality ones being last.
//
alglib::minnlcoptimize(state, nlcfunc2_jac);
minnlcresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [-0.70710,-0.70710,0.49306]
return 0;
}
minns subpackage| minns_d_bc | Nonsmooth box constrained optimization | |
| minns_d_diff | Nonsmooth unconstrained optimization with numerical differentiation | |
| minns_d_nlc | Nonsmooth nonlinearly constrained optimization | |
| minns_d_unconstrained | Nonsmooth unconstrained optimization |
minnsreport class/************************************************************************* This structure stores optimization report: * IterationsCount total number of inner iterations * NFEV number of gradient evaluations * TerminationType termination type (see below) * CErr maximum violation of all types of constraints * LCErr maximum violation of linear constraints * NLCErr maximum violation of nonlinear constraints TERMINATION CODES TerminationType field contains completion code, which can be: -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. -3 box constraints are inconsistent -1 inconsistent parameters were passed: * penalty parameter for minnssetalgoags() is zero, but we have nonlinear constraints set by minnssetnlc() 2 sampling radius decreased below epsx 5 MaxIts steps was taken 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. 8 User requested termination via MinNSRequestTermination() Other fields of this structure are not documented and should not be used! *************************************************************************/class minnsreport { ae_int_t iterationscount; ae_int_t nfev; double cerr; double lcerr; double nlcerr; ae_int_t terminationtype; ae_int_t varidx; ae_int_t funcidx; };
minnsstate class/************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinNS subpackage to work with this object *************************************************************************/class minnsstate { };
minnscreate function/************************************************************************* NONSMOOTH NONCONVEX OPTIMIZATION SUBJECT TO BOX/LINEAR/NONLINEAR-NONSMOOTH CONSTRAINTS DESCRIPTION: The subroutine minimizes function F(x) of N arguments subject to any combination of: * bound constraints * linear inequality constraints * linear equality constraints * nonlinear equality constraints Gi(x)=0 * nonlinear inequality constraints Hi(x)<=0 IMPORTANT: see MinNSSetAlgoAGS for important information on performance restrictions of AGS solver. REQUIREMENTS: * starting point X0 must be feasible or not too far away from the feasible set * F(), G(), H() are continuous, locally Lipschitz and continuously (but not necessarily twice) differentiable in an open dense subset of R^N. Functions F(), G() and H() may be nonsmooth and non-convex. Informally speaking, it means that functions are composed of large differentiable "patches" with nonsmoothness having place only at the boundaries between these "patches". Most real-life nonsmooth functions satisfy these requirements. Say, anything which involves finite number of abs(), min() and max() is very likely to pass the test. Say, it is possible to optimize anything of the following: * f=abs(x0)+2*abs(x1) * f=max(x0,x1) * f=sin(max(x0,x1)+abs(x2)) * for nonlinearly constrained problems: F() must be bounded from below without nonlinear constraints (this requirement is due to the fact that, contrary to box and linear constraints, nonlinear ones require special handling). * user must provide function value and gradient for F(), H(), G() at all points where function/gradient can be calculated. If optimizer requires value exactly at the boundary between "patches" (say, at x=0 for f=abs(x)), where gradient is not defined, user may resolve tie arbitrarily (in our case - return +1 or -1 at its discretion). * NS solver supports numerical differentiation, i.e. it may differentiate your function for you, but it results in 2N increase of function evaluations. Not recommended unless you solve really small problems. See minnscreatef() for more information on this functionality. USAGE: 1. User initializes algorithm state with MinNSCreate() call and chooses what NLC solver to use. There is some solver which is used by default, with default settings, but you should NOT rely on default choice. It may change in future releases of ALGLIB without notice, and no one can guarantee that new solver will be able to solve your problem with default settings. From the other side, if you choose solver explicitly, you can be pretty sure that it will work with new ALGLIB releases. In the current release following solvers can be used: * AGS solver (activated with MinNSSetAlgoAGS() function) 2. User adds boundary and/or linear and/or nonlinear constraints by means of calling one of the following functions: a) MinNSSetBC() for boundary constraints b) MinNSSetLC() for linear constraints c) MinNSSetNLC() for nonlinear constraints You may combine (a), (b) and (c) in one optimization problem. 3. User sets scale of the variables with MinNSSetScale() function. It is VERY important to set scale of the variables, because nonlinearly constrained problems are hard to solve when variables are badly scaled. 4. User sets stopping conditions with MinNSSetCond(). 5. Finally, user calls MinNSOptimize() function which takes algorithm state and pointer (delegate, etc) to callback function which calculates F/G/H. 7. User calls MinNSResults() to get solution 8. Optionally user may call MinNSRestartFrom() to solve another problem with same N but another starting point. MinNSRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. OUTPUT PARAMETERS: State - structure stores algorithm state NOTE: minnscreatef() function may be used if you do not have analytic gradient. This function creates solver which uses numerical differentiation with user-specified step. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnscreate(real_1d_array x, minnsstate& state); void alglib::minnscreate(ae_int_t n, real_1d_array x, minnsstate& state);
minnscreatef function/************************************************************************* Version of minnscreatef() which uses numerical differentiation. I.e., you do not have to calculate derivatives yourself. However, this version needs 2N times more function evaluations. 2-point differentiation formula is used, because more precise 4-point formula is unstable when used on non-smooth functions. INPUT PARAMETERS: N - problem dimension, N>0: * if given, only leading N elements of X are used * if not given, automatically determined from size of X X - starting point, array[N]: * it is better to set X to a feasible point * but X can be infeasible, in which case algorithm will try to find feasible point first, using X as initial approximation. DiffStep- differentiation step, DiffStep>0. Algorithm performs numerical differentiation with step for I-th variable being equal to DiffStep*S[I] (here S[] is a scale vector, set by minnssetscale() function). Do not use too small steps, because it may lead to catastrophic cancellation during intermediate calculations. OUTPUT PARAMETERS: State - structure stores algorithm state -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnscreatef( real_1d_array x, double diffstep, minnsstate& state); void alglib::minnscreatef( ae_int_t n, real_1d_array x, double diffstep, minnsstate& state);
Examples: [1]
minnsoptimize function/************************************************************************* This family of functions is used to launcn iterations of nonlinear optimizer These functions accept following parameters: state - algorithm state fvec - callback which calculates function vector fi[] at given point x jac - callback which calculates function vector fi[] and Jacobian jac at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL NOTES: 1. This function has two different implementations: one which uses exact (analytical) user-supplied Jacobian, and one which uses only function vector and numerically differentiates function in order to obtain gradient. Depending on the specific function used to create optimizer object you should choose appropriate variant of minnsoptimize() - one which accepts function AND Jacobian or one which accepts ONLY function. Be careful to choose variant of minnsoptimize() which corresponds to your optimization scheme! Table below lists different combinations of callback (function/gradient) passed to minnsoptimize() and specific function used to create optimizer. | USER PASSED TO minnsoptimize() CREATED WITH | function only | function and gradient ------------------------------------------------------------ minnscreatef() | works FAILS minnscreate() | FAILS works Here "FAILS" denotes inappropriate combinations of optimizer creation function and minnsoptimize() version. Attemps to use such combination will lead to exception. Either you did not pass gradient when it WAS needed or you passed gradient when it was NOT needed. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void minnsoptimize(minnsstate &state, void (*fvec)(const real_1d_array &x, real_1d_array &fi, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL); void minnsoptimize(minnsstate &state, void (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
minnsrequesttermination function/************************************************************************* This subroutine submits request for termination of running optimizer. It should be called from user-supplied callback when user decides that it is time to "smoothly" terminate optimization process. As result, optimizer stops at point which was "current accepted" when termination request was submitted and returns error code 8 (successful termination). INPUT PARAMETERS: State - optimizer structure NOTE: after request for termination optimizer may perform several additional calls to user-supplied callbacks. It does NOT guarantee to stop immediately - it just guarantees that these additional calls will be discarded later. NOTE: calling this function on optimizer which is NOT running will have no effect. NOTE: multiple calls to this function are possible. First call is counted, subsequent calls are silently ignored. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnsrequesttermination(minnsstate state);
minnsrestartfrom function/************************************************************************* This subroutine restarts algorithm from new point. All optimization parameters (including constraints) are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure previously allocated with minnscreate() call. X - new starting point. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnsrestartfrom(minnsstate state, real_1d_array x);
minnsresults function/************************************************************************* MinNS results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report. You should check Rep.TerminationType in order to distinguish successful termination from unsuccessful one: * -8 internal integrity control detected infinite or NAN values in function/gradient. Abnormal termination signalled. * -3 box constraints are inconsistent * -1 inconsistent parameters were passed: * penalty parameter for minnssetalgoags() is zero, but we have nonlinear constraints set by minnssetnlc() * 2 sampling radius decreased below epsx * 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. * 8 User requested termination via minnsrequesttermination() -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnsresults( minnsstate state, real_1d_array& x, minnsreport& rep);
minnsresultsbuf function/************************************************************************* Buffered implementation of minnsresults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnsresultsbuf( minnsstate state, real_1d_array& x, minnsreport& rep);
minnssetalgoags function/************************************************************************* This function tells MinNS unit to use AGS (adaptive gradient sampling) algorithm for nonsmooth constrained optimization. This algorithm is a slight modification of one described in "An Adaptive Gradient Sampling Algorithm for Nonsmooth Optimization" by Frank E. Curtisy and Xiaocun Quez. This optimizer has following benefits and drawbacks: + robustness; it can be used with nonsmooth and nonconvex functions. + relatively easy tuning; most of the metaparameters are easy to select. - it has convergence of steepest descent, slower than CG/LBFGS. - each iteration involves evaluation of ~2N gradient values and solution of 2Nx2N quadratic programming problem, which limits applicability of algorithm by small-scale problems (up to 50-100). IMPORTANT: this algorithm has convergence guarantees, i.e. it will steadily move towards some stationary point of the function. However, "stationary point" does not always mean "solution". Nonsmooth problems often have "flat spots", i.e. areas where function do not change at all. Such "flat spots" are stationary points by definition, and algorithm may be caught here. Nonsmooth CONVEX tasks are not prone to this problem. Say, if your function has form f()=MAX(f0,f1,...), and f_i are convex, then f() is convex too and you have guaranteed convergence to solution. INPUT PARAMETERS: State - structure which stores algorithm state Radius - initial sampling radius, >=0. Internally multiplied by vector of per-variable scales specified by minnssetscale()). You should select relatively large sampling radius, roughly proportional to scaled length of the first steps of the algorithm. Something close to 0.1 in magnitude should be good for most problems. AGS solver can automatically decrease radius, so too large radius is not a problem (assuming that you won't choose so large radius that algorithm will sample function in too far away points, where gradient value is irrelevant). Too small radius won't cause algorithm to fail, but it may slow down algorithm (it may have to perform too short steps). Penalty - penalty coefficient for nonlinear constraints: * for problem with nonlinear constraints should be some problem-specific positive value, large enough that penalty term changes shape of the function. Starting from some problem-specific value penalty coefficient becomes large enough to exactly enforce nonlinear constraints; larger values do not improve precision. Increasing it too much may slow down convergence, so you should choose it carefully. * can be zero for problems WITHOUT nonlinear constraints (i.e. for unconstrained ones or ones with just box or linear constraints) * if you specify zero value for problem with at least one nonlinear constraint, algorithm will terminate with error code -1. ALGORITHM OUTLINE The very basic outline of unconstrained AGS algorithm is given below: 0. If sampling radius is below EpsX or we performed more then MaxIts iterations - STOP. 1. sample O(N) gradient values at random locations around current point; informally speaking, this sample is an implicit piecewise linear model of the function, although algorithm formulation does not mention that explicitly 2. solve quadratic programming problem in order to find descent direction 3. if QP solver tells us that we are near solution, decrease sampling radius and move to (0) 4. perform backtracking line search 5. after moving to new point, goto (0) As for the constraints: * box constraints are handled exactly by modification of the function being minimized * linear/nonlinear constraints are handled by adding L1 penalty. Because our solver can handle nonsmoothness, we can use L1 penalty function, which is an exact one (i.e. exact solution is returned under such penalty). * penalty coefficient for linear constraints is chosen automatically; however, penalty coefficient for nonlinear constraints must be specified by user. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnssetalgoags( minnsstate state, double radius, double penalty);
minnssetbc function/************************************************************************* This function sets boundary constraints. Boundary constraints are inactive by default (after initial creation). They are preserved after algorithm restart with minnsrestartfrom(). INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF. BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF. NOTE 1: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. NOTE 2: AGS solver has following useful properties: * bound constraints are always satisfied exactly * function is evaluated only INSIDE area specified by bound constraints, even when numerical differentiation is used (algorithm adjusts nodes according to boundary constraints) -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnssetbc( minnsstate state, real_1d_array bndl, real_1d_array bndu);
Examples: [1]
minnssetcond function/************************************************************************* This function sets stopping conditions for iterations of optimizer. INPUT PARAMETERS: State - structure which stores algorithm state EpsX - >=0 The AGS solver finishes its work if on k+1-th iteration sampling radius decreases below EpsX. MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection. We do not recommend you to rely on default choice in production code. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnssetcond(minnsstate state, double epsx, ae_int_t maxits);
minnssetlc function/************************************************************************* This function sets linear constraints. Linear constraints are inactive by default (after initial creation). They are preserved after algorithm restart with minnsrestartfrom(). INPUT PARAMETERS: State - structure previously allocated with minnscreate() call. C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT NOTE: linear (non-bound) constraints are satisfied only approximately: * there always exists some minor violation (about current sampling radius in magnitude during optimization, about EpsX in the solution) due to use of penalty method to handle constraints. * numerical differentiation, if used, may lead to function evaluations outside of the feasible area, because algorithm does NOT change numerical differentiation formula according to linear constraints. If you want constraints to be satisfied exactly, try to reformulate your problem in such manner that all constraints will become boundary ones (this kind of constraints is always satisfied exactly, both in the final solution and in all intermediate points). -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnssetlc( minnsstate state, real_2d_array c, integer_1d_array ct); void alglib::minnssetlc( minnsstate state, real_2d_array c, integer_1d_array ct, ae_int_t k);
minnssetnlc function/************************************************************************* This function sets nonlinear constraints. In fact, this function sets NUMBER of nonlinear constraints. Constraints itself (constraint functions) are passed to minnsoptimize() method. This method requires user-defined vector function F[] and its Jacobian J[], where: * first component of F[] and first row of Jacobian J[] correspond to function being minimized * next NLEC components of F[] (and rows of J) correspond to nonlinear equality constraints G_i(x)=0 * next NLIC components of F[] (and rows of J) correspond to nonlinear inequality constraints H_i(x)<=0 NOTE: you may combine nonlinear constraints with linear/boundary ones. If your problem has mixed constraints, you may explicitly specify some of them as linear ones. It may help optimizer to handle them more efficiently. INPUT PARAMETERS: State - structure previously allocated with minnscreate() call. NLEC - number of Non-Linear Equality Constraints (NLEC), >=0 NLIC - number of Non-Linear Inquality Constraints (NLIC), >=0 NOTE 1: nonlinear constraints are satisfied only approximately! It is possible that algorithm will evaluate function outside of the feasible area! NOTE 2: algorithm scales variables according to scale specified by minnssetscale() function, so it can handle problems with badly scaled variables (as long as we KNOW their scales). However, there is no way to automatically scale nonlinear constraints Gi(x) and Hi(x). Inappropriate scaling of Gi/Hi may ruin convergence. Solving problem with constraint "1000*G0(x)=0" is NOT same as solving it with constraint "0.001*G0(x)=0". It means that YOU are the one who is responsible for correct scaling of nonlinear constraints Gi(x) and Hi(x). We recommend you to scale nonlinear constraints in such way that I-th component of dG/dX (or dH/dx) has approximately unit magnitude (for problems with unit scale) or has magnitude approximately equal to 1/S[i] (where S is a scale set by minnssetscale() function). NOTE 3: nonlinear constraints are always hard to handle, no matter what algorithm you try to use. Even basic box/linear constraints modify function curvature by adding valleys and ridges. However, nonlinear constraints add valleys which are very hard to follow due to their "curved" nature. It means that optimization with single nonlinear constraint may be significantly slower than optimization with multiple linear ones. It is normal situation, and we recommend you to carefully choose Rho parameter of minnssetalgoags(), because too large value may slow down convergence. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnssetnlc(minnsstate state, ae_int_t nlec, ae_int_t nlic);
Examples: [1]
minnssetscale function/************************************************************************* This function sets scaling coefficients for NLC optimizer. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances). Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function Scaling is also used by finite difference variant of the optimizer - step along I-th axis is equal to DiffStep*S[I]. INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 18.05.2015 by Bochkanov Sergey *************************************************************************/void alglib::minnssetscale(minnsstate state, real_1d_array s);
minnssetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to minnsoptimize(). -- ALGLIB -- Copyright 28.11.2010 by Bochkanov Sergey *************************************************************************/void alglib::minnssetxrep(minnsstate state, bool needxrep);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nsfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates
//
// f0(x0,x1) = 2*|x0|+x1
//
// and Jacobian matrix J = [df0/dx0 df0/dx1]
//
fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
jac[0][0] = 2*alglib::sign(x[0]);
jac[0][1] = alglib::sign(x[1]);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = 2*|x0|+|x1|
//
// subject to box constraints
//
// 1 <= x0 < +INF
// -INF <= x1 < +INF
//
// using nonsmooth nonlinear optimizer.
//
real_1d_array x0 = "[1,1]";
real_1d_array s = "[1,1]";
real_1d_array bndl = "[1,-inf]";
real_1d_array bndu = "[+inf,+inf]";
double epsx = 0.00001;
double radius = 0.1;
double rho = 0.0;
ae_int_t maxits = 0;
minnsstate state;
minnsreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AGS algorithm and tune its settings:
// * radius=0.1 good initial value; will be automatically decreased later.
// * rho=0.0 penalty coefficient for nonlinear constraints; can be zero
// because we do not have such constraints
// * epsx=0.000001 stopping conditions
// * s=[1,1] all variables have unit scale
//
minnscreate(2, x0, state);
minnssetalgoags(state, radius, rho);
minnssetcond(state, epsx, maxits);
minnssetscale(state, s);
//
// Set box constraints.
//
// General linear constraints are set in similar way (see comments on
// minnssetlc() function for more information).
//
// You may combine box, linear and nonlinear constraints in one optimization
// problem.
//
minnssetbc(state, bndl, bndu);
//
// Optimize and test results.
//
// Optimizer object accepts vector function and its Jacobian, with first
// component (Jacobian row) being target function, and next components
// (Jacobian rows) being nonlinear equality and inequality constraints
// (box/linear ones are passed separately by means of minnssetbc() and
// minnssetlc() calls).
//
// If you do not have nonlinear constraints (exactly our situation), then
// you will have one-component function vector and 1xN Jacobian matrix.
//
// So, our vector function has form
//
// {f0} = { 2*|x0|+|x1| }
//
// with Jacobian
//
// [ ]
// J = [ 2*sign(x0) sign(x1) ]
// [ ]
//
// NOTE: nonsmooth optimizer requires considerably more function
// evaluations than smooth solver - about 2N times more. Using
// numerical differentiation introduces additional (multiplicative)
// 2N speedup.
//
// It means that if smooth optimizer WITH user-supplied gradient
// needs 100 function evaluations to solve 50-dimensional problem,
// then AGS solver with user-supplied gradient will need about 10.000
// function evaluations, and with numerical gradient about 1.000.000
// function evaluations will be performed.
//
// NOTE: AGS solver used by us can handle nonsmooth and nonconvex
// optimization problems. It has convergence guarantees, i.e. it will
// converge to stationary point of the function after running for some
// time.
//
// However, it is important to remember that "stationary point" is not
// equal to "solution". If your problem is convex, everything is OK.
// But nonconvex optimization problems may have "flat spots" - large
// areas where gradient is exactly zero, but function value is far away
// from optimal. Such areas are stationary points too, and optimizer
// may be trapped here.
//
// "Flat spots" are nonsmooth equivalent of the saddle points, but with
// orders of magnitude worse properties - they may be quite large and
// hard to avoid. All nonsmooth optimizers are prone to this kind of the
// problem, because it is impossible to automatically distinguish "flat
// spot" from true solution.
//
// This note is here to warn you that you should be very careful when
// you solve nonsmooth optimization problems. Visual inspection of
// results is essential.
//
//
alglib::minnsoptimize(state, nsfunc1_jac);
minnsresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nsfunc1_fvec(const real_1d_array &x, real_1d_array &fi, void *ptr)
{
//
// this callback calculates
//
// f0(x0,x1) = 2*|x0|+x1
//
fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = 2*|x0|+|x1|
//
// using nonsmooth nonlinear optimizer with numerical
// differentiation provided by ALGLIB.
//
// NOTE: nonsmooth optimizer requires considerably more function
// evaluations than smooth solver - about 2N times more. Using
// numerical differentiation introduces additional (multiplicative)
// 2N speedup.
//
// It means that if smooth optimizer WITH user-supplied gradient
// needs 100 function evaluations to solve 50-dimensional problem,
// then AGS solver with user-supplied gradient will need about 10.000
// function evaluations, and with numerical gradient about 1.000.000
// function evaluations will be performed.
//
real_1d_array x0 = "[1,1]";
real_1d_array s = "[1,1]";
double epsx = 0.00001;
double diffstep = 0.000001;
double radius = 0.1;
double rho = 0.0;
ae_int_t maxits = 0;
minnsstate state;
minnsreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AGS algorithm and tune its settings:
// * radius=0.1 good initial value; will be automatically decreased later.
// * rho=0.0 penalty coefficient for nonlinear constraints; can be zero
// because we do not have such constraints
// * epsx=0.000001 stopping conditions
// * s=[1,1] all variables have unit scale
//
minnscreatef(2, x0, diffstep, state);
minnssetalgoags(state, radius, rho);
minnssetcond(state, epsx, maxits);
minnssetscale(state, s);
//
// Optimize and test results.
//
// Optimizer object accepts vector function, with first component
// being target function, and next components being nonlinear equality
// and inequality constraints (box/linear ones are passed separately
// by means of minnssetbc() and minnssetlc() calls).
//
// If you do not have nonlinear constraints (exactly our situation), then
// you will have one-component function vector.
//
// So, our vector function has form
//
// {f0} = { 2*|x0|+|x1| }
//
alglib::minnsoptimize(state, nsfunc1_fvec);
minnsresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.0000,0.0000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nsfunc2_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates function vector
//
// f0(x0,x1) = 2*|x0|+x1
// f1(x0,x1) = x0-1
// f2(x0,x1) = -x1-1
//
// and Jacobian matrix J
//
// [ df0/dx0 df0/dx1 ]
// J = [ df1/dx0 df1/dx1 ]
// [ df2/dx0 df2/dx1 ]
//
fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
jac[0][0] = 2*alglib::sign(x[0]);
jac[0][1] = alglib::sign(x[1]);
fi[1] = x[0]-1;
jac[1][0] = 1;
jac[1][1] = 0;
fi[2] = -x[1]-1;
jac[2][0] = 0;
jac[2][1] = -1;
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = 2*|x0|+|x1|
//
// subject to combination of equality and inequality constraints
//
// x0 = 1
// x1 >= -1
//
// using nonsmooth nonlinear optimizer. Although these constraints
// are linear, we treat them as general nonlinear ones in order to
// demonstrate nonlinearly constrained optimization setup.
//
real_1d_array x0 = "[1,1]";
real_1d_array s = "[1,1]";
double epsx = 0.00001;
double radius = 0.1;
double rho = 50.0;
ae_int_t maxits = 0;
minnsstate state;
minnsreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AGS algorithm and tune its settings:
// * radius=0.1 good initial value; will be automatically decreased later.
// * rho=50.0 penalty coefficient for nonlinear constraints. It is your
// responsibility to choose good one - large enough that it
// enforces constraints, but small enough in order to avoid
// extreme slowdown due to ill-conditioning.
// * epsx=0.000001 stopping conditions
// * s=[1,1] all variables have unit scale
//
minnscreate(2, x0, state);
minnssetalgoags(state, radius, rho);
minnssetcond(state, epsx, maxits);
minnssetscale(state, s);
//
// Set general nonlinear constraints.
//
// This part is more tricky than working with box/linear constraints - you
// can not "pack" general nonlinear function into double precision array.
// That's why minnssetnlc() does not accept constraints itself - only
// constraint COUNTS are passed: first parameter is number of equality
// constraints, second one is number of inequality constraints.
//
// As for constraining functions - these functions are passed as part
// of problem Jacobian (see below).
//
// NOTE: MinNS optimizer supports arbitrary combination of boundary, general
// linear and general nonlinear constraints. This example does not
// show how to work with general linear constraints, but you can
// easily find it in documentation on minnlcsetlc() function.
//
minnssetnlc(state, 1, 1);
//
// Optimize and test results.
//
// Optimizer object accepts vector function and its Jacobian, with first
// component (Jacobian row) being target function, and next components
// (Jacobian rows) being nonlinear equality and inequality constraints
// (box/linear ones are passed separately by means of minnssetbc() and
// minnssetlc() calls).
//
// Nonlinear equality constraints have form Gi(x)=0, inequality ones
// have form Hi(x)<=0, so we may have to "normalize" constraints prior
// to passing them to optimizer (right side is zero, constraints are
// sorted, multiplied by -1 when needed).
//
// So, our vector function has form
//
// {f0,f1,f2} = { 2*|x0|+|x1|, x0-1, -x1-1 }
//
// with Jacobian
//
// [ 2*sign(x0) sign(x1) ]
// J = [ 1 0 ]
// [ 0 -1 ]
//
// which means that we have optimization problem
//
// min{f0} subject to f1=0, f2<=0
//
// which is essentially same as
//
// min { 2*|x0|+|x1| } subject to x0=1, x1>=-1
//
// NOTE: AGS solver used by us can handle nonsmooth and nonconvex
// optimization problems. It has convergence guarantees, i.e. it will
// converge to stationary point of the function after running for some
// time.
//
// However, it is important to remember that "stationary point" is not
// equal to "solution". If your problem is convex, everything is OK.
// But nonconvex optimization problems may have "flat spots" - large
// areas where gradient is exactly zero, but function value is far away
// from optimal. Such areas are stationary points too, and optimizer
// may be trapped here.
//
// "Flat spots" are nonsmooth equivalent of the saddle points, but with
// orders of magnitude worse properties - they may be quite large and
// hard to avoid. All nonsmooth optimizers are prone to this kind of the
// problem, because it is impossible to automatically distinguish "flat
// spot" from true solution.
//
// This note is here to warn you that you should be very careful when
// you solve nonsmooth optimization problems. Visual inspection of
// results is essential.
//
alglib::minnsoptimize(state, nsfunc2_jac);
minnsresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [1.0000,0.0000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
void nsfunc1_jac(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr)
{
//
// this callback calculates
//
// f0(x0,x1) = 2*|x0|+x1
//
// and Jacobian matrix J = [df0/dx0 df0/dx1]
//
fi[0] = 2*fabs(double(x[0]))+fabs(double(x[1]));
jac[0][0] = 2*alglib::sign(x[0]);
jac[0][1] = alglib::sign(x[1]);
}
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of
//
// f(x0,x1) = 2*|x0|+|x1|
//
// using nonsmooth nonlinear optimizer.
//
real_1d_array x0 = "[1,1]";
real_1d_array s = "[1,1]";
double epsx = 0.00001;
double radius = 0.1;
double rho = 0.0;
ae_int_t maxits = 0;
minnsstate state;
minnsreport rep;
real_1d_array x1;
//
// Create optimizer object, choose AGS algorithm and tune its settings:
// * radius=0.1 good initial value; will be automatically decreased later.
// * rho=0.0 penalty coefficient for nonlinear constraints; can be zero
// because we do not have such constraints
// * epsx=0.000001 stopping conditions
// * s=[1,1] all variables have unit scale
//
minnscreate(2, x0, state);
minnssetalgoags(state, radius, rho);
minnssetcond(state, epsx, maxits);
minnssetscale(state, s);
//
// Optimize and test results.
//
// Optimizer object accepts vector function and its Jacobian, with first
// component (Jacobian row) being target function, and next components
// (Jacobian rows) being nonlinear equality and inequality constraints
// (box/linear ones are passed separately by means of minnssetbc() and
// minnssetlc() calls).
//
// If you do not have nonlinear constraints (exactly our situation), then
// you will have one-component function vector and 1xN Jacobian matrix.
//
// So, our vector function has form
//
// {f0} = { 2*|x0|+|x1| }
//
// with Jacobian
//
// [ ]
// J = [ 2*sign(x0) sign(x1) ]
// [ ]
//
// NOTE: nonsmooth optimizer requires considerably more function
// evaluations than smooth solver - about 2N times more. Using
// numerical differentiation introduces additional (multiplicative)
// 2N speedup.
//
// It means that if smooth optimizer WITH user-supplied gradient
// needs 100 function evaluations to solve 50-dimensional problem,
// then AGS solver with user-supplied gradient will need about 10.000
// function evaluations, and with numerical gradient about 1.000.000
// function evaluations will be performed.
//
// NOTE: AGS solver used by us can handle nonsmooth and nonconvex
// optimization problems. It has convergence guarantees, i.e. it will
// converge to stationary point of the function after running for some
// time.
//
// However, it is important to remember that "stationary point" is not
// equal to "solution". If your problem is convex, everything is OK.
// But nonconvex optimization problems may have "flat spots" - large
// areas where gradient is exactly zero, but function value is far away
// from optimal. Such areas are stationary points too, and optimizer
// may be trapped here.
//
// "Flat spots" are nonsmooth equivalent of the saddle points, but with
// orders of magnitude worse properties - they may be quite large and
// hard to avoid. All nonsmooth optimizers are prone to this kind of the
// problem, because it is impossible to automatically distinguish "flat
// spot" from true solution.
//
// This note is here to warn you that you should be very careful when
// you solve nonsmooth optimization problems. Visual inspection of
// results is essential.
//
alglib::minnsoptimize(state, nsfunc1_jac);
minnsresults(state, x1, rep);
printf("%s\n", x1.tostring(2).c_str()); // EXPECTED: [0.0000,0.0000]
return 0;
}
minqp subpackage| minqp_d_bc1 | Bound constrained dense quadratic programming | |
| minqp_d_lc1 | Linearly constrained dense quadratic programming | |
| minqp_d_nonconvex | Nonconvex quadratic programming | |
| minqp_d_u1 | Unconstrained dense quadratic programming | |
| minqp_d_u2 | Unconstrained sparse quadratic programming |
minqpreport class/************************************************************************* This structure stores optimization report: * InnerIterationsCount number of inner iterations * OuterIterationsCount number of outer iterations * NCholesky number of Cholesky decomposition * NMV number of matrix-vector products (only products calculated as part of iterative process are counted) * TerminationType completion code (see below) Completion codes: * -5 inappropriate solver was used: * QuickQP solver for problem with general linear constraints (dense/sparse) * -4 BLEIC-QP or QuickQP solver found unconstrained direction of negative curvature (function is unbounded from below even under constraints), no meaningful minimum can be found. * -3 inconsistent constraints (or, maybe, feasible point is too hard to find). If you are sure that constraints are feasible, try to restart optimizer with better initial approximation. * -1 solver error * 1..4 successful completion * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible, X contains best point found so far. *************************************************************************/class minqpreport { ae_int_t inneriterationscount; ae_int_t outeriterationscount; ae_int_t nmv; ae_int_t ncholesky; ae_int_t terminationtype; };
minqpstate class/************************************************************************* This object stores nonlinear optimizer state. You should use functions provided by MinQP subpackage to work with this object *************************************************************************/class minqpstate { };
minqpcreate function/************************************************************************* CONSTRAINED QUADRATIC PROGRAMMING The subroutine creates QP optimizer. After initial creation, it contains default optimization problem with zero quadratic and linear terms and no constraints. You should set quadratic/linear terms with calls to functions provided by MinQP subpackage. You should also choose appropriate QP solver and set it and its stopping criteria by means of MinQPSetAlgo??????() function. Then, you should start solution process by means of MinQPOptimize() call. Solution itself can be obtained with MinQPResults() function. Following solvers are recommended: * QuickQP for dense problems with box-only constraints (or no constraints at all) * QP-BLEIC for dense/sparse problems with moderate (up to 50) number of general linear constraints * DENSE-AUL-QP for dense problems with any (small or large) number of general linear constraints INPUT PARAMETERS: N - problem size OUTPUT PARAMETERS: State - optimizer with zero quadratic/linear terms and no constraints -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpcreate(ae_int_t n, minqpstate& state);
minqpoptimize function/************************************************************************* This function solves quadratic programming problem. Prior to calling this function you should choose solver by means of one of the following functions: * minqpsetalgoquickqp() - for QuickQP solver * minqpsetalgobleic() - for BLEIC-QP solver * minqpsetalgodenseaul() - for Dense-AUL-QP solver These functions also allow you to control stopping criteria of the solver. If you did not set solver, MinQP subpackage will automatically select solver for your problem and will run it with default stopping criteria. However, it is better to set explicitly solver and its stopping criteria. INPUT PARAMETERS: State - algorithm state You should use MinQPResults() function to access results after calls to this function. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey. Special thanks to Elvira Illarionova for important suggestions on the linearly constrained QP algorithm. *************************************************************************/void alglib::minqpoptimize(minqpstate state);
minqpresults function/************************************************************************* QP solver results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: X - array[0..N-1], solution. This array is allocated and initialized only when Rep.TerminationType parameter is positive (success). Rep - optimization report. You should check Rep.TerminationType, which contains completion code, and you may check another fields which contain another information about algorithm functioning. Failure codes returned by algorithm are: * -5 inappropriate solver was used: * QuickQP solver for problem with general linear constraints * -4 BLEIC-QP/QuickQP solver found unconstrained direction of negative curvature (function is unbounded from below even under constraints), no meaningful minimum can be found. * -3 inconsistent constraints (or maybe feasible point is too hard to find). If you are sure that constraints are feasible, try to restart optimizer with better initial approximation. Completion codes specific for Cholesky algorithm: * 4 successful completion Completion codes specific for BLEIC/QuickQP algorithms: * 1 relative function improvement is no more than EpsF. * 2 scaled step is no more than EpsX. * 4 scaled gradient norm is no more than EpsG. * 5 MaxIts steps was taken -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpresults( minqpstate state, real_1d_array& x, minqpreport& rep);
minqpresultsbuf function/************************************************************************* QP results Buffered implementation of MinQPResults() which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpresultsbuf( minqpstate state, real_1d_array& x, minqpreport& rep);
minqpsetalgobleic function/************************************************************************* This function tells solver to use BLEIC-based algorithm and sets stopping criteria for the algorithm. ALGORITHM FEATURES: * supports dense and sparse QP problems * supports box and general linear equality/inequality constraints * can solve all types of problems (convex, semidefinite, nonconvex) as long as they are bounded from below under constraints. Say, it is possible to solve "min{-x^2} subject to -1<=x<=+1". Of course, global minimum is found only for positive definite and semidefinite problems. As for indefinite ones - only local minimum is found. ALGORITHM OUTLINE: * BLEIC-QP solver is just a driver function for MinBLEIC solver; it solves quadratic programming problem as general linearly constrained optimization problem, which is solved by means of BLEIC solver (part of ALGLIB, active set method). ALGORITHM LIMITATIONS: * this algorithm is fast enough for large-scale problems with small amount of general linear constraints (say, up to 50), but it is inefficient for problems with several hundreds of constraints. Iteration cost is roughly quadratic w.r.t. constraint count. Furthermore, it can not efficiently handle sparse constraints (they are converted to dense format prior to solution). Thus, if you have large and/or sparse constraint matrix and convex QP problem, Dense-AUL-QP solver may be better solution. * unlike QuickQP solver, this algorithm does not perform Newton steps and does not use Level 3 BLAS. Being general-purpose active set method, it can activate constraints only one-by-one. Thus, its performance is lower than that of QuickQP. * its precision is also a bit inferior to that of QuickQP. BLEIC-QP performs only LBFGS steps (no Newton steps), which are good at detecting neighborhood of the solution, buy needs many iterations to find solution with more than 6 digits of precision. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled constrained gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinQPSetScale() EpsF - >=0 The subroutine finishes its work if exploratory steepest descent step on k+1-th iteration satisfies following condition: |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} EpsX - >=0 The subroutine finishes its work if exploratory steepest descent step on k+1-th iteration satisfies following condition: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - step vector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinQPSetScale() MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. NOTE: this algorithm uses LBFGS iterations, which are relatively cheap, but improve function value only a bit. So you will need many iterations to converge - from 0.1*N to 10*N, depending on problem's condition number. IT IS VERY IMPORTANT TO CALL MinQPSetScale() WHEN YOU USE THIS ALGORITHM BECAUSE ITS STOPPING CRITERIA ARE SCALE-DEPENDENT! Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (presently it is small step length, but it may change in the future versions of ALGLIB). -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetalgobleic( minqpstate state, double epsg, double epsf, double epsx, ae_int_t maxits);
minqpsetalgocholesky function/************************************************************************* DEPRECATED DEPRECATED DEPRECATED DEPRECATED DEPRECATED DEPRECATED This function tells solver to use Cholesky-based algorithm. This algorithm was deprecated in ALGLIB 3.9.0 because its performance is inferior to that of BLEIC-QP or QuickQP on high-dimensional problems. Furthermore, it supports only dense convex QP problems. This solver is no longer active by default. We recommend you to switch to AUL-QP, BLEIC-QP or QuickQP solver. DEPRECATED DEPRECATED DEPRECATED DEPRECATED DEPRECATED DEPRECATED -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetalgocholesky(minqpstate state);
minqpsetalgodenseaul function/************************************************************************* This function tells QP solver to use Dense-AUL algorithm and sets stopping criteria for the algorithm. ALGORITHM FEATURES: * supports dense and sparse QP problems; although it uses dense Cholesky to build preconditioner, it still works faster for sparse problems. * supports box and dense/sparse general linear equality/inequality constraints * convergence is theoretically proved for positive-definite (convex) QP problems. Semidefinite and non-convex problems can be solved as long as they are bounded from below under constraints, although without theoretical guarantees. * this solver is better than QP-BLEIC on problems with large number of general linear constraints. ALGORITHM OUTLINE: * this algorithm is an augmented Lagrangian method with dense preconditioner (hence its name). It is similar to barrier/penalty methods, but much more precise and faster. * it performs several outer iterations in order to refine values of the Lagrange multipliers. Single outer iteration is a solution of some unconstrained optimization problem: first it performs dense Cholesky factorization of the Hessian in order to build preconditioner (adaptive regularization is applied to enforce positive definiteness), and then it uses L-BFGS optimizer to solve optimization problem. * typically you need about 5-10 outer iterations to converge to solution ALGORITHM LIMITATIONS: * because dense Cholesky driver is used, this algorithm has O(N^2) memory requirements and O(OuterIterations*N^3) minimum running time. From the practical point of view, it limits its applicability by several thousands of variables. From the other side, variables count is the most limiting factor, and dependence on constraint count is much more lower. Assuming that constraint matrix is sparse, it may handle tens of thousands of general linear constraints. * its precision is lower than that of BLEIC-QP and QuickQP. It is hard to find solution with more than 6 digits of precision. INPUT PARAMETERS: State - structure which stores algorithm state EpsX - >=0, stopping criteria for inner optimizer. Inner iterations are stopped when step length (with variable scaling being applied) is less than EpsX. See minqpsetscale() for more information on variable scaling. Rho - penalty coefficient, Rho>0: * large enough that algorithm converges with desired precision. * not TOO large to prevent ill-conditioning * recommended values are 100, 1000 or 10000 ItsCnt - number of outer iterations: * recommended values: 10-15 (although in most cases it converges within 5 iterations, you may need a few more to be sure). * ItsCnt=0 means that small number of outer iterations is automatically chosen (10 iterations in current version). * ItsCnt=1 means that AUL algorithm performs just as usual barrier method. * ItsCnt>1 means that AUL algorithm performs specified number of outer iterations IT IS VERY IMPORTANT TO CALL minqpsetscale() WHEN YOU USE THIS ALGORITHM BECAUSE ITS CONVERGENCE PROPERTIES AND STOPPING CRITERIA ARE SCALE-DEPENDENT! NOTE: Passing EpsX=0 will lead to automatic step length selection (specific step length chosen may change in the future versions of ALGLIB, so it is better to specify step length explicitly). -- ALGLIB -- Copyright 20.08.2016 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetalgodenseaul( minqpstate state, double epsx, double rho, ae_int_t itscnt);
minqpsetalgoquickqp function/************************************************************************* This function tells solver to use QuickQP algorithm: special extra-fast algorithm for problems with box-only constrants. It may solve non-convex problems as long as they are bounded from below under constraints. ALGORITHM FEATURES: * many times (from 5x to 50x!) faster than BLEIC-based QP solver; utilizes accelerated methods for activation of constraints. * supports dense and sparse QP problems * supports ONLY box constraints; general linear constraints are NOT supported by this solver * can solve all types of problems (convex, semidefinite, nonconvex) as long as they are bounded from below under constraints. Say, it is possible to solve "min{-x^2} subject to -1<=x<=+1". In convex/semidefinite case global minimum is returned, in nonconvex case - algorithm returns one of the local minimums. ALGORITHM OUTLINE: * algorithm performs two kinds of iterations: constrained CG iterations and constrained Newton iterations * initially it performs small number of constrained CG iterations, which can efficiently activate/deactivate multiple constraints * after CG phase algorithm tries to calculate Cholesky decomposition and to perform several constrained Newton steps. If Cholesky decomposition failed (matrix is indefinite even under constraints), we perform more CG iterations until we converge to such set of constraints that system matrix becomes positive definite. Constrained Newton steps greatly increase convergence speed and precision. * algorithm interleaves CG and Newton iterations which allows to handle indefinite matrices (CG phase) and quickly converge after final set of constraints is found (Newton phase). Combination of CG and Newton phases is called "outer iteration". * it is possible to turn off Newton phase (beneficial for semidefinite problems - Cholesky decomposition will fail too often) ALGORITHM LIMITATIONS: * algorithm does not support general linear constraints; only box ones are supported * Cholesky decomposition for sparse problems is performed with Skyline Cholesky solver, which is intended for low-profile matrices. No profile- reducing reordering of variables is performed in this version of ALGLIB. * problems with near-zero negative eigenvalues (or exacty zero ones) may experience about 2-3x performance penalty. The reason is that Cholesky decomposition can not be performed until we identify directions of zero and negative curvature and activate corresponding boundary constraints - but we need a lot of trial and errors because these directions are hard to notice in the matrix spectrum. In this case you may turn off Newton phase of algorithm. Large negative eigenvalues are not an issue, so highly non-convex problems can be solved very efficiently. INPUT PARAMETERS: State - structure which stores algorithm state EpsG - >=0 The subroutine finishes its work if the condition |v|<EpsG is satisfied, where: * |.| means Euclidian norm * v - scaled constrained gradient vector, v[i]=g[i]*s[i] * g - gradient * s - scaling coefficients set by MinQPSetScale() EpsF - >=0 The subroutine finishes its work if exploratory steepest descent step on k+1-th iteration satisfies following condition: |F(k+1)-F(k)|<=EpsF*max{|F(k)|,|F(k+1)|,1} EpsX - >=0 The subroutine finishes its work if exploratory steepest descent step on k+1-th iteration satisfies following condition: * |.| means Euclidian norm * v - scaled step vector, v[i]=dx[i]/s[i] * dx - step vector, dx=X(k+1)-X(k) * s - scaling coefficients set by MinQPSetScale() MaxOuterIts-maximum number of OUTER iterations. One outer iteration includes some amount of CG iterations (from 5 to ~N) and one or several (usually small amount) Newton steps. Thus, one outer iteration has high cost, but can greatly reduce funcation value. Use 0 if you do not want to limit number of outer iterations. UseNewton- use Newton phase or not: * Newton phase improves performance of positive definite dense problems (about 2 times improvement can be observed) * can result in some performance penalty on semidefinite or slightly negative definite problems - each Newton phase will bring no improvement (Cholesky failure), but still will require computational time. * if you doubt, you can turn off this phase - optimizer will retain its most of its high speed. IT IS VERY IMPORTANT TO CALL MinQPSetScale() WHEN YOU USE THIS ALGORITHM BECAUSE ITS STOPPING CRITERIA ARE SCALE-DEPENDENT! Passing EpsG=0, EpsF=0 and EpsX=0 and MaxIts=0 (simultaneously) will lead to automatic stopping criterion selection (presently it is small step length, but it may change in the future versions of ALGLIB). -- ALGLIB -- Copyright 22.05.2014 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetalgoquickqp( minqpstate state, double epsg, double epsf, double epsx, ae_int_t maxouterits, bool usenewton);
minqpsetbc function/************************************************************************* This function sets box constraints for QP solver Box constraints are inactive by default (after initial creation). After being set, they are preserved until explicitly turned off with another SetBC() call. All QP solvers may handle box constraints. INPUT PARAMETERS: State - structure stores algorithm state BndL - lower bounds, array[N]. If some (all) variables are unbounded, you may specify very small number or -INF (latter is recommended because it will allow solver to use better algorithm). BndU - upper bounds, array[N]. If some (all) variables are unbounded, you may specify very large number or +INF (latter is recommended because it will allow solver to use better algorithm). NOTE: it is possible to specify BndL[i]=BndU[i]. In this case I-th variable will be "frozen" at X[i]=BndL[i]=BndU[i]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetbc( minqpstate state, real_1d_array bndl, real_1d_array bndu);
Examples: [1]
minqpsetlc function/************************************************************************* This function sets dense linear constraints for QP optimizer. This function overrides results of previous calls to minqpsetlc(), minqpsetlcsparse() and minqpsetlcmixed(). After call to this function sparse constraints are dropped, and you have only those constraints which were specified in the present call. If you want to specify mixed (with dense and sparse terms) linear constraints, you should call minqpsetlcmixed(). SUPPORT BY QP SOLVERS: Following QP solvers can handle dense linear constraints: * BLEIC-QP - handles them with high precision, but may be inefficient for problems with hundreds of constraints * Dense-AUL-QP - handles them with moderate precision (approx. 10^-6), may efficiently handle thousands of constraints. Following QP solvers can NOT handle dense linear constraints: * QuickQP - can not handle general linear constraints INPUT PARAMETERS: State - structure previously allocated with MinQPCreate call. C - linear constraints, array[K,N+1]. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0: * if given, only leading K elements of C/CT are used * if not given, automatically determined from sizes of C/CT NOTE 1: linear (non-bound) constraints are satisfied only approximately - there always exists some violation due to numerical errors and algorithmic limitations (BLEIC-QP solver is most precise, AUL-QP solver is less precise). -- ALGLIB -- Copyright 19.06.2012 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetlc( minqpstate state, real_2d_array c, integer_1d_array ct); void alglib::minqpsetlc( minqpstate state, real_2d_array c, integer_1d_array ct, ae_int_t k);
Examples: [1]
minqpsetlcmixed function/************************************************************************* This function sets mixed linear constraints, which include a set of dense rows, and a set of sparse rows. This function overrides results of previous calls to minqpsetlc(), minqpsetlcsparse() and minqpsetlcmixed(). This function may be useful if constraint matrix includes large number of both types of rows - dense and sparse. If you have just a few sparse rows, you may represent them in dense format without loosing performance. Similarly, if you have just a few dense rows, you may store them in sparse format with almost same performance. SUPPORT BY QP SOLVERS: Following QP solvers can handle mixed dense/sparse linear constraints: * BLEIC-QP - handles them with high precision, but can not utilize their sparsity - sparse constraint matrix is silently converted to dense format. Thus, it may be inefficient for problems with hundreds of constraints. * Dense-AUL-QP - although this solver uses dense linear algebra to calculate Cholesky preconditioner, it may efficiently handle sparse constraints. It may solve problems with hundreds and thousands of constraints. The only drawback is that precision of constraint handling is typically within 1E-4... ..1E-6 range. Following QP solvers can NOT handle mixed linear constraints: * QuickQP - can not handle general linear constraints at all INPUT PARAMETERS: State - structure previously allocated with MinQPCreate call. DenseC - dense linear constraints, array[K,N+1]. Each row of DenseC represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of DenseC (including right part) must be finite. DenseCT - type of constraints, array[K]: * if DenseCT[i]>0, then I-th constraint is DenseC[i,*]*x >= DenseC[i,n+1] * if DenseCT[i]=0, then I-th constraint is DenseC[i,*]*x = DenseC[i,n+1] * if DenseCT[i]<0, then I-th constraint is DenseC[i,*]*x <= DenseC[i,n+1] DenseK - number of equality/inequality constraints, DenseK>=0 SparseC - linear constraints, sparse matrix with dimensions at least [SparseK,N+1]. If matrix has larger size, only leading SPARSEKx(N+1) rectangle is used. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. SparseCT- type of sparse constraints, array[K]: * if SparseCT[i]>0, then I-th constraint is SparseC[i,*]*x >= SparseC[i,n+1] * if SparseCT[i]=0, then I-th constraint is SparseC[i,*]*x = SparseC[i,n+1] * if SparseCT[i]<0, then I-th constraint is SparseC[i,*]*x <= SparseC[i,n+1] SparseK - number of sparse equality/inequality constraints, K>=0 NOTE 1: linear (non-bound) constraints are satisfied only approximately - there always exists some violation due to numerical errors and algorithmic limitations (BLEIC-QP solver is most precise, AUL-QP solver is less precise). -- ALGLIB -- Copyright 22.08.2016 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetlcmixed( minqpstate state, real_2d_array densec, integer_1d_array densect, ae_int_t densek, sparsematrix sparsec, integer_1d_array sparsect, ae_int_t sparsek);
minqpsetlcsparse function/************************************************************************* This function sets sparse linear constraints for QP optimizer. This function overrides results of previous calls to minqpsetlc(), minqpsetlcsparse() and minqpsetlcmixed(). After call to this function dense constraints are dropped, and you have only those constraints which were specified in the present call. If you want to specify mixed (with dense and sparse terms) linear constraints, you should call minqpsetlcmixed(). SUPPORT BY QP SOLVERS: Following QP solvers can handle sparse linear constraints: * BLEIC-QP - handles them with high precision, but can not utilize their sparsity - sparse constraint matrix is silently converted to dense format. Thus, it may be inefficient for problems with hundreds of constraints. * Dense-AUL-QP - although this solver uses dense linear algebra to calculate Cholesky preconditioner, it may efficiently handle sparse constraints. It may solve problems with hundreds and thousands of constraints. The only drawback is that precision of constraint handling is typically within 1E-4... ..1E-6 range. Following QP solvers can NOT handle sparse linear constraints: * QuickQP - can not handle general linear constraints INPUT PARAMETERS: State - structure previously allocated with MinQPCreate call. C - linear constraints, sparse matrix with dimensions at least [K,N+1]. If matrix has larger size, only leading Kx(N+1) rectangle is used. Each row of C represents one constraint, either equality or inequality (see below): * first N elements correspond to coefficients, * last element corresponds to the right part. All elements of C (including right part) must be finite. CT - type of constraints, array[K]: * if CT[i]>0, then I-th constraint is C[i,*]*x >= C[i,n+1] * if CT[i]=0, then I-th constraint is C[i,*]*x = C[i,n+1] * if CT[i]<0, then I-th constraint is C[i,*]*x <= C[i,n+1] K - number of equality/inequality constraints, K>=0 NOTE 1: linear (non-bound) constraints are satisfied only approximately - there always exists some violation due to numerical errors and algorithmic limitations (BLEIC-QP solver is most precise, AUL-QP solver is less precise). -- ALGLIB -- Copyright 22.08.2016 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetlcsparse( minqpstate state, sparsematrix c, integer_1d_array ct, ae_int_t k);
minqpsetlinearterm function/************************************************************************* This function sets linear term for QP solver. By default, linear term is zero. INPUT PARAMETERS: State - structure which stores algorithm state B - linear term, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetlinearterm(minqpstate state, real_1d_array b);
minqpsetorigin function/************************************************************************* This function sets origin for QP solver. By default, following QP program is solved: min(0.5*x'*A*x+b'*x) This function allows to solve different problem: min(0.5*(x-x_origin)'*A*(x-x_origin)+b'*(x-x_origin)) Specification of non-zero origin affects function being minimized, but not constraints. Box and linear constraints are still calculated without origin. INPUT PARAMETERS: State - structure which stores algorithm state XOrigin - origin, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetorigin(minqpstate state, real_1d_array xorigin);
minqpsetquadraticterm function/************************************************************************* This function sets dense quadratic term for QP solver. By default, quadratic term is zero. SUPPORT BY QP SOLVERS: Dense quadratic term can be handled by following QP solvers: * QuickQP * BLEIC-QP * Dense-AUL-QP IMPORTANT: This solver minimizes following function: f(x) = 0.5*x'*A*x + b'*x. Note that quadratic term has 0.5 before it. So if you want to minimize f(x) = x^2 + x you should rewrite your problem as follows: f(x) = 0.5*(2*x^2) + x and your matrix A will be equal to [[2.0]], not to [[1.0]] INPUT PARAMETERS: State - structure which stores algorithm state A - matrix, array[N,N] IsUpper - (optional) storage type: * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used * if not given, both lower and upper triangles must be filled. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetquadraticterm(minqpstate state, real_2d_array a); void alglib::minqpsetquadraticterm( minqpstate state, real_2d_array a, bool isupper);
minqpsetquadratictermsparse function/************************************************************************* This function sets sparse quadratic term for QP solver. By default, quadratic term is zero. This function overrides previous calls to minqpsetquadraticterm() or minqpsetquadratictermsparse(). SUPPORT BY QP SOLVERS: Sparse quadratic term can be handled by following QP solvers: * QuickQP * BLEIC-QP * Dense-AUL-QP (internally converts sparse matrix to dense format) IMPORTANT: This solver minimizes following function: f(x) = 0.5*x'*A*x + b'*x. Note that quadratic term has 0.5 before it. So if you want to minimize f(x) = x^2 + x you should rewrite your problem as follows: f(x) = 0.5*(2*x^2) + x and your matrix A will be equal to [[2.0]], not to [[1.0]] INPUT PARAMETERS: State - structure which stores algorithm state A - matrix, array[N,N] IsUpper - (optional) storage type: * if True, symmetric matrix A is given by its upper triangle, and the lower triangle isn't used * if False, symmetric matrix A is given by its lower triangle, and the upper triangle isn't used * if not given, both lower and upper triangles must be filled. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetquadratictermsparse( minqpstate state, sparsematrix a, bool isupper);
Examples: [1]
minqpsetscale function/************************************************************************* This function sets scaling coefficients. ALGLIB optimizers use scaling matrices to test stopping conditions (step size and gradient are scaled before comparison with tolerances) and as preconditioner. Scale of the I-th variable is a translation invariant measure of: a) "how large" the variable is b) how large the step should be to make significant changes in the function INPUT PARAMETERS: State - structure stores algorithm state S - array[N], non-zero scaling coefficients S[i] may be negative, sign doesn't matter. -- ALGLIB -- Copyright 14.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetscale(minqpstate state, real_1d_array s);
minqpsetstartingpoint function/************************************************************************* This function sets starting point for QP solver. It is useful to have good initial approximation to the solution, because it will increase speed of convergence and identification of active constraints. INPUT PARAMETERS: State - structure which stores algorithm state X - starting point, array[N]. -- ALGLIB -- Copyright 11.01.2011 by Bochkanov Sergey *************************************************************************/void alglib::minqpsetstartingpoint(minqpstate state, real_1d_array x);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
// subject to bound constraints 0<=x0<=2.5, 0<=x1<=2.5
//
// Exact solution is [x0,x1] = [2.5,2]
//
// We provide algorithm with starting point. With such small problem good starting
// point is not really necessary, but with high-dimensional problem it can save us
// a lot of time.
//
// Several QP solvers are tried: QuickQP, BLEIC, DENSE-AUL.
//
// IMPORTANT: this solver minimizes following function:
// f(x) = 0.5*x'*A*x + b'*x.
// Note that quadratic term has 0.5 before it. So if you want to minimize
// quadratic function, you should rewrite it in such way that quadratic term
// is multiplied by 0.5 too.
// For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as
// f(x) = 0.5*(2*x0^2+2*x1^2) + ....
// and pass diag(2,2) as quadratic term - NOT diag(1,1)!
//
real_2d_array a = "[[2,0],[0,2]]";
real_1d_array b = "[-6,-4]";
real_1d_array x0 = "[0,1]";
real_1d_array s = "[1,1]";
real_1d_array bndl = "[0.0,0.0]";
real_1d_array bndu = "[2.5,2.5]";
real_1d_array x;
minqpstate state;
minqpreport rep;
// create solver, set quadratic/linear terms
minqpcreate(2, state);
minqpsetquadraticterm(state, a);
minqpsetlinearterm(state, b);
minqpsetstartingpoint(state, x0);
minqpsetbc(state, bndl, bndu);
// Set scale of the parameters.
// It is strongly recommended that you set scale of your variables.
// Knowing their scales is essential for evaluation of stopping criteria
// and for preconditioning of the algorithm steps.
// You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
minqpsetscale(state, s);
//
// Solve problem with QuickQP solver.
//
// This solver is intended for medium and large-scale problems with box
// constraints (general linear constraints are not supported).
//
// Default stopping criteria are used, Newton phase is active.
//
minqpsetalgoquickqp(state, 0.0, 0.0, 0.0, 0, true);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 4
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]
//
// Solve problem with BLEIC-based QP solver.
//
// This solver is intended for problems with moderate (up to 50) number
// of general linear constraints and unlimited number of box constraints.
//
// Default stopping criteria are used.
//
minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]
//
// Solve problem with DENSE-AUL solver.
//
// This solver is optimized for problems with up to several thousands of
// variables and large amount of general linear constraints. Problems with
// less than 50 general linear constraints can be efficiently solved with
// BLEIC, problems with box-only constraints can be solved with QuickQP.
// However, DENSE-AUL will work in any (including unconstrained) case.
//
// Default stopping criteria are used.
//
minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2.5,2]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
// subject to linear constraint x0+x1<=2
//
// Exact solution is [x0,x1] = [1.5,0.5]
//
// IMPORTANT: this solver minimizes following function:
// f(x) = 0.5*x'*A*x + b'*x.
// Note that quadratic term has 0.5 before it. So if you want to minimize
// quadratic function, you should rewrite it in such way that quadratic term
// is multiplied by 0.5 too.
// For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as
// f(x) = 0.5*(2*x0^2+2*x1^2) + ....
// and pass diag(2,2) as quadratic term - NOT diag(1,1)!
//
real_2d_array a = "[[2,0],[0,2]]";
real_1d_array b = "[-6,-4]";
real_1d_array s = "[1,1]";
real_2d_array c = "[[1.0,1.0,2.0]]";
integer_1d_array ct = "[-1]";
real_1d_array x;
minqpstate state;
minqpreport rep;
// create solver, set quadratic/linear terms
minqpcreate(2, state);
minqpsetquadraticterm(state, a);
minqpsetlinearterm(state, b);
minqpsetlc(state, c, ct);
// Set scale of the parameters.
// It is strongly recommended that you set scale of your variables.
// Knowing their scales is essential for evaluation of stopping criteria
// and for preconditioning of the algorithm steps.
// You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
minqpsetscale(state, s);
//
// Solve problem with BLEIC-based QP solver.
//
// This solver is intended for problems with moderate (up to 50) number
// of general linear constraints and unlimited number of box constraints.
//
// Default stopping criteria are used.
//
minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(1).c_str()); // EXPECTED: [1.500,0.500]
//
// Solve problem with DENSE-AUL solver.
//
// This solver is optimized for problems with up to several thousands of
// variables and large amount of general linear constraints. Problems with
// less than 50 general linear constraints can be efficiently solved with
// BLEIC, problems with box-only constraints can be solved with QuickQP.
// However, DENSE-AUL will work in any (including unconstrained) case.
//
// Default stopping criteria are used.
//
minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(1).c_str()); // EXPECTED: [1.500,0.500]
//
// Solve problem with QuickQP solver.
//
// This solver is intended for medium and large-scale problems with box
// constraints, and...
//
// ...Oops! It does not support general linear constraints, -5 returned as completion code!
//
minqpsetalgoquickqp(state, 0.0, 0.0, 0.0, 0, true);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: -5
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of nonconvex function
// F(x0,x1) = -(x0^2+x1^2)
// subject to constraints x0,x1 in [1.0,2.0]
// Exact solution is [x0,x1] = [2,2].
//
// Non-convex problems are harded to solve than convex ones, and they
// may have more than one local minimum. However, ALGLIB solves may deal
// with such problems (altough they do not guarantee convergence to
// global minimum).
//
// IMPORTANT: this solver minimizes following function:
// f(x) = 0.5*x'*A*x + b'*x.
// Note that quadratic term has 0.5 before it. So if you want to minimize
// quadratic function, you should rewrite it in such way that quadratic term
// is multiplied by 0.5 too.
//
// For example, our function is f(x)=-(x0^2+x1^2), but we rewrite it as
// f(x) = 0.5*(-2*x0^2-2*x1^2)
// and pass diag(-2,-2) as quadratic term - NOT diag(-1,-1)!
//
real_2d_array a = "[[-2,0],[0,-2]]";
real_1d_array x0 = "[1,1]";
real_1d_array s = "[1,1]";
real_1d_array bndl = "[1.0,1.0]";
real_1d_array bndu = "[2.0,2.0]";
real_1d_array x;
minqpstate state;
minqpreport rep;
// create solver, set quadratic/linear terms, constraints
minqpcreate(2, state);
minqpsetquadraticterm(state, a);
minqpsetstartingpoint(state, x0);
minqpsetbc(state, bndl, bndu);
// Set scale of the parameters.
// It is strongly recommended that you set scale of your variables.
// Knowing their scales is essential for evaluation of stopping criteria
// and for preconditioning of the algorithm steps.
// You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
minqpsetscale(state, s);
//
// Solve problem with BLEIC-based QP solver.
//
// This solver is intended for problems with moderate (up to 50) number
// of general linear constraints and unlimited number of box constraints.
//
// It may solve non-convex problems as long as they are bounded from
// below under constraints.
//
// Default stopping criteria are used.
//
minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,2]
//
// Solve problem with DENSE-AUL solver.
//
// This solver is optimized for problems with up to several thousands of
// variables and large amount of general linear constraints. Problems with
// less than 50 general linear constraints can be efficiently solved with
// BLEIC, problems with box-only constraints can be solved with QuickQP.
// However, DENSE-AUL will work in any (including unconstrained) case.
//
// Algorithm convergence is guaranteed only for convex case, but you may
// expect that it will work for non-convex problems too (because near the
// solution they are locally convex).
//
// Default stopping criteria are used.
//
minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [2,2]
// Hmm... this problem is bounded from below (has solution) only under constraints.
// What it we remove them?
//
// You may see that BLEIC algorithm detects unboundedness of the problem,
// -4 is returned as completion code. However, DENSE-AUL is unable to detect
// such situation and it will cycle forever (we do not test it here).
real_1d_array nobndl = "[-inf,-inf]";
real_1d_array nobndu = "[+inf,+inf]";
minqpsetbc(state, nobndl, nobndu);
minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: -4
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1
//
// Exact solution is [x0,x1] = [3,2]
//
// We provide algorithm with starting point, although in this case
// (dense matrix, no constraints) it can work without such information.
//
// Several QP solvers are tried: QuickQP, BLEIC, DENSE-AUL.
//
// IMPORTANT: this solver minimizes following function:
// f(x) = 0.5*x'*A*x + b'*x.
// Note that quadratic term has 0.5 before it. So if you want to minimize
// quadratic function, you should rewrite it in such way that quadratic term
// is multiplied by 0.5 too.
//
// For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as
// f(x) = 0.5*(2*x0^2+2*x1^2) + ....
// and pass diag(2,2) as quadratic term - NOT diag(1,1)!
//
real_2d_array a = "[[2,0],[0,2]]";
real_1d_array b = "[-6,-4]";
real_1d_array x0 = "[0,1]";
real_1d_array s = "[1,1]";
real_1d_array x;
minqpstate state;
minqpreport rep;
// create solver, set quadratic/linear terms
minqpcreate(2, state);
minqpsetquadraticterm(state, a);
minqpsetlinearterm(state, b);
minqpsetstartingpoint(state, x0);
// Set scale of the parameters.
// It is strongly recommended that you set scale of your variables.
// Knowing their scales is essential for evaluation of stopping criteria
// and for preconditioning of the algorithm steps.
// You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
minqpsetscale(state, s);
//
// Solve problem with QuickQP solver.
//
// This solver is intended for medium and large-scale problems with box
// constraints (general linear constraints are not supported), but it can
// also be efficiently used on unconstrained problems.
//
// Default stopping criteria are used, Newton phase is active.
//
minqpsetalgoquickqp(state, 0.0, 0.0, 0.0, 0, true);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
//
// Solve problem with BLEIC-based QP solver.
//
// This solver is intended for problems with moderate (up to 50) number
// of general linear constraints and unlimited number of box constraints.
// Of course, unconstrained problems can be solved too.
//
// Default stopping criteria are used.
//
minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
//
// Solve problem with DENSE-AUL solver.
//
// This solver is optimized for problems with up to several thousands of
// variables and large amount of general linear constraints. Problems with
// less than 50 general linear constraints can be efficiently solved with
// BLEIC, problems with box-only constraints can be solved with QuickQP.
// However, DENSE-AUL will work in any (including unconstrained) case.
//
// Default stopping criteria are used.
//
minqpsetalgodenseaul(state, 1.0e-9, 1.0e+4, 5);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "optimization.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates minimization of F(x0,x1) = x0^2 + x1^2 -6*x0 - 4*x1,
// with quadratic term given by sparse matrix structure.
//
// Exact solution is [x0,x1] = [3,2]
//
// We provide algorithm with starting point, although in this case
// (dense matrix, no constraints) it can work without such information.
//
// IMPORTANT: this solver minimizes following function:
// f(x) = 0.5*x'*A*x + b'*x.
// Note that quadratic term has 0.5 before it. So if you want to minimize
// quadratic function, you should rewrite it in such way that quadratic term
// is multiplied by 0.5 too.
//
// For example, our function is f(x)=x0^2+x1^2+..., but we rewrite it as
// f(x) = 0.5*(2*x0^2+2*x1^2) + ....
// and pass diag(2,2) as quadratic term - NOT diag(1,1)!
//
sparsematrix a;
real_1d_array b = "[-6,-4]";
real_1d_array x0 = "[0,1]";
real_1d_array s = "[1,1]";
real_1d_array x;
minqpstate state;
minqpreport rep;
// initialize sparsematrix structure
sparsecreate(2, 2, 0, a);
sparseset(a, 0, 0, 2.0);
sparseset(a, 1, 1, 2.0);
// create solver, set quadratic/linear terms
minqpcreate(2, state);
minqpsetquadratictermsparse(state, a, true);
minqpsetlinearterm(state, b);
minqpsetstartingpoint(state, x0);
// Set scale of the parameters.
// It is strongly recommended that you set scale of your variables.
// Knowing their scales is essential for evaluation of stopping criteria
// and for preconditioning of the algorithm steps.
// You can find more information on scaling at http://www.alglib.net/optimization/scaling.php
minqpsetscale(state, s);
//
// Solve problem with BLEIC-based QP solver.
//
// This solver is intended for problems with moderate (up to 50) number
// of general linear constraints and unlimited number of box constraints.
// It also supports sparse problems.
//
// Default stopping criteria are used.
//
minqpsetalgobleic(state, 0.0, 0.0, 0.0, 0);
minqpoptimize(state);
minqpresults(state, x, rep);
printf("%s\n", x.tostring(2).c_str()); // EXPECTED: [3,2]
return 0;
}
mlpbase subpackagemodelerrors class/************************************************************************* Model's errors: * RelCLSError - fraction of misclassified cases. * AvgCE - acerage cross-entropy * RMSError - root-mean-square error * AvgError - average error * AvgRelError - average relative error NOTE 1: RelCLSError/AvgCE are zero on regression problems. NOTE 2: on classification problems RMSError/AvgError/AvgRelError contain errors in prediction of posterior probabilities *************************************************************************/class modelerrors { double relclserror; double avgce; double rmserror; double avgerror; double avgrelerror; };
multilayerperceptron class/************************************************************************* *************************************************************************/class multilayerperceptron { };
mlpactivationfunction function/************************************************************************* Neural network activation function INPUT PARAMETERS: NET - neuron input K - function index (zero for linear function) OUTPUT PARAMETERS: F - function DF - its derivative D2F - its second derivative -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpactivationfunction( double net, ae_int_t k, double& f, double& df, double& d2f);
mlpallerrorssparsesubset function/************************************************************************* Calculation of all types of errors on subset of dataset. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - original dataset given by sparse matrix; one sample = one row; first NIn columns contain inputs, next NOut columns - desired outputs. SetSize - real size of XY, SetSize>=0; Subset - subset of SubsetSize elements, array[SubsetSize]; SubsetSize- number of elements in Subset[] array: * if SubsetSize>0, rows of XY with indices Subset[0]... ...Subset[SubsetSize-1] are processed * if SubsetSize=0, zeros are returned * if SubsetSize<0, entire dataset is processed; Subset[] array is ignored in this case. OUTPUT PARAMETERS: Rep - it contains all type of errors. -- ALGLIB -- Copyright 04.09.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpallerrorssparsesubset( multilayerperceptron network, sparsematrix xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize, modelerrors& rep); void alglib::smp_mlpallerrorssparsesubset( multilayerperceptron network, sparsematrix xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize, modelerrors& rep);
mlpallerrorssubset function/************************************************************************* Calculation of all types of errors on subset of dataset. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - original dataset; one sample = one row; first NIn columns contain inputs, next NOut columns - desired outputs. SetSize - real size of XY, SetSize>=0; Subset - subset of SubsetSize elements, array[SubsetSize]; SubsetSize- number of elements in Subset[] array: * if SubsetSize>0, rows of XY with indices Subset[0]... ...Subset[SubsetSize-1] are processed * if SubsetSize=0, zeros are returned * if SubsetSize<0, entire dataset is processed; Subset[] array is ignored in this case. OUTPUT PARAMETERS: Rep - it contains all type of errors. -- ALGLIB -- Copyright 04.09.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpallerrorssubset( multilayerperceptron network, real_2d_array xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize, modelerrors& rep); void alglib::smp_mlpallerrorssubset( multilayerperceptron network, real_2d_array xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize, modelerrors& rep);
mlpavgce function/************************************************************************* Average cross-entropy (in bits per element) on the test set. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: CrossEntropy/(NPoints*LN(2)). Zero if network solves regression task. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 08.01.2009 by Bochkanov Sergey *************************************************************************/double alglib::mlpavgce( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); double alglib::smp_mlpavgce( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlpavgcesparse function/************************************************************************* Average cross-entropy (in bits per element) on the test set given by sparse matrix. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Sparse matrix must use CRS format for storage. NPoints - points count, >=0. RESULT: CrossEntropy/(NPoints*LN(2)). Zero if network solves regression task. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 9.08.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlpavgcesparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints); double alglib::smp_mlpavgcesparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints);
mlpavgerror function/************************************************************************* Average absolute error on the test set. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: Its meaning for regression task is obvious. As for classification task, it means average error when estimating posterior probabilities. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 11.03.2008 by Bochkanov Sergey *************************************************************************/double alglib::mlpavgerror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); double alglib::smp_mlpavgerror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlpavgerrorsparse function/************************************************************************* Average absolute error on the test set given by sparse matrix. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Sparse matrix must use CRS format for storage. NPoints - points count, >=0. RESULT: Its meaning for regression task is obvious. As for classification task, it means average error when estimating posterior probabilities. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 09.08.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlpavgerrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints); double alglib::smp_mlpavgerrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints);
mlpavgrelerror function/************************************************************************* Average relative error on the test set. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: Its meaning for regression task is obvious. As for classification task, it means average relative error when estimating posterior probability of belonging to the correct class. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 11.03.2008 by Bochkanov Sergey *************************************************************************/double alglib::mlpavgrelerror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); double alglib::smp_mlpavgrelerror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlpavgrelerrorsparse function/************************************************************************* Average relative error on the test set given by sparse matrix. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Sparse matrix must use CRS format for storage. NPoints - points count, >=0. RESULT: Its meaning for regression task is obvious. As for classification task, it means average relative error when estimating posterior probability of belonging to the correct class. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 09.08.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlpavgrelerrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints); double alglib::smp_mlpavgrelerrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints);
mlpclserror function/************************************************************************* Classification error of the neural network on dataset. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: classification error (number of misclassified cases) DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mlpclserror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); ae_int_t alglib::smp_mlpclserror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlpcopy function/************************************************************************* Copying of neural network INPUT PARAMETERS: Network1 - original OUTPUT PARAMETERS: Network2 - copy -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcopy( multilayerperceptron network1, multilayerperceptron& network2);
mlpcopytunableparameters function/************************************************************************* This function copies tunable parameters (weights/means/sigmas) from one network to another with same architecture. It performs some rudimentary checks that architectures are same, and throws exception if check fails. It is intended for fast copying of states between two network which are known to have same geometry. INPUT PARAMETERS: Network1 - source, must be correctly initialized Network2 - target, must have same architecture OUTPUT PARAMETERS: Network2 - network state is copied from source to target -- ALGLIB -- Copyright 20.06.2013 by Bochkanov Sergey *************************************************************************/void alglib::mlpcopytunableparameters( multilayerperceptron network1, multilayerperceptron network2);
mlpcreate0 function/************************************************************************* Creates neural network with NIn inputs, NOut outputs, without hidden layers, with linear output layer. Network weights are filled with small random values. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreate0( ae_int_t nin, ae_int_t nout, multilayerperceptron& network);
mlpcreate1 function/************************************************************************* Same as MLPCreate0, but with one hidden layer (NHid neurons) with non-linear activation function. Output layer is linear. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreate1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, multilayerperceptron& network);
mlpcreate2 function/************************************************************************* Same as MLPCreate0, but with two hidden layers (NHid1 and NHid2 neurons) with non-linear activation function. Output layer is linear. $ALL -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreate2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, multilayerperceptron& network);
mlpcreateb0 function/************************************************************************* Creates neural network with NIn inputs, NOut outputs, without hidden layers with non-linear output layer. Network weights are filled with small random values. Activation function of the output layer takes values: (B, +INF), if D>=0 or (-INF, B), if D<0. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreateb0( ae_int_t nin, ae_int_t nout, double b, double d, multilayerperceptron& network);
mlpcreateb1 function/************************************************************************* Same as MLPCreateB0 but with non-linear hidden layer. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreateb1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, double b, double d, multilayerperceptron& network);
mlpcreateb2 function/************************************************************************* Same as MLPCreateB0 but with two non-linear hidden layers. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreateb2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, double b, double d, multilayerperceptron& network);
mlpcreatec0 function/************************************************************************* Creates classifier network with NIn inputs and NOut possible classes. Network contains no hidden layers and linear output layer with SOFTMAX- normalization (so outputs sums up to 1.0 and converge to posterior probabilities). -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreatec0( ae_int_t nin, ae_int_t nout, multilayerperceptron& network);
mlpcreatec1 function/************************************************************************* Same as MLPCreateC0, but with one non-linear hidden layer. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreatec1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, multilayerperceptron& network);
mlpcreatec2 function/************************************************************************* Same as MLPCreateC0, but with two non-linear hidden layers. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreatec2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, multilayerperceptron& network);
mlpcreater0 function/************************************************************************* Creates neural network with NIn inputs, NOut outputs, without hidden layers with non-linear output layer. Network weights are filled with small random values. Activation function of the output layer takes values [A,B]. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreater0( ae_int_t nin, ae_int_t nout, double a, double b, multilayerperceptron& network);
mlpcreater1 function/************************************************************************* Same as MLPCreateR0, but with non-linear hidden layer. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreater1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, double a, double b, multilayerperceptron& network);
mlpcreater2 function/************************************************************************* Same as MLPCreateR0, but with two non-linear hidden layers. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreater2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, double a, double b, multilayerperceptron& network);
mlperror function/************************************************************************* Error of the neural network on dataset. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x, depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2) DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/double alglib::mlperror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); double alglib::smp_mlperror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlperrorn function/************************************************************************* Natural error function for neural network, internal subroutine. NOTE: this function is single-threaded. Unlike other error function, it receives no speed-up from being executed in SMP mode. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/double alglib::mlperrorn( multilayerperceptron network, real_2d_array xy, ae_int_t ssize);
mlperrorsparse function/************************************************************************* Error of the neural network on dataset given by sparse matrix. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x, depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Sparse matrix must use CRS format for storage. NPoints - points count, >=0 RESULT: sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2) DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlperrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints); double alglib::smp_mlperrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints);
mlperrorsparsesubset function/************************************************************************* Error of the neural network on subset of sparse dataset. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Sparse matrix must use CRS format for storage. SetSize - real size of XY, SetSize>=0; it is used when SubsetSize<0; Subset - subset of SubsetSize elements, array[SubsetSize]; SubsetSize- number of elements in Subset[] array: * if SubsetSize>0, rows of XY with indices Subset[0]... ...Subset[SubsetSize-1] are processed * if SubsetSize=0, zeros are returned * if SubsetSize<0, entire dataset is processed; Subset[] array is ignored in this case. RESULT: sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2) DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 04.09.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlperrorsparsesubset( multilayerperceptron network, sparsematrix xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize); double alglib::smp_mlperrorsparsesubset( multilayerperceptron network, sparsematrix xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize);
mlperrorsubset function/************************************************************************* Error of the neural network on subset of dataset. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; SetSize - real size of XY, SetSize>=0; Subset - subset of SubsetSize elements, array[SubsetSize]; SubsetSize- number of elements in Subset[] array: * if SubsetSize>0, rows of XY with indices Subset[0]... ...Subset[SubsetSize-1] are processed * if SubsetSize=0, zeros are returned * if SubsetSize<0, entire dataset is processed; Subset[] array is ignored in this case. RESULT: sum-of-squares error, SUM(sqr(y[i]-desired_y[i])/2) DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 04.09.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlperrorsubset( multilayerperceptron network, real_2d_array xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize); double alglib::smp_mlperrorsubset( multilayerperceptron network, real_2d_array xy, ae_int_t setsize, integer_1d_array subset, ae_int_t subsetsize);
mlpgetinputscaling function/************************************************************************* This function returns offset/scaling coefficients for I-th input of the network. INPUT PARAMETERS: Network - network I - input index OUTPUT PARAMETERS: Mean - mean term Sigma - sigma term, guaranteed to be nonzero. I-th input is passed through linear transformation IN[i] = (IN[i]-Mean)/Sigma before feeding to the network -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpgetinputscaling( multilayerperceptron network, ae_int_t i, double& mean, double& sigma);
mlpgetinputscount function/************************************************************************* Returns number of inputs. -- ALGLIB -- Copyright 19.10.2011 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mlpgetinputscount(multilayerperceptron network);
mlpgetlayerscount function/************************************************************************* This function returns total number of layers (including input, hidden and output layers). -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mlpgetlayerscount(multilayerperceptron network);
mlpgetlayersize function/************************************************************************* This function returns size of K-th layer. K=0 corresponds to input layer, K=CNT-1 corresponds to output layer. Size of the output layer is always equal to the number of outputs, although when we have softmax-normalized network, last neuron doesn't have any connections - it is just zero. -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mlpgetlayersize( multilayerperceptron network, ae_int_t k);
mlpgetneuroninfo function/************************************************************************* This function returns information about Ith neuron of Kth layer INPUT PARAMETERS: Network - network K - layer index I - neuron index (within layer) OUTPUT PARAMETERS: FKind - activation function type (used by MLPActivationFunction()) this value is zero for input or linear neurons Threshold - also called offset, bias zero for input neurons NOTE: this function throws exception if layer or neuron with given index do not exists. -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpgetneuroninfo( multilayerperceptron network, ae_int_t k, ae_int_t i, ae_int_t& fkind, double& threshold);
mlpgetoutputscaling function/************************************************************************* This function returns offset/scaling coefficients for I-th output of the network. INPUT PARAMETERS: Network - network I - input index OUTPUT PARAMETERS: Mean - mean term Sigma - sigma term, guaranteed to be nonzero. I-th output is passed through linear transformation OUT[i] = OUT[i]*Sigma+Mean before returning it to user. In case we have SOFTMAX-normalized network, we return (Mean,Sigma)=(0.0,1.0). -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpgetoutputscaling( multilayerperceptron network, ae_int_t i, double& mean, double& sigma);
mlpgetoutputscount function/************************************************************************* Returns number of outputs. -- ALGLIB -- Copyright 19.10.2011 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mlpgetoutputscount(multilayerperceptron network);
mlpgetweight function/************************************************************************* This function returns information about connection from I0-th neuron of K0-th layer to I1-th neuron of K1-th layer. INPUT PARAMETERS: Network - network K0 - layer index I0 - neuron index (within layer) K1 - layer index I1 - neuron index (within layer) RESULT: connection weight (zero for non-existent connections) This function: 1. throws exception if layer or neuron with given index do not exists. 2. returns zero if neurons exist, but there is no connection between them -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/double alglib::mlpgetweight( multilayerperceptron network, ae_int_t k0, ae_int_t i0, ae_int_t k1, ae_int_t i1);
mlpgetweightscount function/************************************************************************* Returns number of weights. -- ALGLIB -- Copyright 19.10.2011 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::mlpgetweightscount(multilayerperceptron network);
mlpgrad function/************************************************************************* Gradient calculation INPUT PARAMETERS: Network - network initialized with one of the network creation funcs X - input vector, length of array must be at least NIn DesiredY- desired outputs, length of array must be at least NOut Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, SUM(sqr(y[i]-desiredy[i])/2,i) Grad - gradient of E with respect to weights of network, array[WCount] -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpgrad( multilayerperceptron network, real_1d_array x, real_1d_array desiredy, double& e, real_1d_array& grad);
mlpgradbatch function/************************************************************************* Batch gradient calculation for a set of inputs/outputs FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - original dataset in dense format; one sample = one row: * first NIn columns contain inputs, * for regression problem, next NOut columns store desired outputs. * for classification problem, next column (just one!) stores class number. SSize - number of elements in XY Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, SUM(sqr(y[i]-desiredy[i])/2,i) Grad - gradient of E with respect to weights of network, array[WCount] -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpgradbatch( multilayerperceptron network, real_2d_array xy, ae_int_t ssize, double& e, real_1d_array& grad); void alglib::smp_mlpgradbatch( multilayerperceptron network, real_2d_array xy, ae_int_t ssize, double& e, real_1d_array& grad);
mlpgradbatchsparse function/************************************************************************* Batch gradient calculation for a set of inputs/outputs given by sparse matrices FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - original dataset in sparse format; one sample = one row: * MATRIX MUST BE STORED IN CRS FORMAT * first NIn columns contain inputs. * for regression problem, next NOut columns store desired outputs. * for classification problem, next column (just one!) stores class number. SSize - number of elements in XY Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, SUM(sqr(y[i]-desiredy[i])/2,i) Grad - gradient of E with respect to weights of network, array[WCount] -- ALGLIB -- Copyright 26.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpgradbatchsparse( multilayerperceptron network, sparsematrix xy, ae_int_t ssize, double& e, real_1d_array& grad); void alglib::smp_mlpgradbatchsparse( multilayerperceptron network, sparsematrix xy, ae_int_t ssize, double& e, real_1d_array& grad);
mlpgradbatchsparsesubset function/************************************************************************* Batch gradient calculation for a set of inputs/outputs for a subset of dataset given by set of indexes. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - original dataset in sparse format; one sample = one row: * MATRIX MUST BE STORED IN CRS FORMAT * first NIn columns contain inputs, * for regression problem, next NOut columns store desired outputs. * for classification problem, next column (just one!) stores class number. SetSize - real size of XY, SetSize>=0; Idx - subset of SubsetSize elements, array[SubsetSize]: * Idx[I] stores row index in the original dataset which is given by XY. Gradient is calculated with respect to rows whose indexes are stored in Idx[]. * Idx[] must store correct indexes; this function throws an exception in case incorrect index (less than 0 or larger than rows(XY)) is given * Idx[] may store indexes in any order and even with repetitions. SubsetSize- number of elements in Idx[] array: * positive value means that subset given by Idx[] is processed * zero value results in zero gradient * negative value means that full dataset is processed Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, SUM(sqr(y[i]-desiredy[i])/2,i) Grad - gradient of E with respect to weights of network, array[WCount] NOTE: when SubsetSize<0 is used full dataset by call MLPGradBatchSparse function. -- ALGLIB -- Copyright 26.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpgradbatchsparsesubset( multilayerperceptron network, sparsematrix xy, ae_int_t setsize, integer_1d_array idx, ae_int_t subsetsize, double& e, real_1d_array& grad); void alglib::smp_mlpgradbatchsparsesubset( multilayerperceptron network, sparsematrix xy, ae_int_t setsize, integer_1d_array idx, ae_int_t subsetsize, double& e, real_1d_array& grad);
mlpgradbatchsubset function/************************************************************************* Batch gradient calculation for a subset of dataset FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - original dataset in dense format; one sample = one row: * first NIn columns contain inputs, * for regression problem, next NOut columns store desired outputs. * for classification problem, next column (just one!) stores class number. SetSize - real size of XY, SetSize>=0; Idx - subset of SubsetSize elements, array[SubsetSize]: * Idx[I] stores row index in the original dataset which is given by XY. Gradient is calculated with respect to rows whose indexes are stored in Idx[]. * Idx[] must store correct indexes; this function throws an exception in case incorrect index (less than 0 or larger than rows(XY)) is given * Idx[] may store indexes in any order and even with repetitions. SubsetSize- number of elements in Idx[] array: * positive value means that subset given by Idx[] is processed * zero value results in zero gradient * negative value means that full dataset is processed Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, SUM(sqr(y[i]-desiredy[i])/2,i) Grad - gradient of E with respect to weights of network, array[WCount] -- ALGLIB -- Copyright 26.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpgradbatchsubset( multilayerperceptron network, real_2d_array xy, ae_int_t setsize, integer_1d_array idx, ae_int_t subsetsize, double& e, real_1d_array& grad); void alglib::smp_mlpgradbatchsubset( multilayerperceptron network, real_2d_array xy, ae_int_t setsize, integer_1d_array idx, ae_int_t subsetsize, double& e, real_1d_array& grad);
mlpgradn function/************************************************************************* Gradient calculation (natural error function is used) INPUT PARAMETERS: Network - network initialized with one of the network creation funcs X - input vector, length of array must be at least NIn DesiredY- desired outputs, length of array must be at least NOut Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, sum-of-squares for regression networks, cross-entropy for classification networks. Grad - gradient of E with respect to weights of network, array[WCount] -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpgradn( multilayerperceptron network, real_1d_array x, real_1d_array desiredy, double& e, real_1d_array& grad);
mlpgradnbatch function/************************************************************************* Batch gradient calculation for a set of inputs/outputs (natural error function is used) INPUT PARAMETERS: Network - network initialized with one of the network creation funcs XY - set of inputs/outputs; one sample = one row; first NIn columns contain inputs, next NOut columns - desired outputs. SSize - number of elements in XY Grad - possibly preallocated array. If size of array is smaller than WCount, it will be reallocated. It is recommended to reuse previously allocated array to reduce allocation overhead. OUTPUT PARAMETERS: E - error function, sum-of-squares for regression networks, cross-entropy for classification networks. Grad - gradient of E with respect to weights of network, array[WCount] -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpgradnbatch( multilayerperceptron network, real_2d_array xy, ae_int_t ssize, double& e, real_1d_array& grad);
mlphessianbatch function/************************************************************************* Batch Hessian calculation using R-algorithm. Internal subroutine. -- ALGLIB -- Copyright 26.01.2008 by Bochkanov Sergey. Hessian calculation based on R-algorithm described in "Fast Exact Multiplication by the Hessian", B. A. Pearlmutter, Neural Computation, 1994. *************************************************************************/void alglib::mlphessianbatch( multilayerperceptron network, real_2d_array xy, ae_int_t ssize, double& e, real_1d_array& grad, real_2d_array& h);
mlphessiannbatch function/************************************************************************* Batch Hessian calculation (natural error function) using R-algorithm. Internal subroutine. -- ALGLIB -- Copyright 26.01.2008 by Bochkanov Sergey. Hessian calculation based on R-algorithm described in "Fast Exact Multiplication by the Hessian", B. A. Pearlmutter, Neural Computation, 1994. *************************************************************************/void alglib::mlphessiannbatch( multilayerperceptron network, real_2d_array xy, ae_int_t ssize, double& e, real_1d_array& grad, real_2d_array& h);
mlpinitpreprocessor function/************************************************************************* Internal subroutine. -- ALGLIB -- Copyright 30.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlpinitpreprocessor( multilayerperceptron network, real_2d_array xy, ae_int_t ssize);
mlpissoftmax function/************************************************************************* Tells whether network is SOFTMAX-normalized (i.e. classifier) or not. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/bool alglib::mlpissoftmax(multilayerperceptron network);
mlpprocess function/************************************************************************* Procesing INPUT PARAMETERS: Network - neural network X - input vector, array[0..NIn-1]. OUTPUT PARAMETERS: Y - result. Regression estimate when solving regression task, vector of posterior probabilities for classification task. See also MLPProcessI -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpprocess( multilayerperceptron network, real_1d_array x, real_1d_array& y);
mlpprocessi function/************************************************************************* 'interactive' variant of MLPProcess for languages like Python which support constructs like "Y = MLPProcess(NN,X)" and interactive mode of the interpreter This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 21.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::mlpprocessi( multilayerperceptron network, real_1d_array x, real_1d_array& y);
mlpproperties function/************************************************************************* Returns information about initialized network: number of inputs, outputs, weights. -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpproperties( multilayerperceptron network, ae_int_t& nin, ae_int_t& nout, ae_int_t& wcount);
mlprandomize function/************************************************************************* Randomization of neural network weights -- ALGLIB -- Copyright 06.11.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlprandomize(multilayerperceptron network);
mlprandomizefull function/************************************************************************* Randomization of neural network weights and standartisator -- ALGLIB -- Copyright 10.03.2008 by Bochkanov Sergey *************************************************************************/void alglib::mlprandomizefull(multilayerperceptron network);
mlprelclserror function/************************************************************************* Relative classification error on the test set. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: Percent of incorrectly classified cases. Works both for classifier networks and general purpose networks used as classifiers. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 25.12.2008 by Bochkanov Sergey *************************************************************************/double alglib::mlprelclserror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); double alglib::smp_mlprelclserror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlprelclserrorsparse function/************************************************************************* Relative classification error on the test set given by sparse matrix. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format. Sparse matrix must use CRS format for storage. NPoints - points count, >=0. RESULT: Percent of incorrectly classified cases. Works both for classifier networks and general purpose networks used as classifiers. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 09.08.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlprelclserrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints); double alglib::smp_mlprelclserrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints);
mlprmserror function/************************************************************************* RMS error on the test set given. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format; NPoints - points count. RESULT: Root mean square error. Its meaning for regression task is obvious. As for classification task, RMS error means error when estimating posterior probabilities. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 04.11.2007 by Bochkanov Sergey *************************************************************************/double alglib::mlprmserror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints); double alglib::smp_mlprmserror( multilayerperceptron network, real_2d_array xy, ae_int_t npoints);
mlprmserrorsparse function/************************************************************************* RMS error on the test set given by sparse matrix. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support ! ! First improvement gives close-to-linear speedup on multicore systems. ! Second improvement gives constant speedup (2-3x depending on your CPU) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: Network - neural network; XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Sparse matrix must use CRS format for storage. NPoints - points count, >=0. RESULT: Root mean square error. Its meaning for regression task is obvious. As for classification task, RMS error means error when estimating posterior probabilities. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following dataset format is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 09.08.2012 by Bochkanov Sergey *************************************************************************/double alglib::mlprmserrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints); double alglib::smp_mlprmserrorsparse( multilayerperceptron network, sparsematrix xy, ae_int_t npoints);
mlpserialize function/************************************************************************* This function serializes data structure to string. Important properties of s_out: * it contains alphanumeric characters, dots, underscores, minus signs * these symbols are grouped into words, which are separated by spaces and Windows-style (CR+LF) newlines * although serializer uses spaces and CR+LF as separators, you can replace any separator character by arbitrary combination of spaces, tabs, Windows or Unix newlines. It allows flexible reformatting of the string in case you want to include it into text or XML file. But you should not insert separators into the middle of the "words" nor you should change case of letters. * s_out can be freely moved between 32-bit and 64-bit systems, little and big endian machines, and so on. You can serialize structure on 32-bit machine and unserialize it on 64-bit one (or vice versa), or serialize it on SPARC and unserialize on x86. You can also serialize it in C++ version of ALGLIB and unserialize in C# one, and vice versa. *************************************************************************/void mlpserialize(multilayerperceptron &obj, std::string &s_out); void mlpserialize(multilayerperceptron &obj, std::ostream &s_out);
mlpsetinputscaling function/************************************************************************* This function sets offset/scaling coefficients for I-th input of the network. INPUT PARAMETERS: Network - network I - input index Mean - mean term Sigma - sigma term (if zero, will be replaced by 1.0) NTE: I-th input is passed through linear transformation IN[i] = (IN[i]-Mean)/Sigma before feeding to the network. This function sets Mean and Sigma. -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetinputscaling( multilayerperceptron network, ae_int_t i, double mean, double sigma);
mlpsetneuroninfo function/************************************************************************* This function modifies information about Ith neuron of Kth layer INPUT PARAMETERS: Network - network K - layer index I - neuron index (within layer) FKind - activation function type (used by MLPActivationFunction()) this value must be zero for input neurons (you can not set activation function for input neurons) Threshold - also called offset, bias this value must be zero for input neurons (you can not set threshold for input neurons) NOTES: 1. this function throws exception if layer or neuron with given index do not exists. 2. this function also throws exception when you try to set non-linear activation function for input neurons (any kind of network) or for output neurons of classifier network. 3. this function throws exception when you try to set non-zero threshold for input neurons (any kind of network). -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetneuroninfo( multilayerperceptron network, ae_int_t k, ae_int_t i, ae_int_t fkind, double threshold);
mlpsetoutputscaling function/************************************************************************* This function sets offset/scaling coefficients for I-th output of the network. INPUT PARAMETERS: Network - network I - input index Mean - mean term Sigma - sigma term (if zero, will be replaced by 1.0) OUTPUT PARAMETERS: NOTE: I-th output is passed through linear transformation OUT[i] = OUT[i]*Sigma+Mean before returning it to user. This function sets Sigma/Mean. In case we have SOFTMAX-normalized network, you can not set (Sigma,Mean) to anything other than(0.0,1.0) - this function will throw exception. -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetoutputscaling( multilayerperceptron network, ae_int_t i, double mean, double sigma);
mlpsetweight function/************************************************************************* This function modifies information about connection from I0-th neuron of K0-th layer to I1-th neuron of K1-th layer. INPUT PARAMETERS: Network - network K0 - layer index I0 - neuron index (within layer) K1 - layer index I1 - neuron index (within layer) W - connection weight (must be zero for non-existent connections) This function: 1. throws exception if layer or neuron with given index do not exists. 2. throws exception if you try to set non-zero weight for non-existent connection -- ALGLIB -- Copyright 25.03.2011 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetweight( multilayerperceptron network, ae_int_t k0, ae_int_t i0, ae_int_t k1, ae_int_t i1, double w);
mlpunserialize function/************************************************************************* This function unserializes data structure from string. *************************************************************************/void mlpunserialize(const std::string &s_in, multilayerperceptron &obj); void mlpunserialize(const std::istream &s_in, multilayerperceptron &obj);
mlpe subpackagemlpensemble class/************************************************************************* Neural networks ensemble *************************************************************************/class mlpensemble { };
mlpeavgce function/************************************************************************* Average cross-entropy (in bits per element) on the test set INPUT PARAMETERS: Ensemble- ensemble XY - test set NPoints - test set size RESULT: CrossEntropy/(NPoints*LN(2)). Zero if ensemble solves regression task. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::mlpeavgce( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints);
mlpeavgerror function/************************************************************************* Average error on the test set INPUT PARAMETERS: Ensemble- ensemble XY - test set NPoints - test set size RESULT: Its meaning for regression task is obvious. As for classification task it means average error when estimating posterior probabilities. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::mlpeavgerror( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints);
mlpeavgrelerror function/************************************************************************* Average relative error on the test set INPUT PARAMETERS: Ensemble- ensemble XY - test set NPoints - test set size RESULT: Its meaning for regression task is obvious. As for classification task it means average relative error when estimating posterior probabilities. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::mlpeavgrelerror( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints);
mlpecreate0 function/************************************************************************* Like MLPCreate0, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreate0( ae_int_t nin, ae_int_t nout, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreate1 function/************************************************************************* Like MLPCreate1, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreate1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreate2 function/************************************************************************* Like MLPCreate2, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreate2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreateb0 function/************************************************************************* Like MLPCreateB0, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreateb0( ae_int_t nin, ae_int_t nout, double b, double d, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreateb1 function/************************************************************************* Like MLPCreateB1, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreateb1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, double b, double d, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreateb2 function/************************************************************************* Like MLPCreateB2, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreateb2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, double b, double d, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreatec0 function/************************************************************************* Like MLPCreateC0, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreatec0( ae_int_t nin, ae_int_t nout, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreatec1 function/************************************************************************* Like MLPCreateC1, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreatec1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreatec2 function/************************************************************************* Like MLPCreateC2, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreatec2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreatefromnetwork function/************************************************************************* Creates ensemble from network. Only network geometry is copied. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreatefromnetwork( multilayerperceptron network, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreater0 function/************************************************************************* Like MLPCreateR0, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreater0( ae_int_t nin, ae_int_t nout, double a, double b, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreater1 function/************************************************************************* Like MLPCreateR1, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreater1( ae_int_t nin, ae_int_t nhid, ae_int_t nout, double a, double b, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpecreater2 function/************************************************************************* Like MLPCreateR2, but for ensembles. -- ALGLIB -- Copyright 18.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpecreater2( ae_int_t nin, ae_int_t nhid1, ae_int_t nhid2, ae_int_t nout, double a, double b, ae_int_t ensemblesize, mlpensemble& ensemble);
mlpeissoftmax function/************************************************************************* Return normalization type (whether ensemble is SOFTMAX-normalized or not). -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/bool alglib::mlpeissoftmax(mlpensemble ensemble);
mlpeprocess function/************************************************************************* Procesing INPUT PARAMETERS: Ensemble- neural networks ensemble X - input vector, array[0..NIn-1]. Y - (possibly) preallocated buffer; if size of Y is less than NOut, it will be reallocated. If it is large enough, it is NOT reallocated, so we can save some time on reallocation. OUTPUT PARAMETERS: Y - result. Regression estimate when solving regression task, vector of posterior probabilities for classification task. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpeprocess( mlpensemble ensemble, real_1d_array x, real_1d_array& y);
mlpeprocessi function/************************************************************************* 'interactive' variant of MLPEProcess for languages like Python which support constructs like "Y = MLPEProcess(LM,X)" and interactive mode of the interpreter This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpeprocessi( mlpensemble ensemble, real_1d_array x, real_1d_array& y);
mlpeproperties function/************************************************************************* Return ensemble properties (number of inputs and outputs). -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpeproperties( mlpensemble ensemble, ae_int_t& nin, ae_int_t& nout);
mlperandomize function/************************************************************************* Randomization of MLP ensemble -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlperandomize(mlpensemble ensemble);
mlperelclserror function/************************************************************************* Relative classification error on the test set INPUT PARAMETERS: Ensemble- ensemble XY - test set NPoints - test set size RESULT: percent of incorrectly classified cases. Works both for classifier betwork and for regression networks which are used as classifiers. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::mlperelclserror( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints);
mlpermserror function/************************************************************************* RMS error on the test set INPUT PARAMETERS: Ensemble- ensemble XY - test set NPoints - test set size RESULT: root mean square error. Its meaning for regression task is obvious. As for classification task RMS error means error when estimating posterior probabilities. -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/double alglib::mlpermserror( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints);
mlpeserialize function/************************************************************************* This function serializes data structure to string. Important properties of s_out: * it contains alphanumeric characters, dots, underscores, minus signs * these symbols are grouped into words, which are separated by spaces and Windows-style (CR+LF) newlines * although serializer uses spaces and CR+LF as separators, you can replace any separator character by arbitrary combination of spaces, tabs, Windows or Unix newlines. It allows flexible reformatting of the string in case you want to include it into text or XML file. But you should not insert separators into the middle of the "words" nor you should change case of letters. * s_out can be freely moved between 32-bit and 64-bit systems, little and big endian machines, and so on. You can serialize structure on 32-bit machine and unserialize it on 64-bit one (or vice versa), or serialize it on SPARC and unserialize on x86. You can also serialize it in C++ version of ALGLIB and unserialize in C# one, and vice versa. *************************************************************************/void mlpeserialize(mlpensemble &obj, std::string &s_out); void mlpeserialize(mlpensemble &obj, std::ostream &s_out);
mlpeunserialize function/************************************************************************* This function unserializes data structure from string. *************************************************************************/void mlpeunserialize(const std::string &s_in, mlpensemble &obj); void mlpeunserialize(const std::istream &s_in, mlpensemble &obj);
mlptrain subpackage| nn_cls2 | Binary classification problem | |
| nn_cls3 | Multiclass classification problem | |
| nn_crossvalidation | Cross-validation | |
| nn_ensembles_es | Early stopping ensembles | |
| nn_parallel | Parallel training | |
| nn_regr | Regression problem with one output (2=>1) | |
| nn_regr_n | Regression problem with multiple outputs (2=>2) | |
| nn_trainerobject | Advanced example on trainer object |
mlpcvreport class/************************************************************************* Cross-validation estimates of generalization error *************************************************************************/class mlpcvreport { double relclserror; double avgce; double rmserror; double avgerror; double avgrelerror; };
mlpreport class/************************************************************************* Training report: * RelCLSError - fraction of misclassified cases. * AvgCE - acerage cross-entropy * RMSError - root-mean-square error * AvgError - average error * AvgRelError - average relative error * NGrad - number of gradient calculations * NHess - number of Hessian calculations * NCholesky - number of Cholesky decompositions NOTE 1: RelCLSError/AvgCE are zero on regression problems. NOTE 2: on classification problems RMSError/AvgError/AvgRelError contain errors in prediction of posterior probabilities *************************************************************************/class mlpreport { double relclserror; double avgce; double rmserror; double avgerror; double avgrelerror; ae_int_t ngrad; ae_int_t nhess; ae_int_t ncholesky; };
mlptrainer class/************************************************************************* Trainer object for neural network. You should not try to access fields of this object directly - use ALGLIB functions to work with this object. *************************************************************************/class mlptrainer { };
mlpcontinuetraining function/************************************************************************* IMPORTANT: this is an "expert" version of the MLPTrain() function. We do not recommend you to use it unless you are pretty sure that you need ability to monitor training progress. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support (C++ computational core) ! ! Second improvement gives constant speedup (2-3X). First improvement ! gives close-to-linear speedup on multicore systems. Following ! operations can be executed in parallel: ! * gradient calculation over large dataset (if dataset is large enough) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. This function performs step-by-step training of the neural network. Here "step-by-step" means that training starts with MLPStartTraining() call, and then user subsequently calls MLPContinueTraining() to perform one more iteration of the training. This function performs one more iteration of the training and returns either True (training continues) or False (training stopped). In case True was returned, Network weights are updated according to the current state of the optimization progress. In case False was returned, no additional updates is performed (previous update of the network weights moved us to the final point, and no additional updates is needed). EXAMPLE: > > [initialize network and trainer object] > > MLPStartTraining(Trainer, Network, True) > while MLPContinueTraining(Trainer, Network) do > [visualize training progress] > INPUT PARAMETERS: S - trainer object Network - neural network structure, which is used to store current state of the training process. OUTPUT PARAMETERS: Network - weights of the neural network are rewritten by the current approximation. NOTE: this method uses sum-of-squares error function for training. NOTE: it is expected that trainer object settings are NOT changed during step-by-step training, i.e. no one changes stopping criteria or training set during training. It is possible and there is no defense against such actions, but algorithm behavior in such cases is undefined and can be unpredictable. NOTE: It is expected that Network is the same one which was passed to MLPStartTraining() function. However, THIS function checks only following: * that number of network inputs is consistent with trainer object settings * that number of network outputs/classes is consistent with trainer object settings * that number of network weights is the same as number of weights in the network passed to MLPStartTraining() function Exception is thrown when these conditions are violated. It is also expected that you do not change state of the network on your own - the only party who has right to change network during its training is a trainer object. Any attempt to interfere with trainer may lead to unpredictable results. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/bool alglib::mlpcontinuetraining( mlptrainer s, multilayerperceptron network); bool alglib::smp_mlpcontinuetraining( mlptrainer s, multilayerperceptron network);
mlpcreatetrainer function/************************************************************************* Creation of the network trainer object for regression networks INPUT PARAMETERS: NIn - number of inputs, NIn>=1 NOut - number of outputs, NOut>=1 OUTPUT PARAMETERS: S - neural network trainer object. This structure can be used to train any regression network with NIn inputs and NOut outputs. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreatetrainer(ae_int_t nin, ae_int_t nout, mlptrainer& s);
Examples: [1] [2] [3] [4] [5] [6]
mlpcreatetrainercls function/************************************************************************* Creation of the network trainer object for classification networks INPUT PARAMETERS: NIn - number of inputs, NIn>=1 NClasses - number of classes, NClasses>=2 OUTPUT PARAMETERS: S - neural network trainer object. This structure can be used to train any classification network with NIn inputs and NOut outputs. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpcreatetrainercls( ae_int_t nin, ae_int_t nclasses, mlptrainer& s);
mlpebagginglbfgs function/************************************************************************* Training neural networks ensemble using bootstrap aggregating (bagging). L-BFGS algorithm is used as base training method. INPUT PARAMETERS: Ensemble - model with initialized geometry XY - training set NPoints - training set size Decay - weight decay coefficient, >=0.001 Restarts - restarts, >0. WStep - stopping criterion, same as in MLPTrainLBFGS MaxIts - stopping criterion, same as in MLPTrainLBFGS OUTPUT PARAMETERS: Ensemble - trained model Info - return code: * -8, if both WStep=0 and MaxIts=0 * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<0, Restarts<1). * 2, if task has been solved. Rep - training report. OOBErrors - out-of-bag generalization error estimate -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpebagginglbfgs( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, double wstep, ae_int_t maxits, ae_int_t& info, mlpreport& rep, mlpcvreport& ooberrors);
mlpebagginglm function/************************************************************************* Training neural networks ensemble using bootstrap aggregating (bagging). Modified Levenberg-Marquardt algorithm is used as base training method. INPUT PARAMETERS: Ensemble - model with initialized geometry XY - training set NPoints - training set size Decay - weight decay coefficient, >=0.001 Restarts - restarts, >0. OUTPUT PARAMETERS: Ensemble - trained model Info - return code: * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<0, Restarts<1). * 2, if task has been solved. Rep - training report. OOBErrors - out-of-bag generalization error estimate -- ALGLIB -- Copyright 17.02.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpebagginglm( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, ae_int_t& info, mlpreport& rep, mlpcvreport& ooberrors);
mlpetraines function/************************************************************************* Training neural networks ensemble using early stopping. INPUT PARAMETERS: Ensemble - model with initialized geometry XY - training set NPoints - training set size Decay - weight decay coefficient, >=0.001 Restarts - restarts, >0. OUTPUT PARAMETERS: Ensemble - trained model Info - return code: * -2, if there is a point with class number outside of [0..NClasses-1]. * -1, if incorrect parameters was passed (NPoints<0, Restarts<1). * 6, if task has been solved. Rep - training report. OOBErrors - out-of-bag generalization error estimate -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlpetraines( mlpensemble ensemble, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, ae_int_t& info, mlpreport& rep);
mlpkfoldcv function/************************************************************************* This function estimates generalization error using cross-validation on the current dataset with current training settings. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support (C++ computational core) ! ! Second improvement gives constant speedup (2-3X). First improvement ! gives close-to-linear speedup on multicore systems. Following ! operations can be executed in parallel: ! * FoldsCount cross-validation rounds (always) ! * NRestarts training sessions performed within each of ! cross-validation rounds (if NRestarts>1) ! * gradient calculation over large dataset (if dataset is large enough) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - trainer object Network - neural network. It must have same number of inputs and output/classes as was specified during creation of the trainer object. Network is not changed during cross- validation and is not trained - it is used only as representative of its architecture. I.e., we estimate generalization properties of ARCHITECTURE, not some specific network. NRestarts - number of restarts, >=0: * NRestarts>0 means that for each cross-validation round specified number of random restarts is performed, with best network being chosen after training. * NRestarts=0 is same as NRestarts=1 FoldsCount - number of folds in k-fold cross-validation: * 2<=FoldsCount<=size of dataset * recommended value: 10. * values larger than dataset size will be silently truncated down to dataset size OUTPUT PARAMETERS: Rep - structure which contains cross-validation estimates: * Rep.RelCLSError - fraction of misclassified cases. * Rep.AvgCE - acerage cross-entropy * Rep.RMSError - root-mean-square error * Rep.AvgError - average error * Rep.AvgRelError - average relative error NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(), or subset with only one point was given, zeros are returned as estimates. NOTE: this method performs FoldsCount cross-validation rounds, each one with NRestarts random starts. Thus, FoldsCount*NRestarts networks are trained in total. NOTE: Rep.RelCLSError/Rep.AvgCE are zero on regression problems. NOTE: on classification problems Rep.RMSError/Rep.AvgError/Rep.AvgRelError contain errors in prediction of posterior probabilities. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpkfoldcv( mlptrainer s, multilayerperceptron network, ae_int_t nrestarts, ae_int_t foldscount, mlpreport& rep); void alglib::smp_mlpkfoldcv( mlptrainer s, multilayerperceptron network, ae_int_t nrestarts, ae_int_t foldscount, mlpreport& rep);
mlpkfoldcvlbfgs function/************************************************************************* Cross-validation estimate of generalization error. Base algorithm - L-BFGS. INPUT PARAMETERS: Network - neural network with initialized geometry. Network is not changed during cross-validation - it is used only as a representative of its architecture. XY - training set. SSize - training set size Decay - weight decay, same as in MLPTrainLBFGS Restarts - number of restarts, >0. restarts are counted for each partition separately, so total number of restarts will be Restarts*FoldsCount. WStep - stopping criterion, same as in MLPTrainLBFGS MaxIts - stopping criterion, same as in MLPTrainLBFGS FoldsCount - number of folds in k-fold cross-validation, 2<=FoldsCount<=SSize. recommended value: 10. OUTPUT PARAMETERS: Info - return code, same as in MLPTrainLBFGS Rep - report, same as in MLPTrainLM/MLPTrainLBFGS CVRep - generalization error estimates -- ALGLIB -- Copyright 09.12.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpkfoldcvlbfgs( multilayerperceptron network, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, double wstep, ae_int_t maxits, ae_int_t foldscount, ae_int_t& info, mlpreport& rep, mlpcvreport& cvrep);
mlpkfoldcvlm function/************************************************************************* Cross-validation estimate of generalization error. Base algorithm - Levenberg-Marquardt. INPUT PARAMETERS: Network - neural network with initialized geometry. Network is not changed during cross-validation - it is used only as a representative of its architecture. XY - training set. SSize - training set size Decay - weight decay, same as in MLPTrainLBFGS Restarts - number of restarts, >0. restarts are counted for each partition separately, so total number of restarts will be Restarts*FoldsCount. FoldsCount - number of folds in k-fold cross-validation, 2<=FoldsCount<=SSize. recommended value: 10. OUTPUT PARAMETERS: Info - return code, same as in MLPTrainLBFGS Rep - report, same as in MLPTrainLM/MLPTrainLBFGS CVRep - generalization error estimates -- ALGLIB -- Copyright 09.12.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlpkfoldcvlm( multilayerperceptron network, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, ae_int_t foldscount, ae_int_t& info, mlpreport& rep, mlpcvreport& cvrep);
mlpsetalgobatch function/************************************************************************* This function sets training algorithm: batch training using L-BFGS will be used. This algorithm: * the most robust for small-scale problems, but may be too slow for large scale ones. * perfoms full pass through the dataset before performing step * uses conditions specified by MLPSetCond() for stopping * is default one used by trainer object INPUT PARAMETERS: S - trainer object -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetalgobatch(mlptrainer s);
mlpsetcond function/************************************************************************* This function sets stopping criteria for the optimizer. INPUT PARAMETERS: S - trainer object WStep - stopping criterion. Algorithm stops if step size is less than WStep. Recommended value - 0.01. Zero step size means stopping after MaxIts iterations. WStep>=0. MaxIts - stopping criterion. Algorithm stops after MaxIts epochs (full passes over entire dataset). Zero MaxIts means stopping when step is sufficiently small. MaxIts>=0. NOTE: by default, WStep=0.005 and MaxIts=0 are used. These values are also used when MLPSetCond() is called with WStep=0 and MaxIts=0. NOTE: these stopping criteria are used for all kinds of neural training - from "conventional" networks to early stopping ensembles. When used for "conventional" networks, they are used as the only stopping criteria. When combined with early stopping, they used as ADDITIONAL stopping criteria which can terminate early stopping algorithm. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetcond(mlptrainer s, double wstep, ae_int_t maxits);
mlpsetdataset function/************************************************************************* This function sets "current dataset" of the trainer object to one passed by user. INPUT PARAMETERS: S - trainer object XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. NPoints - points count, >=0. DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following datasetformat is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetdataset( mlptrainer s, real_2d_array xy, ae_int_t npoints);
Examples: [1] [2] [3] [4] [5] [6] [7] [8]
mlpsetdecay function/************************************************************************* This function sets weight decay coefficient which is used for training. INPUT PARAMETERS: S - trainer object Decay - weight decay coefficient, >=0. Weight decay term 'Decay*||Weights||^2' is added to error function. If you don't know what Decay to choose, use 1.0E-3. Weight decay can be set to zero, in this case network is trained without weight decay. NOTE: by default network uses some small nonzero value for weight decay. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetdecay(mlptrainer s, double decay);
mlpsetsparsedataset function/************************************************************************* This function sets "current dataset" of the trainer object to one passed by user (sparse matrix is used to store dataset). INPUT PARAMETERS: S - trainer object XY - training set, see below for information on the training set format. This function checks correctness of the dataset (no NANs/INFs, class numbers are correct) and throws exception when incorrect dataset is passed. Any sparse storage format can be used: Hash-table, CRS... NPoints - points count, >=0 DATASET FORMAT: This function uses two different dataset formats - one for regression networks, another one for classification networks. For regression networks with NIn inputs and NOut outputs following dataset format is used: * dataset is given by NPoints*(NIn+NOut) matrix * each row corresponds to one example * first NIn columns are inputs, next NOut columns are outputs For classification networks with NIn inputs and NClasses clases following datasetformat is used: * dataset is given by NPoints*(NIn+1) matrix * each row corresponds to one example * first NIn columns are inputs, last column stores class number (from 0 to NClasses-1). -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpsetsparsedataset( mlptrainer s, sparsematrix xy, ae_int_t npoints);
mlpstarttraining function/************************************************************************* IMPORTANT: this is an "expert" version of the MLPTrain() function. We do not recommend you to use it unless you are pretty sure that you need ability to monitor training progress. This function performs step-by-step training of the neural network. Here "step-by-step" means that training starts with MLPStartTraining() call, and then user subsequently calls MLPContinueTraining() to perform one more iteration of the training. After call to this function trainer object remembers network and is ready to train it. However, no training is performed until first call to MLPContinueTraining() function. Subsequent calls to MLPContinueTraining() will advance training progress one iteration further. EXAMPLE: > > ...initialize network and trainer object.... > > MLPStartTraining(Trainer, Network, True) > while MLPContinueTraining(Trainer, Network) do > ...visualize training progress... > INPUT PARAMETERS: S - trainer object Network - neural network. It must have same number of inputs and output/classes as was specified during creation of the trainer object. RandomStart - randomize network before training or not: * True means that network is randomized and its initial state (one which was passed to the trainer object) is lost. * False means that training is started from the current state of the network OUTPUT PARAMETERS: Network - neural network which is ready to training (weights are initialized, preprocessor is initialized using current training set) NOTE: this method uses sum-of-squares error function for training. NOTE: it is expected that trainer object settings are NOT changed during step-by-step training, i.e. no one changes stopping criteria or training set during training. It is possible and there is no defense against such actions, but algorithm behavior in such cases is undefined and can be unpredictable. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlpstarttraining( mlptrainer s, multilayerperceptron network, bool randomstart);
mlptrainensemblees function/************************************************************************* This function trains neural network ensemble passed to this function using current dataset and early stopping training algorithm. Each early stopping round performs NRestarts random restarts (thus, EnsembleSize*NRestarts training rounds is performed in total). FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support (C++ computational core) ! ! Second improvement gives constant speedup (2-3X). First improvement ! gives close-to-linear speedup on multicore systems. Following ! operations can be executed in parallel: ! * EnsembleSize training sessions performed for each of ensemble ! members (always parallelized) ! * NRestarts training sessions performed within each of training ! sessions (if NRestarts>1) ! * gradient calculation over large dataset (if dataset is large enough) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - trainer object; Ensemble - neural network ensemble. It must have same number of inputs and outputs/classes as was specified during creation of the trainer object. NRestarts - number of restarts, >=0: * NRestarts>0 means that specified number of random restarts are performed during each ES round; * NRestarts=0 is silently replaced by 1. OUTPUT PARAMETERS: Ensemble - trained ensemble; Rep - it contains all type of errors. NOTE: this training method uses BOTH early stopping and weight decay! So, you should select weight decay before starting training just as you select it before training "conventional" networks. NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(), or single-point dataset was passed, ensemble is filled by zero values. NOTE: this method uses sum-of-squares error function for training. -- ALGLIB -- Copyright 22.08.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlptrainensemblees( mlptrainer s, mlpensemble ensemble, ae_int_t nrestarts, mlpreport& rep); void alglib::smp_mlptrainensemblees( mlptrainer s, mlpensemble ensemble, ae_int_t nrestarts, mlpreport& rep);
mlptraines function/************************************************************************* Neural network training using early stopping (base algorithm - L-BFGS with regularization). INPUT PARAMETERS: Network - neural network with initialized geometry TrnXY - training set TrnSize - training set size, TrnSize>0 ValXY - validation set ValSize - validation set size, ValSize>0 Decay - weight decay constant, >=0.001 Decay term 'Decay*||Weights||^2' is added to error function. If you don't know what Decay to choose, use 0.001. Restarts - number of restarts, either: * strictly positive number - algorithm make specified number of restarts from random position. * -1, in which case algorithm makes exactly one run from the initial state of the network (no randomization). If you don't know what Restarts to choose, choose one one the following: * -1 (deterministic start) * +1 (one random restart) * +5 (moderate amount of random restarts) OUTPUT PARAMETERS: Network - trained neural network. Info - return code: * -2, if there is a point with class number outside of [0..NOut-1]. * -1, if wrong parameters specified (NPoints<0, Restarts<1, ...). * 2, task has been solved, stopping criterion met - sufficiently small step size. Not expected (we use EARLY stopping) but possible and not an error. * 6, task has been solved, stopping criterion met - increasing of validation set error. Rep - training report NOTE: Algorithm stops if validation set error increases for a long enough or step size is small enought (there are task where validation set may decrease for eternity). In any case solution returned corresponds to the minimum of validation set error. -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlptraines( multilayerperceptron network, real_2d_array trnxy, ae_int_t trnsize, real_2d_array valxy, ae_int_t valsize, double decay, ae_int_t restarts, ae_int_t& info, mlpreport& rep);
mlptrainlbfgs function/************************************************************************* Neural network training using L-BFGS algorithm with regularization. Subroutine trains neural network with restarts from random positions. Algorithm is well suited for problems of any dimensionality (memory requirements and step complexity are linear by weights number). INPUT PARAMETERS: Network - neural network with initialized geometry XY - training set NPoints - training set size Decay - weight decay constant, >=0.001 Decay term 'Decay*||Weights||^2' is added to error function. If you don't know what Decay to choose, use 0.001. Restarts - number of restarts from random position, >0. If you don't know what Restarts to choose, use 2. WStep - stopping criterion. Algorithm stops if step size is less than WStep. Recommended value - 0.01. Zero step size means stopping after MaxIts iterations. MaxIts - stopping criterion. Algorithm stops after MaxIts iterations (NOT gradient calculations). Zero MaxIts means stopping when step is sufficiently small. OUTPUT PARAMETERS: Network - trained neural network. Info - return code: * -8, if both WStep=0 and MaxIts=0 * -2, if there is a point with class number outside of [0..NOut-1]. * -1, if wrong parameters specified (NPoints<0, Restarts<1). * 2, if task has been solved. Rep - training report -- ALGLIB -- Copyright 09.12.2007 by Bochkanov Sergey *************************************************************************/void alglib::mlptrainlbfgs( multilayerperceptron network, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, double wstep, ae_int_t maxits, ae_int_t& info, mlpreport& rep);
mlptrainlm function/************************************************************************* Neural network training using modified Levenberg-Marquardt with exact Hessian calculation and regularization. Subroutine trains neural network with restarts from random positions. Algorithm is well suited for small and medium scale problems (hundreds of weights). INPUT PARAMETERS: Network - neural network with initialized geometry XY - training set NPoints - training set size Decay - weight decay constant, >=0.001 Decay term 'Decay*||Weights||^2' is added to error function. If you don't know what Decay to choose, use 0.001. Restarts - number of restarts from random position, >0. If you don't know what Restarts to choose, use 2. OUTPUT PARAMETERS: Network - trained neural network. Info - return code: * -9, if internal matrix inverse subroutine failed * -2, if there is a point with class number outside of [0..NOut-1]. * -1, if wrong parameters specified (NPoints<0, Restarts<1). * 2, if task has been solved. Rep - training report -- ALGLIB -- Copyright 10.03.2009 by Bochkanov Sergey *************************************************************************/void alglib::mlptrainlm( multilayerperceptron network, real_2d_array xy, ae_int_t npoints, double decay, ae_int_t restarts, ae_int_t& info, mlpreport& rep);
mlptrainnetwork function/************************************************************************* This function trains neural network passed to this function, using current dataset (one which was passed to MLPSetDataset() or MLPSetSparseDataset()) and current training settings. Training from NRestarts random starting positions is performed, best network is chosen. Training is performed using current training algorithm. FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes two important improvements of ! this function: ! * multicore support (C++ and C# computational cores) ! * SSE support (C++ computational core) ! ! Second improvement gives constant speedup (2-3X). First improvement ! gives close-to-linear speedup on multicore systems. Following ! operations can be executed in parallel: ! * NRestarts training sessions performed within each of ! cross-validation rounds (if NRestarts>1) ! * gradient calculation over large dataset (if dataset is large enough) ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! In order to use SSE features you have to: ! * use commercial version of ALGLIB on Intel processors ! * use C++ computational core ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - trainer object Network - neural network. It must have same number of inputs and output/classes as was specified during creation of the trainer object. NRestarts - number of restarts, >=0: * NRestarts>0 means that specified number of random restarts are performed, best network is chosen after training * NRestarts=0 means that current state of the network is used for training. OUTPUT PARAMETERS: Network - trained network NOTE: when no dataset was specified with MLPSetDataset/SetSparseDataset(), network is filled by zero values. Same behavior for functions MLPStartTraining and MLPContinueTraining. NOTE: this method uses sum-of-squares error function for training. -- ALGLIB -- Copyright 23.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::mlptrainnetwork( mlptrainer s, multilayerperceptron network, ae_int_t nrestarts, mlpreport& rep); void alglib::smp_mlptrainnetwork( mlptrainer s, multilayerperceptron network, ae_int_t nrestarts, mlpreport& rep);
Examples: [1] [2] [3] [4] [5] [6]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Suppose that we want to classify numbers as positive (class 0) and negative
// (class 1). We have training set which includes several strictly positive
// or negative numbers - and zero.
//
// The problem is that we are not sure how to classify zero, so from time to
// time we mark it as positive or negative (with equal probability). Other
// numbers are marked in pure deterministic setting. How will neural network
// cope with such classification task?
//
// NOTE: we use network with excessive amount of neurons, which guarantees
// almost exact reproduction of the training set. Generalization ability
// of such network is rather low, but we are not concerned with such
// questions in this basic demo.
//
mlptrainer trn;
multilayerperceptron network;
mlpreport rep;
real_1d_array x = "[0]";
real_1d_array y = "[0,0]";
//
// Training set. One row corresponds to one record [A => class(A)].
//
// Classes are denoted by numbers from 0 to 1, where 0 corresponds to positive
// numbers and 1 to negative numbers.
//
// [ +1 0]
// [ +2 0]
// [ -1 1]
// [ -2 1]
// [ 0 0] !! sometimes we classify 0 as positive, sometimes as negative
// [ 0 1] !!
//
real_2d_array xy = "[[+1,0],[+2,0],[-1,1],[-2,1],[0,0],[0,1]]";
//
//
// When we solve classification problems, everything is slightly different from
// the regression ones:
//
// 1. Network is created. Because we solve classification problem, we use
// mlpcreatec1() function instead of mlpcreate1(). This function creates
// classifier network with SOFTMAX-normalized outputs. This network returns
// vector of class membership probabilities which are normalized to be
// non-negative and sum to 1.0
//
// 2. We use mlpcreatetrainercls() function instead of mlpcreatetrainer() to
// create trainer object. Trainer object process dataset and neural network
// slightly differently to account for specifics of the classification
// problems.
//
// 3. Dataset is attached to trainer object. Note that dataset format is slightly
// different from one used for regression.
//
mlpcreatetrainercls(1, 2, trn);
mlpcreatec1(1, 5, 2, network);
mlpsetdataset(trn, xy, 6);
//
// Network is trained with 5 restarts from random positions
//
mlptrainnetwork(trn, network, 5, rep);
//
// Test our neural network on strictly positive and strictly negative numbers.
//
// IMPORTANT! Classifier network returns class membership probabilities instead
// of class indexes. Network returns two values (probabilities) instead of one
// (class index).
//
// Thus, for +1 we expect to get [P0,P1] = [1,0], where P0 is probability that
// number is positive (belongs to class 0), and P1 is probability that number
// is negative (belongs to class 1).
//
// For -1 we expect to get [P0,P1] = [0,1]
//
// Following properties are guaranteed by network architecture:
// * P0>=0, P1>=0 non-negativity
// * P0+P1=1 normalization
//
x = "[1]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [1.000,0.000]
x = "[-1]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,1.000]
//
// But what our network will return for 0, which is between classes 0 and 1?
//
// In our dataset it has two different marks assigned (class 0 AND class 1).
// So network will return something average between class 0 and class 1:
// 0 => [0.5, 0.5]
//
x = "[0]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.500,0.500]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Suppose that we want to classify numbers as positive (class 0) and negative
// (class 1). We also have one more class for zero (class 2).
//
// NOTE: we use network with excessive amount of neurons, which guarantees
// almost exact reproduction of the training set. Generalization ability
// of such network is rather low, but we are not concerned with such
// questions in this basic demo.
//
mlptrainer trn;
multilayerperceptron network;
mlpreport rep;
real_1d_array x = "[0]";
real_1d_array y = "[0,0,0]";
//
// Training set. One row corresponds to one record [A => class(A)].
//
// Classes are denoted by numbers from 0 to 2, where 0 corresponds to positive
// numbers, 1 to negative numbers, 2 to zero
//
// [ +1 0]
// [ +2 0]
// [ -1 1]
// [ -2 1]
// [ 0 2]
//
real_2d_array xy = "[[+1,0],[+2,0],[-1,1],[-2,1],[0,2]]";
//
//
// When we solve classification problems, everything is slightly different from
// the regression ones:
//
// 1. Network is created. Because we solve classification problem, we use
// mlpcreatec1() function instead of mlpcreate1(). This function creates
// classifier network with SOFTMAX-normalized outputs. This network returns
// vector of class membership probabilities which are normalized to be
// non-negative and sum to 1.0
//
// 2. We use mlpcreatetrainercls() function instead of mlpcreatetrainer() to
// create trainer object. Trainer object process dataset and neural network
// slightly differently to account for specifics of the classification
// problems.
//
// 3. Dataset is attached to trainer object. Note that dataset format is slightly
// different from one used for regression.
//
mlpcreatetrainercls(1, 3, trn);
mlpcreatec1(1, 5, 3, network);
mlpsetdataset(trn, xy, 5);
//
// Network is trained with 5 restarts from random positions
//
mlptrainnetwork(trn, network, 5, rep);
//
// Test our neural network on strictly positive and strictly negative numbers.
//
// IMPORTANT! Classifier network returns class membership probabilities instead
// of class indexes. Network returns three values (probabilities) instead of one
// (class index).
//
// Thus, for +1 we expect to get [P0,P1,P2] = [1,0,0],
// for -1 we expect to get [P0,P1,P2] = [0,1,0],
// and for 0 we will get [P0,P1,P2] = [0,0,1].
//
// Following properties are guaranteed by network architecture:
// * P0>=0, P1>=0, P2>=0 non-negativity
// * P0+P1+P2=1 normalization
//
x = "[1]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [1.000,0.000,0.000]
x = "[-1]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,1.000,0.000]
x = "[0]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [0.000,0.000,1.000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example shows how to perform cross-validation with ALGLIB
//
mlptrainer trn;
multilayerperceptron network;
mlpreport rep;
//
// Training set: f(x)=1/(x^2+1)
// One row corresponds to one record [x,f(x)]
//
real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";
//
// Trainer object is created.
// Dataset is attached to trainer object.
//
// NOTE: it is not good idea to perform cross-validation on sample
// as small as ours (13 examples). It is done for demonstration
// purposes only. Generalization error estimates won't be
// precise enough for practical purposes.
//
mlpcreatetrainer(1, 1, trn);
mlpsetdataset(trn, xy, 13);
//
// The key property of the cross-validation is that it estimates
// generalization properties of neural ARCHITECTURE. It does NOT
// estimates generalization error of some specific network which
// is passed to the k-fold CV routine.
//
// In our example we create 1x4x1 neural network and pass it to
// CV routine without training it. Original state of the network
// is not used for cross-validation - each round is restarted from
// random initial state. Only geometry of network matters.
//
// We perform 5 restarts from different random positions for each
// of the 10 cross-validation rounds.
//
mlpcreate1(1, 4, 1, network);
mlpkfoldcv(trn, network, 5, 10, rep);
//
// Cross-validation routine stores estimates of the generalization
// error to MLP report structure. You may examine its fields and
// see estimates of different errors (RMS, CE, Avg).
//
// Because cross-validation is non-deterministic, in our manual we
// can not say what values will be stored to rep after call to
// mlpkfoldcv(). Every CV round will return slightly different
// estimates.
//
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example shows how to train early stopping ensebles.
//
mlptrainer trn;
mlpensemble ensemble;
mlpreport rep;
//
// Training set: f(x)=1/(x^2+1)
// One row corresponds to one record [x,f(x)]
//
real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";
//
// Trainer object is created.
// Dataset is attached to trainer object.
//
// NOTE: it is not good idea to use early stopping ensemble on sample
// as small as ours (13 examples). It is done for demonstration
// purposes only. Ensemble training algorithm won't find good
// solution on such small sample.
//
mlpcreatetrainer(1, 1, trn);
mlpsetdataset(trn, xy, 13);
//
// Ensemble is created and trained. Each of 50 network is trained
// with 5 restarts.
//
mlpecreate1(1, 4, 1, 50, ensemble);
mlptrainensemblees(trn, ensemble, 5, rep);
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example shows how to use parallel functionality of ALGLIB.
// We generate simple 1-dimensional regression problem and show how
// to use parallel training, parallel cross-validation, parallel
// training of neural ensembles.
//
// We assume that you already know how to use ALGLIB in serial mode
// and concentrate on its parallel capabilities.
//
// NOTE: it is not good idea to use parallel features on sample as small
// as ours (13 examples). It is done only for demonstration purposes.
//
mlptrainer trn;
multilayerperceptron network;
mlpensemble ensemble;
mlpreport rep;
real_2d_array xy = "[[-2.0,0.2],[-1.6,0.3],[-1.3,0.4],[-1,0.5],[-0.6,0.7],[-0.3,0.9],[0,1],[2.0,0.2],[1.6,0.3],[1.3,0.4],[1,0.5],[0.6,0.7],[0.3,0.9]]";
mlpcreatetrainer(1, 1, trn);
mlpsetdataset(trn, xy, 13);
mlpcreate1(1, 4, 1, network);
mlpecreate1(1, 4, 1, 50, ensemble);
//
// Below we demonstrate how to perform:
// * parallel training of individual networks
// * parallel cross-validation
// * parallel training of neural ensembles
//
// In order to use multithreading, you have to:
// 1) Install SMP edition of ALGLIB.
// 2) This step is specific for C++ users: you should activate OS-specific
// capabilities of ALGLIB by defining AE_OS=AE_POSIX (for *nix systems)
// or AE_OS=AE_WINDOWS (for Windows systems).
// C# users do not have to perform this step because C# programs are
// portable across different systems without OS-specific tuning.
// 3) Allow ALGLIB to know about number of worker threads to use:
// a) autodetection (C++, C#):
// ALGLIB will automatically determine number of CPU cores and
// (by default) will use all cores except for one. Say, on 4-core
// system it will use three cores - unless you manually told it
// to use more or less. It will keep your system responsive during
// lengthy computations.
// 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)
// b) manual specification (C++, C#):
// You may want to specify maximum number of worker threads during
// compile time by means of preprocessor definition AE_NWORKERS.
// For C++ it will be "AE_NWORKERS=X" where X can be any positive number.
// For C# it is "AE_NWORKERSX", where X should be replaced by number of
// workers (AE_NWORKERS2, AE_NWORKERS3, AE_NWORKERS4, ...).
// You can add this definition to compiler command line or change
// corresponding project settings in your IDE.
//
// After you installed and configured SMP edition of ALGLIB, you may choose
// between serial and multithreaded versions of SMP-capable functions:
// * serial version works as usual, in the context of the calling thread
// * multithreaded version (with "smp_" prefix) creates (or wakes up) worker
// threads, inserts task in the worker queue, and waits for completion of
// the task. All processing is done in context of worker thread(s).
//
// NOTE: because starting/stopping worker threads costs thousands of CPU cycles,
// you should not use multithreading for lightweight computational problems.
//
// NOTE: some old POSIX-compatible operating systems do not support
// sysconf(_SC_NPROCESSORS_ONLN) system call which is required in order
// to automatically determine number of active cores. On these systems
// you should specify number of cores manually at compile time.
// Without it ALGLIB will run in single-threaded mode.
//
//
// First, we perform parallel training of individual network with 5
// restarts from random positions. These 5 rounds of training are
// executed in parallel manner, with best network chosen after
// training.
//
// ALGLIB can use additional way to speed up computations - divide
// dataset into smaller subsets and process these subsets
// simultaneously. It allows us to efficiently parallelize even
// single training round. This operation is performed automatically
// for large datasets, but our toy dataset is too small.
//
smp_mlptrainnetwork(trn, network, 5, rep);
//
// Then, we perform parallel 10-fold cross-validation, with 5 random
// restarts per each CV round. I.e., 5*10=50 networks are trained
// in total. All these operations can be parallelized.
//
// NOTE: again, ALGLIB can parallelize calculation of gradient
// over entire dataset - but our dataset is too small.
//
smp_mlpkfoldcv(trn, network, 5, 10, rep);
//
// Finally, we train early stopping ensemble of 50 neural networks,
// each of them is trained with 5 random restarts. I.e., 5*50=250
// networks aretrained in total.
//
smp_mlptrainensemblees(trn, ensemble, 5, rep);
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// The very simple example on neural network: network is trained to reproduce
// small 2x2 multiplication table.
//
// NOTE: we use network with excessive amount of neurons, which guarantees
// almost exact reproduction of the training set. Generalization ability
// of such network is rather low, but we are not concerned with such
// questions in this basic demo.
//
mlptrainer trn;
multilayerperceptron network;
mlpreport rep;
//
// Training set:
// * one row corresponds to one record A*B=C in the multiplication table
// * first two columns store A and B, last column stores C
//
// [1 * 1 = 1]
// [1 * 2 = 2]
// [2 * 1 = 2]
// [2 * 2 = 4]
//
real_2d_array xy = "[[1,1,1],[1,2,2],[2,1,2],[2,2,4]]";
//
// Network is created.
// Trainer object is created.
// Dataset is attached to trainer object.
//
mlpcreatetrainer(2, 1, trn);
mlpcreate1(2, 5, 1, network);
mlpsetdataset(trn, xy, 4);
//
// Network is trained with 5 restarts from random positions
//
mlptrainnetwork(trn, network, 5, rep);
//
// 2*2=?
//
real_1d_array x = "[2,2]";
real_1d_array y = "[0]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [4.000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Network with 2 inputs and 2 outputs is trained to reproduce vector function:
// (x0,x1) => (x0+x1, x0*x1)
//
// Informally speaking, we want neural network to simultaneously calculate
// both sum of two numbers and their product.
//
// NOTE: we use network with excessive amount of neurons, which guarantees
// almost exact reproduction of the training set. Generalization ability
// of such network is rather low, but we are not concerned with such
// questions in this basic demo.
//
mlptrainer trn;
multilayerperceptron network;
mlpreport rep;
//
// Training set. One row corresponds to one record [A,B,A+B,A*B].
//
// [ 1 1 1+1 1*1 ]
// [ 1 2 1+2 1*2 ]
// [ 2 1 2+1 2*1 ]
// [ 2 2 2+2 2*2 ]
//
real_2d_array xy = "[[1,1,2,1],[1,2,3,2],[2,1,3,2],[2,2,4,4]]";
//
// Network is created.
// Trainer object is created.
// Dataset is attached to trainer object.
//
mlpcreatetrainer(2, 2, trn);
mlpcreate1(2, 5, 2, network);
mlpsetdataset(trn, xy, 4);
//
// Network is trained with 5 restarts from random positions
//
mlptrainnetwork(trn, network, 5, rep);
//
// 2+1=?
// 2*1=?
//
real_1d_array x = "[2,1]";
real_1d_array y = "[0,0]";
mlpprocess(network, x, y);
printf("%s\n", y.tostring(1).c_str()); // EXPECTED: [3.000,2.000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "dataanalysis.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Trainer object is used to train network. It stores dataset, training settings,
// and other information which is NOT part of neural network. You should use
// trainer object as follows:
// (1) you create trainer object and specify task type (classification/regression)
// and number of inputs/outputs
// (2) you add dataset to the trainer object
// (3) you may change training settings (stopping criteria or weight decay)
// (4) finally, you may train one or more networks
//
// You may interleave stages 2...4 and repeat them many times. Trainer object
// remembers its internal state and can be used several times after its creation
// and initialization.
//
mlptrainer trn;
//
// Stage 1: object creation.
//
// We have to specify number of inputs and outputs. Trainer object can be used
// only for problems with same number of inputs/outputs as was specified during
// its creation.
//
// In case you want to train SOFTMAX-normalized network which solves classification
// problems, you must use another function to create trainer object:
// mlpcreatetrainercls().
//
// Below we create trainer object which can be used to train regression networks
// with 2 inputs and 1 output.
//
mlpcreatetrainer(2, 1, trn);
//
// Stage 2: specification of the training set
//
// By default trainer object stores empty dataset. So to solve your non-empty problem
// you have to set dataset by passing to trainer dense or sparse matrix.
//
// One row of the matrix corresponds to one record A*B=C in the multiplication table.
// First two columns store A and B, last column stores C
//
// [1 * 1 = 1] [ 1 1 1 ]
// [1 * 2 = 2] [ 1 2 2 ]
// [2 * 1 = 2] = [ 2 1 2 ]
// [2 * 2 = 4] [ 2 2 4 ]
//
real_2d_array xy = "[[1,1,1],[1,2,2],[2,1,2],[2,2,4]]";
mlpsetdataset(trn, xy, 4);
//
// Stage 3: modification of the training parameters.
//
// You may modify parameters like weights decay or stopping criteria:
// * we set moderate weight decay
// * we choose iterations limit as stopping condition (another condition - step size -
// is zero, which means than this condition is not active)
//
double wstep = 0.000;
ae_int_t maxits = 100;
mlpsetdecay(trn, 0.01);
mlpsetcond(trn, wstep, maxits);
//
// Stage 4: training.
//
// We will train several networks with different architecture using same trainer object.
// We may change training parameters or even dataset, so different networks are trained
// differently. But in this simple example we will train all networks with same settings.
//
// We create and train three networks:
// * network 1 has 2x1 architecture (2 inputs, no hidden neurons, 1 output)
// * network 2 has 2x5x1 architecture (2 inputs, 5 hidden neurons, 1 output)
// * network 3 has 2x5x5x1 architecture (2 inputs, two hidden layers, 1 output)
//
// NOTE: these networks solve regression problems. For classification problems you
// should use mlpcreatec0/c1/c2 to create neural networks which have SOFTMAX-
// normalized outputs.
//
multilayerperceptron net1;
multilayerperceptron net2;
multilayerperceptron net3;
mlpreport rep;
mlpcreate0(2, 1, net1);
mlpcreate1(2, 5, 1, net2);
mlpcreate2(2, 5, 5, 1, net3);
mlptrainnetwork(trn, net1, 5, rep);
mlptrainnetwork(trn, net2, 5, rep);
mlptrainnetwork(trn, net3, 5, rep);
return 0;
}
nearestneighbor subpackage| nneighbor_d_1 | Nearest neighbor search, KNN queries | |
| nneighbor_d_2 | Serialization of KD-trees |
kdtree class/************************************************************************* KD-tree object. *************************************************************************/class kdtree { };
kdtreerequestbuffer class/************************************************************************* Buffer object which is used to perform nearest neighbor requests in the multithreaded mode (multiple threads working with same KD-tree object). This object should be created with KDTreeCreateBuffer(). *************************************************************************/class kdtreerequestbuffer { };
kdtreebuild function/************************************************************************* KD-tree creation This subroutine creates KD-tree from set of X-values and optional Y-values INPUT PARAMETERS XY - dataset, array[0..N-1,0..NX+NY-1]. one row corresponds to one point. first NX columns contain X-values, next NY (NY may be zero) columns may contain associated Y-values N - number of points, N>=0. NX - space dimension, NX>=1. NY - number of optional Y-values, NY>=0. NormType- norm type: * 0 denotes infinity-norm * 1 denotes 1-norm * 2 denotes 2-norm (Euclidean norm) OUTPUT PARAMETERS KDT - KD-tree NOTES 1. KD-tree creation have O(N*logN) complexity and O(N*(2*NX+NY)) memory requirements. 2. Although KD-trees may be used with any combination of N and NX, they are more efficient than brute-force search only when N >> 4^NX. So they are most useful in low-dimensional tasks (NX=2, NX=3). NX=1 is another inefficient case, because simple binary search (without additional structures) is much more efficient in such tasks than KD-trees. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreebuild( real_2d_array xy, ae_int_t nx, ae_int_t ny, ae_int_t normtype, kdtree& kdt); void alglib::kdtreebuild( real_2d_array xy, ae_int_t n, ae_int_t nx, ae_int_t ny, ae_int_t normtype, kdtree& kdt);
kdtreebuildtagged function/************************************************************************* KD-tree creation This subroutine creates KD-tree from set of X-values, integer tags and optional Y-values INPUT PARAMETERS XY - dataset, array[0..N-1,0..NX+NY-1]. one row corresponds to one point. first NX columns contain X-values, next NY (NY may be zero) columns may contain associated Y-values Tags - tags, array[0..N-1], contains integer tags associated with points. N - number of points, N>=0 NX - space dimension, NX>=1. NY - number of optional Y-values, NY>=0. NormType- norm type: * 0 denotes infinity-norm * 1 denotes 1-norm * 2 denotes 2-norm (Euclidean norm) OUTPUT PARAMETERS KDT - KD-tree NOTES 1. KD-tree creation have O(N*logN) complexity and O(N*(2*NX+NY)) memory requirements. 2. Although KD-trees may be used with any combination of N and NX, they are more efficient than brute-force search only when N >> 4^NX. So they are most useful in low-dimensional tasks (NX=2, NX=3). NX=1 is another inefficient case, because simple binary search (without additional structures) is much more efficient in such tasks than KD-trees. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreebuildtagged( real_2d_array xy, integer_1d_array tags, ae_int_t nx, ae_int_t ny, ae_int_t normtype, kdtree& kdt); void alglib::kdtreebuildtagged( real_2d_array xy, integer_1d_array tags, ae_int_t n, ae_int_t nx, ae_int_t ny, ae_int_t normtype, kdtree& kdt);
Examples: [1]
kdtreecreaterequestbuffer function/************************************************************************* This function creates buffer structure which can be used to perform parallel KD-tree requests. KD-tree subpackage provides two sets of request functions - ones which use internal buffer of KD-tree object (these functions are single-threaded because they use same buffer, which can not shared between threads), and ones which use external buffer. This function is used to initialize external buffer. INPUT PARAMETERS KDT - KD-tree which is associated with newly created buffer OUTPUT PARAMETERS Buf - external buffer. IMPORTANT: KD-tree buffer should be used only with KD-tree object which was used to initialize buffer. Any attempt to use biffer with different object is dangerous - you may get integrity check failure (exception) because sizes of internal arrays do not fit to dimensions of KD-tree structure. -- ALGLIB -- Copyright 18.03.2016 by Bochkanov Sergey *************************************************************************/void alglib::kdtreecreaterequestbuffer( kdtree kdt, kdtreerequestbuffer& buf);
kdtreequeryaknn function/************************************************************************* K-NN query: approximate K nearest neighbors IMPORTANT: this function can not be used in multithreaded code because it uses internal temporary buffer of kd-tree object, which can not be shared between multiple threads. If you want to perform parallel requests, use function which uses external request buffer: KDTreeTsQueryAKNN() ("Ts" stands for "thread-safe"). INPUT PARAMETERS KDT - KD-tree X - point, array[0..NX-1]. K - number of neighbors to return, K>=1 SelfMatch - whether self-matches are allowed: * if True, nearest neighbor may be the point itself (if it exists in original dataset) * if False, then only points with non-zero distance are returned * if not given, considered True Eps - approximation factor, Eps>=0. eps-approximate nearest neighbor is a neighbor whose distance from X is at most (1+eps) times distance of true nearest neighbor. RESULT number of actual neighbors found (either K or N, if K>N). NOTES significant performance gain may be achieved only when Eps is is on the order of magnitude of 1 or larger. This subroutine performs query and stores its result in the internal structures of the KD-tree. You can use following subroutines to obtain these results: * KDTreeQueryResultsX() to get X-values * KDTreeQueryResultsXY() to get X- and Y-values * KDTreeQueryResultsTags() to get tag values * KDTreeQueryResultsDistances() to get distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreequeryaknn( kdtree kdt, real_1d_array x, ae_int_t k, double eps); ae_int_t alglib::kdtreequeryaknn( kdtree kdt, real_1d_array x, ae_int_t k, bool selfmatch, double eps);
Examples: [1]
kdtreequerybox function/************************************************************************* Box query: all points within user-specified box. IMPORTANT: this function can not be used in multithreaded code because it uses internal temporary buffer of kd-tree object, which can not be shared between multiple threads. If you want to perform parallel requests, use function which uses external request buffer: KDTreeTsQueryBox() ("Ts" stands for "thread-safe"). INPUT PARAMETERS KDT - KD-tree BoxMin - lower bounds, array[0..NX-1]. BoxMax - upper bounds, array[0..NX-1]. RESULT number of actual neighbors found (in [0,N]). This subroutine performs query and stores its result in the internal structures of the KD-tree. You can use following subroutines to obtain these results: * KDTreeQueryResultsX() to get X-values * KDTreeQueryResultsXY() to get X- and Y-values * KDTreeQueryResultsTags() to get tag values * KDTreeQueryResultsDistances() returns zeros for this request NOTE: this particular query returns unordered results, because there is no meaningful way of ordering points. Furthermore, no 'distance' is associated with points - it is either INSIDE or OUTSIDE (so request for distances will return zeros). -- ALGLIB -- Copyright 14.05.2016 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreequerybox( kdtree kdt, real_1d_array boxmin, real_1d_array boxmax);
kdtreequeryknn function/************************************************************************* K-NN query: K nearest neighbors IMPORTANT: this function can not be used in multithreaded code because it uses internal temporary buffer of kd-tree object, which can not be shared between multiple threads. If you want to perform parallel requests, use function which uses external request buffer: KDTreeTsQueryKNN() ("Ts" stands for "thread-safe"). INPUT PARAMETERS KDT - KD-tree X - point, array[0..NX-1]. K - number of neighbors to return, K>=1 SelfMatch - whether self-matches are allowed: * if True, nearest neighbor may be the point itself (if it exists in original dataset) * if False, then only points with non-zero distance are returned * if not given, considered True RESULT number of actual neighbors found (either K or N, if K>N). This subroutine performs query and stores its result in the internal structures of the KD-tree. You can use following subroutines to obtain these results: * KDTreeQueryResultsX() to get X-values * KDTreeQueryResultsXY() to get X- and Y-values * KDTreeQueryResultsTags() to get tag values * KDTreeQueryResultsDistances() to get distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreequeryknn(kdtree kdt, real_1d_array x, ae_int_t k); ae_int_t alglib::kdtreequeryknn( kdtree kdt, real_1d_array x, ae_int_t k, bool selfmatch);
Examples: [1]
kdtreequeryresultsdistances function/************************************************************************* Distances from last query This function retuns results stored in the internal buffer of kd-tree object. If you performed buffered requests (ones which use instances of kdtreerequestbuffer class), you should call buffered version of this function - kdtreetsqueryresultsdistances(). INPUT PARAMETERS KDT - KD-tree R - possibly pre-allocated buffer. If X is too small to store result, it is resized. If size(X) is enough to store result, it is left unchanged. OUTPUT PARAMETERS R - filled with distances (in corresponding norm) NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsX() X-values * KDTreeQueryResultsXY() X- and Y-values * KDTreeQueryResultsTags() tag values -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultsdistances(kdtree kdt, real_1d_array& r);
Examples: [1]
kdtreequeryresultsdistancesi function/************************************************************************* Distances from last query; 'interactive' variant for languages like Python which support constructs like "R = KDTreeQueryResultsDistancesI(KDT)" and interactive mode of interpreter. This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultsdistancesi(kdtree kdt, real_1d_array& r);
kdtreequeryresultstags function/************************************************************************* Tags from last query This function retuns results stored in the internal buffer of kd-tree object. If you performed buffered requests (ones which use instances of kdtreerequestbuffer class), you should call buffered version of this function - kdtreetsqueryresultstags(). INPUT PARAMETERS KDT - KD-tree Tags - possibly pre-allocated buffer. If X is too small to store result, it is resized. If size(X) is enough to store result, it is left unchanged. OUTPUT PARAMETERS Tags - filled with tags associated with points, or, when no tags were supplied, with zeros NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsX() X-values * KDTreeQueryResultsXY() X- and Y-values * KDTreeQueryResultsDistances() distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultstags(kdtree kdt, integer_1d_array& tags);
Examples: [1]
kdtreequeryresultstagsi function/************************************************************************* Tags from last query; 'interactive' variant for languages like Python which support constructs like "Tags = KDTreeQueryResultsTagsI(KDT)" and interactive mode of interpreter. This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultstagsi(kdtree kdt, integer_1d_array& tags);
kdtreequeryresultsx function/************************************************************************* X-values from last query. This function retuns results stored in the internal buffer of kd-tree object. If you performed buffered requests (ones which use instances of kdtreerequestbuffer class), you should call buffered version of this function - kdtreetsqueryresultsx(). INPUT PARAMETERS KDT - KD-tree X - possibly pre-allocated buffer. If X is too small to store result, it is resized. If size(X) is enough to store result, it is left unchanged. OUTPUT PARAMETERS X - rows are filled with X-values NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsXY() X- and Y-values * KDTreeQueryResultsTags() tag values * KDTreeQueryResultsDistances() distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultsx(kdtree kdt, real_2d_array& x);
Examples: [1]
kdtreequeryresultsxi function/************************************************************************* X-values from last query; 'interactive' variant for languages like Python which support constructs like "X = KDTreeQueryResultsXI(KDT)" and interactive mode of interpreter. This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultsxi(kdtree kdt, real_2d_array& x);
kdtreequeryresultsxy function/************************************************************************* X- and Y-values from last query This function retuns results stored in the internal buffer of kd-tree object. If you performed buffered requests (ones which use instances of kdtreerequestbuffer class), you should call buffered version of this function - kdtreetsqueryresultsxy(). INPUT PARAMETERS KDT - KD-tree XY - possibly pre-allocated buffer. If XY is too small to store result, it is resized. If size(XY) is enough to store result, it is left unchanged. OUTPUT PARAMETERS XY - rows are filled with points: first NX columns with X-values, next NY columns - with Y-values. NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsX() X-values * KDTreeQueryResultsTags() tag values * KDTreeQueryResultsDistances() distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultsxy(kdtree kdt, real_2d_array& xy);
Examples: [1]
kdtreequeryresultsxyi function/************************************************************************* XY-values from last query; 'interactive' variant for languages like Python which support constructs like "XY = KDTreeQueryResultsXYI(KDT)" and interactive mode of interpreter. This function allocates new array on each call, so it is significantly slower than its 'non-interactive' counterpart, but it is more convenient when you call it from command line. -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreequeryresultsxyi(kdtree kdt, real_2d_array& xy);
kdtreequeryrnn function/************************************************************************* R-NN query: all points within R-sphere centered at X IMPORTANT: this function can not be used in multithreaded code because it uses internal temporary buffer of kd-tree object, which can not be shared between multiple threads. If you want to perform parallel requests, use function which uses external request buffer: KDTreeTsQueryRNN() ("Ts" stands for "thread-safe"). INPUT PARAMETERS KDT - KD-tree X - point, array[0..NX-1]. R - radius of sphere (in corresponding norm), R>0 SelfMatch - whether self-matches are allowed: * if True, nearest neighbor may be the point itself (if it exists in original dataset) * if False, then only points with non-zero distance are returned * if not given, considered True RESULT number of neighbors found, >=0 This subroutine performs query and stores its result in the internal structures of the KD-tree. You can use following subroutines to obtain actual results: * KDTreeQueryResultsX() to get X-values * KDTreeQueryResultsXY() to get X- and Y-values * KDTreeQueryResultsTags() to get tag values * KDTreeQueryResultsDistances() to get distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreequeryrnn(kdtree kdt, real_1d_array x, double r); ae_int_t alglib::kdtreequeryrnn( kdtree kdt, real_1d_array x, double r, bool selfmatch);
Examples: [1]
kdtreeserialize function/************************************************************************* This function serializes data structure to string. Important properties of s_out: * it contains alphanumeric characters, dots, underscores, minus signs * these symbols are grouped into words, which are separated by spaces and Windows-style (CR+LF) newlines * although serializer uses spaces and CR+LF as separators, you can replace any separator character by arbitrary combination of spaces, tabs, Windows or Unix newlines. It allows flexible reformatting of the string in case you want to include it into text or XML file. But you should not insert separators into the middle of the "words" nor you should change case of letters. * s_out can be freely moved between 32-bit and 64-bit systems, little and big endian machines, and so on. You can serialize structure on 32-bit machine and unserialize it on 64-bit one (or vice versa), or serialize it on SPARC and unserialize on x86. You can also serialize it in C++ version of ALGLIB and unserialize in C# one, and vice versa. *************************************************************************/void kdtreeserialize(kdtree &obj, std::string &s_out); void kdtreeserialize(kdtree &obj, std::ostream &s_out);
kdtreetsqueryaknn function/************************************************************************* K-NN query: approximate K nearest neighbors, using thread-local buffer. You can call this function from multiple threads for same kd-tree instance, assuming that different instances of buffer object are passed to different threads. INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure with kdtreecreaterequestbuffer() function. X - point, array[0..NX-1]. K - number of neighbors to return, K>=1 SelfMatch - whether self-matches are allowed: * if True, nearest neighbor may be the point itself (if it exists in original dataset) * if False, then only points with non-zero distance are returned * if not given, considered True Eps - approximation factor, Eps>=0. eps-approximate nearest neighbor is a neighbor whose distance from X is at most (1+eps) times distance of true nearest neighbor. RESULT number of actual neighbors found (either K or N, if K>N). NOTES significant performance gain may be achieved only when Eps is is on the order of magnitude of 1 or larger. This subroutine performs query and stores its result in the internal structures of the buffer object. You can use following subroutines to obtain these results (pay attention to "buf" in their names): * KDTreeTsQueryResultsX() to get X-values * KDTreeTsQueryResultsXY() to get X- and Y-values * KDTreeTsQueryResultsTags() to get tag values * KDTreeTsQueryResultsDistances() to get distances IMPORTANT: kd-tree buffer should be used only with KD-tree object which was used to initialize buffer. Any attempt to use biffer with different object is dangerous - you may get integrity check failure (exception) because sizes of internal arrays do not fit to dimensions of KD-tree structure. -- ALGLIB -- Copyright 18.03.2016 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreetsqueryaknn( kdtree kdt, kdtreerequestbuffer buf, real_1d_array x, ae_int_t k, double eps); ae_int_t alglib::kdtreetsqueryaknn( kdtree kdt, kdtreerequestbuffer buf, real_1d_array x, ae_int_t k, bool selfmatch, double eps);
kdtreetsquerybox function/************************************************************************* Box query: all points within user-specified box, using thread-local buffer. You can call this function from multiple threads for same kd-tree instance, assuming that different instances of buffer object are passed to different threads. INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure with kdtreecreaterequestbuffer() function. BoxMin - lower bounds, array[0..NX-1]. BoxMax - upper bounds, array[0..NX-1]. RESULT number of actual neighbors found (in [0,N]). This subroutine performs query and stores its result in the internal structures of the buffer object. You can use following subroutines to obtain these results (pay attention to "ts" in their names): * KDTreeTsQueryResultsX() to get X-values * KDTreeTsQueryResultsXY() to get X- and Y-values * KDTreeTsQueryResultsTags() to get tag values * KDTreeTsQueryResultsDistances() returns zeros for this query NOTE: this particular query returns unordered results, because there is no meaningful way of ordering points. Furthermore, no 'distance' is associated with points - it is either INSIDE or OUTSIDE (so request for distances will return zeros). IMPORTANT: kd-tree buffer should be used only with KD-tree object which was used to initialize buffer. Any attempt to use biffer with different object is dangerous - you may get integrity check failure (exception) because sizes of internal arrays do not fit to dimensions of KD-tree structure. -- ALGLIB -- Copyright 14.05.2016 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreetsquerybox( kdtree kdt, kdtreerequestbuffer buf, real_1d_array boxmin, real_1d_array boxmax);
kdtreetsqueryknn function/************************************************************************* K-NN query: K nearest neighbors, using external thread-local buffer. You can call this function from multiple threads for same kd-tree instance, assuming that different instances of buffer object are passed to different threads. INPUT PARAMETERS KDT - kd-tree Buf - request buffer object created for this particular instance of kd-tree structure with kdtreecreaterequestbuffer() function. X - point, array[0..NX-1]. K - number of neighbors to return, K>=1 SelfMatch - whether self-matches are allowed: * if True, nearest neighbor may be the point itself (if it exists in original dataset) * if False, then only points with non-zero distance are returned * if not given, considered True RESULT number of actual neighbors found (either K or N, if K>N). This subroutine performs query and stores its result in the internal structures of the buffer object. You can use following subroutines to obtain these results (pay attention to "buf" in their names): * KDTreeTsQueryResultsX() to get X-values * KDTreeTsQueryResultsXY() to get X- and Y-values * KDTreeTsQueryResultsTags() to get tag values * KDTreeTsQueryResultsDistances() to get distances IMPORTANT: kd-tree buffer should be used only with KD-tree object which was used to initialize buffer. Any attempt to use biffer with different object is dangerous - you may get integrity check failure (exception) because sizes of internal arrays do not fit to dimensions of KD-tree structure. -- ALGLIB -- Copyright 18.03.2016 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreetsqueryknn( kdtree kdt, kdtreerequestbuffer buf, real_1d_array x, ae_int_t k); ae_int_t alglib::kdtreetsqueryknn( kdtree kdt, kdtreerequestbuffer buf, real_1d_array x, ae_int_t k, bool selfmatch);
kdtreetsqueryresultsdistances function/************************************************************************* Distances from last query associated with kdtreerequestbuffer object. This function retuns results stored in the internal buffer of kd-tree object. If you performed buffered requests (ones which use instances of kdtreerequestbuffer class), you should call buffered version of this function - KDTreeTsqueryresultsdistances(). INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure. R - possibly pre-allocated buffer. If X is too small to store result, it is resized. If size(X) is enough to store result, it is left unchanged. OUTPUT PARAMETERS R - filled with distances (in corresponding norm) NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsX() X-values * KDTreeQueryResultsXY() X- and Y-values * KDTreeQueryResultsTags() tag values -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreetsqueryresultsdistances( kdtree kdt, kdtreerequestbuffer buf, real_1d_array& r);
kdtreetsqueryresultstags function/************************************************************************* Tags from last query associated with kdtreerequestbuffer object. This function retuns results stored in the internal buffer of kd-tree object. If you performed buffered requests (ones which use instances of kdtreerequestbuffer class), you should call buffered version of this function - KDTreeTsqueryresultstags(). INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure. Tags - possibly pre-allocated buffer. If X is too small to store result, it is resized. If size(X) is enough to store result, it is left unchanged. OUTPUT PARAMETERS Tags - filled with tags associated with points, or, when no tags were supplied, with zeros NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsX() X-values * KDTreeQueryResultsXY() X- and Y-values * KDTreeQueryResultsDistances() distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreetsqueryresultstags( kdtree kdt, kdtreerequestbuffer buf, integer_1d_array& tags);
kdtreetsqueryresultsx function/************************************************************************* X-values from last query associated with kdtreerequestbuffer object. INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure. X - possibly pre-allocated buffer. If X is too small to store result, it is resized. If size(X) is enough to store result, it is left unchanged. OUTPUT PARAMETERS X - rows are filled with X-values NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsXY() X- and Y-values * KDTreeQueryResultsTags() tag values * KDTreeQueryResultsDistances() distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreetsqueryresultsx( kdtree kdt, kdtreerequestbuffer buf, real_2d_array& x);
kdtreetsqueryresultsxy function/************************************************************************* X- and Y-values from last query associated with kdtreerequestbuffer object. INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure. XY - possibly pre-allocated buffer. If XY is too small to store result, it is resized. If size(XY) is enough to store result, it is left unchanged. OUTPUT PARAMETERS XY - rows are filled with points: first NX columns with X-values, next NY columns - with Y-values. NOTES 1. points are ordered by distance from the query point (first = closest) 2. if XY is larger than required to store result, only leading part will be overwritten; trailing part will be left unchanged. So if on input XY = [[A,B],[C,D]], and result is [1,2], then on exit we will get XY = [[1,2],[C,D]]. This is done purposely to increase performance; if you want function to resize array according to result size, use function with same name and suffix 'I'. SEE ALSO * KDTreeQueryResultsX() X-values * KDTreeQueryResultsTags() tag values * KDTreeQueryResultsDistances() distances -- ALGLIB -- Copyright 28.02.2010 by Bochkanov Sergey *************************************************************************/void alglib::kdtreetsqueryresultsxy( kdtree kdt, kdtreerequestbuffer buf, real_2d_array& xy);
kdtreetsqueryrnn function/************************************************************************* R-NN query: all points within R-sphere centered at X, using external thread-local buffer. You can call this function from multiple threads for same kd-tree instance, assuming that different instances of buffer object are passed to different threads. INPUT PARAMETERS KDT - KD-tree Buf - request buffer object created for this particular instance of kd-tree structure with kdtreecreaterequestbuffer() function. X - point, array[0..NX-1]. R - radius of sphere (in corresponding norm), R>0 SelfMatch - whether self-matches are allowed: * if True, nearest neighbor may be the point itself (if it exists in original dataset) * if False, then only points with non-zero distance are returned * if not given, considered True RESULT number of neighbors found, >=0 This subroutine performs query and stores its result in the internal structures of the buffer object. You can use following subroutines to obtain these results (pay attention to "buf" in their names): * KDTreeTsQueryResultsX() to get X-values * KDTreeTsQueryResultsXY() to get X- and Y-values * KDTreeTsQueryResultsTags() to get tag values * KDTreeTsQueryResultsDistances() to get distances IMPORTANT: kd-tree buffer should be used only with KD-tree object which was used to initialize buffer. Any attempt to use biffer with different object is dangerous - you may get integrity check failure (exception) because sizes of internal arrays do not fit to dimensions of KD-tree structure. -- ALGLIB -- Copyright 18.03.2016 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::kdtreetsqueryrnn( kdtree kdt, kdtreerequestbuffer buf, real_1d_array x, double r); ae_int_t alglib::kdtreetsqueryrnn( kdtree kdt, kdtreerequestbuffer buf, real_1d_array x, double r, bool selfmatch);
kdtreeunserialize function/************************************************************************* This function unserializes data structure from string. *************************************************************************/void kdtreeunserialize(const std::string &s_in, kdtree &obj); void kdtreeunserialize(const std::istream &s_in, kdtree &obj);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "alglibmisc.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array a = "[[0,0],[0,1],[1,0],[1,1]]";
ae_int_t nx = 2;
ae_int_t ny = 0;
ae_int_t normtype = 2;
kdtree kdt;
real_1d_array x;
real_2d_array r = "[[]]";
ae_int_t k;
kdtreebuild(a, nx, ny, normtype, kdt);
x = "[-1,0]";
k = kdtreequeryknn(kdt, x, 1);
printf("%d\n", int(k)); // EXPECTED: 1
kdtreequeryresultsx(kdt, r);
printf("%s\n", r.tostring(1).c_str()); // EXPECTED: [[0,0]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "alglibmisc.h"
using namespace alglib;
int main(int argc, char **argv)
{
real_2d_array a = "[[0,0],[0,1],[1,0],[1,1]]";
ae_int_t nx = 2;
ae_int_t ny = 0;
ae_int_t normtype = 2;
kdtree kdt0;
kdtree kdt1;
std::string s;
real_1d_array x;
real_2d_array r0 = "[[]]";
real_2d_array r1 = "[[]]";
//
// Build tree and serialize it
//
kdtreebuild(a, nx, ny, normtype, kdt0);
alglib::kdtreeserialize(kdt0, s);
alglib::kdtreeunserialize(s, kdt1);
//
// Compare results from KNN queries
//
x = "[-1,0]";
kdtreequeryknn(kdt0, x, 1);
kdtreequeryresultsx(kdt0, r0);
kdtreequeryknn(kdt1, x, 1);
kdtreequeryresultsx(kdt1, r1);
printf("%s\n", r0.tostring(1).c_str()); // EXPECTED: [[0,0]]
printf("%s\n", r1.tostring(1).c_str()); // EXPECTED: [[0,0]]
return 0;
}
nleq subpackagenleqreport class/************************************************************************* *************************************************************************/class nleqreport { ae_int_t iterationscount; ae_int_t nfunc; ae_int_t njac; ae_int_t terminationtype; };
nleqstate class/************************************************************************* *************************************************************************/class nleqstate { };
nleqcreatelm function/************************************************************************* LEVENBERG-MARQUARDT-LIKE NONLINEAR SOLVER DESCRIPTION: This algorithm solves system of nonlinear equations F[0](x[0], ..., x[n-1]) = 0 F[1](x[0], ..., x[n-1]) = 0 ... F[M-1](x[0], ..., x[n-1]) = 0 with M/N do not necessarily coincide. Algorithm converges quadratically under following conditions: * the solution set XS is nonempty * for some xs in XS there exist such neighbourhood N(xs) that: * vector function F(x) and its Jacobian J(x) are continuously differentiable on N * ||F(x)|| provides local error bound on N, i.e. there exists such c1, that ||F(x)||>c1*distance(x,XS) Note that these conditions are much more weaker than usual non-singularity conditions. For example, algorithm will converge for any affine function F (whether its Jacobian singular or not). REQUIREMENTS: Algorithm will request following information during its operation: * function vector F[] and Jacobian matrix at given point X * value of merit function f(x)=F[0]^2(x)+...+F[M-1]^2(x) at given point X USAGE: 1. User initializes algorithm state with NLEQCreateLM() call 2. User tunes solver parameters with NLEQSetCond(), NLEQSetStpMax() and other functions 3. User calls NLEQSolve() function which takes algorithm state and pointers (delegates, etc.) to callback functions which calculate merit function value and Jacobian. 4. User calls NLEQResults() to get solution 5. Optionally, user may call NLEQRestartFrom() to solve another problem with same parameters (N/M) but another starting point and/or another function vector. NLEQRestartFrom() allows to reuse already initialized structure. INPUT PARAMETERS: N - space dimension, N>1: * if provided, only leading N elements of X are used * if not provided, determined automatically from size of X M - system size X - starting point OUTPUT PARAMETERS: State - structure which stores algorithm state NOTES: 1. you may tune stopping conditions with NLEQSetCond() function 2. if target function contains exp() or other fast growing functions, and optimization algorithm makes too large steps which leads to overflow, use NLEQSetStpMax() function to bound algorithm's steps. 3. this algorithm is a slightly modified implementation of the method described in 'Levenberg-Marquardt method for constrained nonlinear equations with strong local convergence properties' by Christian Kanzow Nobuo Yamashita and Masao Fukushima and further developed in 'On the convergence of a New Levenberg-Marquardt Method' by Jin-yan Fan and Ya-Xiang Yuan. -- ALGLIB -- Copyright 20.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::nleqcreatelm(ae_int_t m, real_1d_array x, nleqstate& state); void alglib::nleqcreatelm( ae_int_t n, ae_int_t m, real_1d_array x, nleqstate& state);
nleqrestartfrom function/************************************************************************* This subroutine restarts CG algorithm from new point. All optimization parameters are left unchanged. This function allows to solve multiple optimization problems (which must have same number of dimensions) without object reallocation penalty. INPUT PARAMETERS: State - structure used for reverse communication previously allocated with MinCGCreate call. X - new starting point. BndL - new lower bounds BndU - new upper bounds -- ALGLIB -- Copyright 30.07.2010 by Bochkanov Sergey *************************************************************************/void alglib::nleqrestartfrom(nleqstate state, real_1d_array x);
nleqresults function/************************************************************************* NLEQ solver results INPUT PARAMETERS: State - algorithm state. OUTPUT PARAMETERS: X - array[0..N-1], solution Rep - optimization report: * Rep.TerminationType completetion code: * -4 ERROR: algorithm has converged to the stationary point Xf which is local minimum of f=F[0]^2+...+F[m-1]^2, but is not solution of nonlinear system. * 1 sqrt(f)<=EpsF. * 5 MaxIts steps was taken * 7 stopping conditions are too stringent, further improvement is impossible * Rep.IterationsCount contains iterations count * NFEV countains number of function calculations * ActiveConstraints contains number of active constraints -- ALGLIB -- Copyright 20.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::nleqresults( nleqstate state, real_1d_array& x, nleqreport& rep);
nleqresultsbuf function/************************************************************************* NLEQ solver results Buffered implementation of NLEQResults(), which uses pre-allocated buffer to store X[]. If buffer size is too small, it resizes buffer. It is intended to be used in the inner cycles of performance critical algorithms where array reallocation penalty is too large to be ignored. -- ALGLIB -- Copyright 20.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::nleqresultsbuf( nleqstate state, real_1d_array& x, nleqreport& rep);
nleqsetcond function/************************************************************************* This function sets stopping conditions for the nonlinear solver INPUT PARAMETERS: State - structure which stores algorithm state EpsF - >=0 The subroutine finishes its work if on k+1-th iteration the condition ||F||<=EpsF is satisfied MaxIts - maximum number of iterations. If MaxIts=0, the number of iterations is unlimited. Passing EpsF=0 and MaxIts=0 simultaneously will lead to automatic stopping criterion selection (small EpsF). NOTES: -- ALGLIB -- Copyright 20.08.2010 by Bochkanov Sergey *************************************************************************/void alglib::nleqsetcond(nleqstate state, double epsf, ae_int_t maxits);
nleqsetstpmax function/************************************************************************* This function sets maximum step length INPUT PARAMETERS: State - structure which stores algorithm state StpMax - maximum step length, >=0. Set StpMax to 0.0, if you don't want to limit step length. Use this subroutine when target function contains exp() or other fast growing functions, and algorithm makes too large steps which lead to overflow. This function allows us to reject steps that are too large (and therefore expose us to the possible overflow) without actually calculating function value at the x+stp*d. -- ALGLIB -- Copyright 20.08.2010 by Bochkanov Sergey *************************************************************************/void alglib::nleqsetstpmax(nleqstate state, double stpmax);
nleqsetxrep function/************************************************************************* This function turns on/off reporting. INPUT PARAMETERS: State - structure which stores algorithm state NeedXRep- whether iteration reports are needed or not If NeedXRep is True, algorithm will call rep() callback function if it is provided to NLEQSolve(). -- ALGLIB -- Copyright 20.08.2010 by Bochkanov Sergey *************************************************************************/void alglib::nleqsetxrep(nleqstate state, bool needxrep);
nleqsolve function/************************************************************************* This family of functions is used to launcn iterations of nonlinear solver These functions accept following parameters: state - algorithm state func - callback which calculates function (or merit function) value func at given point x jac - callback which calculates function vector fi[] and Jacobian jac at given point x rep - optional callback which is called after each iteration can be NULL ptr - optional pointer which is passed to func/grad/hess/jac/rep can be NULL -- ALGLIB -- Copyright 20.03.2009 by Bochkanov Sergey *************************************************************************/void nleqsolve(nleqstate &state, void (*func)(const real_1d_array &x, double &func, void *ptr), void (*jac)(const real_1d_array &x, real_1d_array &fi, real_2d_array &jac, void *ptr), void (*rep)(const real_1d_array &x, double func, void *ptr) = NULL, void *ptr = NULL);
normaldistr subpackageerrorfunction function/************************************************************************* Error function The integral is x - 2 | | 2 erf(x) = -------- | exp( - t ) dt. sqrt(pi) | | - 0 For 0 <= |x| < 1, erf(x) = x * P4(x**2)/Q5(x**2); otherwise erf(x) = 1 - erfc(x). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,1 30000 3.7e-16 1.0e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/double alglib::errorfunction(double x);
errorfunctionc function/************************************************************************* Complementary error function 1 - erf(x) = inf. - 2 | | 2 erfc(x) = -------- | exp( - t ) dt sqrt(pi) | | - x For small x, erfc(x) = 1 - erf(x); otherwise rational approximations are computed. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0,26.6417 30000 5.7e-14 1.5e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/double alglib::errorfunctionc(double x);
inverf function/************************************************************************* Inverse of the error function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/double alglib::inverf(double e);
invnormaldistribution function/************************************************************************* Inverse of Normal distribution function Returns the argument, x, for which the area under the Gaussian probability density function (integrated from minus infinity to x) is equal to y. For small arguments 0 < y < exp(-2), the program computes z = sqrt( -2.0 * log(y) ); then the approximation is x = z - log(z)/z - (1/z) P(1/z) / Q(1/z). There are two rational functions P/Q, one for 0 < y < exp(-32) and the other for y up to exp(-2). For larger arguments, w = y - 0.5, and x/sqrt(2pi) = w + w**3 R(w**2)/S(w**2)). ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE 0.125, 1 20000 7.2e-16 1.3e-16 IEEE 3e-308, 0.135 50000 4.6e-16 9.8e-17 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invnormaldistribution(double y0);
normaldistribution function/************************************************************************* Normal distribution function Returns the area under the Gaussian probability density function, integrated from minus infinity to x: x - 1 | | 2 ndtr(x) = --------- | exp( - t /2 ) dt sqrt(2pi) | | - -inf. = ( 1 + erf(z) ) / 2 = erfc(z) / 2 where z = x/sqrt(2). Computation is via the functions erf and erfc. ACCURACY: Relative error: arithmetic domain # trials peak rms IEEE -13,0 30000 3.4e-14 6.7e-15 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1988, 1992, 2000 by Stephen L. Moshier *************************************************************************/double alglib::normaldistribution(double x);
normestimator subpackagenormestimatorstate class/************************************************************************* This object stores state of the iterative norm estimation algorithm. You should use ALGLIB functions to work with this object. *************************************************************************/class normestimatorstate { };
normestimatorcreate function/************************************************************************* This procedure initializes matrix norm estimator. USAGE: 1. User initializes algorithm state with NormEstimatorCreate() call 2. User calls NormEstimatorEstimateSparse() (or NormEstimatorIteration()) 3. User calls NormEstimatorResults() to get solution. INPUT PARAMETERS: M - number of rows in the matrix being estimated, M>0 N - number of columns in the matrix being estimated, N>0 NStart - number of random starting vectors recommended value - at least 5. NIts - number of iterations to do with best starting vector recommended value - at least 5. OUTPUT PARAMETERS: State - structure which stores algorithm state NOTE: this algorithm is effectively deterministic, i.e. it always returns same result when repeatedly called for the same matrix. In fact, algorithm uses randomized starting vectors, but internal random numbers generator always generates same sequence of the random values (it is a feature, not bug). Algorithm can be made non-deterministic with NormEstimatorSetSeed(0) call. -- ALGLIB -- Copyright 06.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::normestimatorcreate( ae_int_t m, ae_int_t n, ae_int_t nstart, ae_int_t nits, normestimatorstate& state);
normestimatorestimatesparse function/************************************************************************* This function estimates norm of the sparse M*N matrix A. INPUT PARAMETERS: State - norm estimator state, must be initialized with a call to NormEstimatorCreate() A - sparse M*N matrix, must be converted to CRS format prior to calling this function. After this function is over you can call NormEstimatorResults() to get estimate of the norm(A). -- ALGLIB -- Copyright 06.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::normestimatorestimatesparse( normestimatorstate state, sparsematrix a);
normestimatorresults function/************************************************************************* Matrix norm estimation results INPUT PARAMETERS: State - algorithm state OUTPUT PARAMETERS: Nrm - estimate of the matrix norm, Nrm>=0 -- ALGLIB -- Copyright 06.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::normestimatorresults(normestimatorstate state, double& nrm);
normestimatorsetseed function/************************************************************************* This function changes seed value used by algorithm. In some cases we need deterministic processing, i.e. subsequent calls must return equal results, in other cases we need non-deterministic algorithm which returns different results for the same matrix on every pass. Setting zero seed will lead to non-deterministic algorithm, while non-zero value will make our algorithm deterministic. INPUT PARAMETERS: State - norm estimator state, must be initialized with a call to NormEstimatorCreate() SeedVal - seed value, >=0. Zero value = non-deterministic algo. -- ALGLIB -- Copyright 06.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::normestimatorsetseed( normestimatorstate state, ae_int_t seedval);
nsfit subpackagensfitspheremcc function/************************************************************************* Fits minimum circumscribed (MCC) circle (or NX-dimensional sphere) to data (a set of points in NX-dimensional space). INPUT PARAMETERS: XY - array[NPoints,NX] (or larger), contains dataset. One row = one point in NX-dimensional space. NPoints - dataset size, NPoints>0 NX - space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on) OUTPUT PARAMETERS: CX - central point for a sphere RHi - radius NOTE: this function is an easy-to-use wrapper around more powerful "expert" function nsfitspherex(). This wrapper is optimized for ease of use and stability - at the cost of somewhat lower performance (we have to use very tight stopping criteria for inner optimizer because we want to make sure that it will converge on any dataset). If you are ready to experiment with settings of "expert" function, you can achieve ~2-4x speedup over standard "bulletproof" settings. -- ALGLIB -- Copyright 14.04.2017 by Bochkanov Sergey *************************************************************************/void alglib::nsfitspheremcc( real_2d_array xy, ae_int_t npoints, ae_int_t nx, real_1d_array& cx, double& rhi);
nsfitspheremic function/************************************************************************* Fits maximum inscribed circle (or NX-dimensional sphere) to data (a set of points in NX-dimensional space). INPUT PARAMETERS: XY - array[NPoints,NX] (or larger), contains dataset. One row = one point in NX-dimensional space. NPoints - dataset size, NPoints>0 NX - space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on) OUTPUT PARAMETERS: CX - central point for a sphere RLo - radius NOTE: this function is an easy-to-use wrapper around more powerful "expert" function nsfitspherex(). This wrapper is optimized for ease of use and stability - at the cost of somewhat lower performance (we have to use very tight stopping criteria for inner optimizer because we want to make sure that it will converge on any dataset). If you are ready to experiment with settings of "expert" function, you can achieve ~2-4x speedup over standard "bulletproof" settings. -- ALGLIB -- Copyright 14.04.2017 by Bochkanov Sergey *************************************************************************/void alglib::nsfitspheremic( real_2d_array xy, ae_int_t npoints, ae_int_t nx, real_1d_array& cx, double& rlo);
nsfitspheremzc function/************************************************************************* Fits minimum zone circle (or NX-dimensional sphere) to data (a set of points in NX-dimensional space). INPUT PARAMETERS: XY - array[NPoints,NX] (or larger), contains dataset. One row = one point in NX-dimensional space. NPoints - dataset size, NPoints>0 NX - space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on) OUTPUT PARAMETERS: CX - central point for a sphere RLo - radius of inscribed circle RHo - radius of circumscribed circle NOTE: this function is an easy-to-use wrapper around more powerful "expert" function nsfitspherex(). This wrapper is optimized for ease of use and stability - at the cost of somewhat lower performance (we have to use very tight stopping criteria for inner optimizer because we want to make sure that it will converge on any dataset). If you are ready to experiment with settings of "expert" function, you can achieve ~2-4x speedup over standard "bulletproof" settings. -- ALGLIB -- Copyright 14.04.2017 by Bochkanov Sergey *************************************************************************/void alglib::nsfitspheremzc( real_2d_array xy, ae_int_t npoints, ae_int_t nx, real_1d_array& cx, double& rlo, double& rhi);
nsfitspherex function/************************************************************************* Fitting minimum circumscribed, maximum inscribed or minimum zone circles (or NX-dimensional spheres) to data (a set of points in NX-dimensional space). This is expert function which allows to tweak many parameters of underlying nonlinear solver: * stopping criteria for inner iterations * number of outer iterations * penalty coefficient used to handle nonlinear constraints (we convert unconstrained nonsmooth optimization problem ivolving max() and/or min() operations to quadratically constrained smooth one). You may tweak all these parameters or only some of them, leaving other ones at their default state - just specify zero value, and solver will fill it with appropriate default one. These comments also include some discussion of approach used to handle such unusual fitting problem, its stability, drawbacks of alternative methods, and convergence properties. INPUT PARAMETERS: XY - array[NPoints,NX] (or larger), contains dataset. One row = one point in NX-dimensional space. NPoints - dataset size, NPoints>0 NX - space dimensionality, NX>0 (1, 2, 3, 4, 5 and so on) ProblemType-used to encode problem type: * 1 for minimum circumscribed circle/sphere fitting (MCC) * 2 for maximum inscribed circle/sphere fitting (MIC) * 3 for minimum zone circle fitting (difference between Rhi and Rlo is minimized), denoted as MZC EpsX - stopping condition for NLC optimizer: * must be non-negative * use 0 to choose default value (1.0E-12 is used by default) * you may specify larger values, up to 1.0E-6, if you want to speed-up solver; NLC solver performs several preconditioned outer iterations, so final result typically has precision much better than EpsX. AULIts - number of outer iterations performed by NLC optimizer: * must be non-negative * use 0 to choose default value (20 is used by default) * you may specify values smaller than 20 if you want to speed up solver; 10 often results in good combination of precision and speed; sometimes you may get good results with just 6 outer iterations. Penalty - penalty coefficient for NLC optimizer: * must be non-negative * use 0 to choose default value (1.0E6 in current version) * it should be really large, 1.0E6...1.0E7 is a good value to start from; * generally, default value is good enough OUTPUT PARAMETERS: CX - central point for a sphere RLo - radius: * for ProblemType=2,3, radius of the inscribed sphere * for ProblemType=1 - zero RHo - radius: * for ProblemType=1,3, radius of the circumscribed sphere * for ProblemType=2 - zero NOTE: ON THE UNIQUENESS OF SOLUTIONS ALGLIB provides solution to several related circle fitting problems: MCC (minimum circumscribed), MIC (maximum inscribed) and MZC (minimum zone) fitting. It is important to note that among these problems only MCC is convex and has unique solution independently from starting point. As for MIC, it may (or may not, depending on dataset properties) have multiple solutions, and it always has one degenerate solution C=infinity which corresponds to infinitely large radius. Thus, there are no guarantees that solution to MIC returned by this solver will be the best one (and no one can provide you with such guarantee because problem is NP-hard). The only guarantee you have is that this solution is locally optimal, i.e. it can not be improved by infinitesimally small tweaks in the parameters. It is also possible to "run away" to infinity when started from bad initial point located outside of point cloud (or when point cloud does not span entire circumference/surface of the sphere). Finally, MZC (minimum zone circle) stands somewhere between MCC and MIC in stability. It is somewhat regularized by "circumscribed" term of the merit function; however, solutions to MZC may be non-unique, and in some unlucky cases it is also possible to "run away to infinity". NOTE: ON THE NONLINEARLY CONSTRAINED PROGRAMMING APPROACH The problem formulation for MCC (minimum circumscribed circle; for the sake of simplicity we omit MZC and MIC here) is: [ [ ]2 ] min [ max [ XY[i]-C ] ] C [ i [ ] ] i.e. it is unconstrained nonsmooth optimization problem of finding "best" central point, with radius R being unambiguously determined from C. In order to move away from non-smoothness we use following reformulation: [ ] [ ]2 min [ R ] subject to R>=0, [ XY[i]-C ] <= R^2 C,R [ ] [ ] i.e. it becomes smooth quadratically constrained optimization problem with linear target function. Such problem statement is 100% equivalent to the original nonsmooth one, but much easier to approach. We solve it with MinNLC solver provided by ALGLIB. NOTE: ON INSTABILITY OF SEQUENTIAL LINEAR PROGRAMMING APPROACJ ALGLIB has nonlinearly constrained solver which proved to be stable on such problems. However, some authors proposed to linearize constraints in the vicinity of current approximation (Ci,Ri) and to get next approximate solution (Ci+1,Ri+1) as solution to linear programming problem. Obviously, LP problems are easier than nonlinearly constrained ones. Indeed, SLP approach to MCC/MIC/MZC resulted in ~10-20x increase in performance (when compared with NLC solver). However, it turned out that in some cases linearized model fails to predict correct direction for next step and tells us that we converged to solution even when we are still 2-4 digits of precision away from it. It is important that it is not failure of LP solver - it is failure of the linear model; even when solved exactly, it fails to handle subtle nonlinearities which arise near the solution. We validated it by comparing results returned by ALGLIB linear solver with that of MATLAB. In our experiments with SLP solver: * MCC failed most often, at both realistic and synthetic datasets * MIC sometimes failed, but sometimes succeeded * MZC often succeeded; our guess is that presence of two independent sets of constraints (one set for Rlo and another one for Rhi) and two terms in the target function (Rlo and Rhi) regularizes task, so when linear model fails to handle nonlinearities from Rlo, it uses Rhi as a hint (and vice versa). Because SLP approach failed to achieve stable results, we do not include it in ALGLIB. -- ALGLIB -- Copyright 14.04.2017 by Bochkanov Sergey *************************************************************************/void alglib::nsfitspherex( real_2d_array xy, ae_int_t npoints, ae_int_t nx, ae_int_t problemtype, double epsx, ae_int_t aulits, double penalty, real_1d_array& cx, double& rlo, double& rhi);
odesolver subpackage| odesolver_d1 | Solving y'=-y with ODE solver |
odesolverreport class/************************************************************************* *************************************************************************/class odesolverreport { ae_int_t nfev; ae_int_t terminationtype; };
odesolverstate class/************************************************************************* *************************************************************************/class odesolverstate { };
odesolverresults function/************************************************************************* ODE solver results Called after OdeSolverIteration returned False. INPUT PARAMETERS: State - algorithm state (used by OdeSolverIteration). OUTPUT PARAMETERS: M - number of tabulated values, M>=1 XTbl - array[0..M-1], values of X YTbl - array[0..M-1,0..N-1], values of Y in X[i] Rep - solver report: * Rep.TerminationType completetion code: * -2 X is not ordered by ascending/descending or there are non-distinct X[], i.e. X[i]=X[i+1] * -1 incorrect parameters were specified * 1 task has been solved * Rep.NFEV contains number of function calculations -- ALGLIB -- Copyright 01.09.2009 by Bochkanov Sergey *************************************************************************/void alglib::odesolverresults( odesolverstate state, ae_int_t& m, real_1d_array& xtbl, real_2d_array& ytbl, odesolverreport& rep);
Examples: [1]
odesolverrkck function/************************************************************************* Cash-Karp adaptive ODE solver. This subroutine solves ODE Y'=f(Y,x) with initial conditions Y(xs)=Ys (here Y may be single variable or vector of N variables). INPUT PARAMETERS: Y - initial conditions, array[0..N-1]. contains values of Y[] at X[0] N - system size X - points at which Y should be tabulated, array[0..M-1] integrations starts at X[0], ends at X[M-1], intermediate values at X[i] are returned too. SHOULD BE ORDERED BY ASCENDING OR BY DESCENDING! M - number of intermediate points + first point + last point: * M>2 means that you need both Y(X[M-1]) and M-2 values at intermediate points * M=2 means that you want just to integrate from X[0] to X[1] and don't interested in intermediate values. * M=1 means that you don't want to integrate :) it is degenerate case, but it will be handled correctly. * M<1 means error Eps - tolerance (absolute/relative error on each step will be less than Eps). When passing: * Eps>0, it means desired ABSOLUTE error * Eps<0, it means desired RELATIVE error. Relative errors are calculated with respect to maximum values of Y seen so far. Be careful to use this criterion when starting from Y[] that are close to zero. H - initial step lenth, it will be adjusted automatically after the first step. If H=0, step will be selected automatically (usualy it will be equal to 0.001 of min(x[i]-x[j])). OUTPUT PARAMETERS State - structure which stores algorithm state between subsequent calls of OdeSolverIteration. Used for reverse communication. This structure should be passed to the OdeSolverIteration subroutine. SEE ALSO AutoGKSmoothW, AutoGKSingular, AutoGKIteration, AutoGKResults. -- ALGLIB -- Copyright 01.09.2009 by Bochkanov Sergey *************************************************************************/void alglib::odesolverrkck( real_1d_array y, real_1d_array x, double eps, double h, odesolverstate& state); void alglib::odesolverrkck( real_1d_array y, ae_int_t n, real_1d_array x, ae_int_t m, double eps, double h, odesolverstate& state);
Examples: [1]
odesolversolve function/************************************************************************* This function is used to launcn iterations of ODE solver It accepts following parameters: diff - callback which calculates dy/dx for given y and x ptr - optional pointer which is passed to diff; can be NULL -- ALGLIB -- Copyright 01.09.2009 by Bochkanov Sergey *************************************************************************/void odesolversolve(odesolverstate &state, void (*diff)(const real_1d_array &y, double x, real_1d_array &dy, void *ptr), void *ptr = NULL);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "diffequations.h"
using namespace alglib;
void ode_function_1_diff(const real_1d_array &y, double x, real_1d_array &dy, void *ptr)
{
// this callback calculates f(y[],x)=-y[0]
dy[0] = -y[0];
}
int main(int argc, char **argv)
{
real_1d_array y = "[1]";
real_1d_array x = "[0, 1, 2, 3]";
double eps = 0.00001;
double h = 0;
odesolverstate s;
ae_int_t m;
real_1d_array xtbl;
real_2d_array ytbl;
odesolverreport rep;
odesolverrkck(y, x, eps, h, s);
alglib::odesolversolve(s, ode_function_1_diff);
odesolverresults(s, m, xtbl, ytbl, rep);
printf("%d\n", int(m)); // EXPECTED: 4
printf("%s\n", xtbl.tostring(2).c_str()); // EXPECTED: [0, 1, 2, 3]
printf("%s\n", ytbl.tostring(2).c_str()); // EXPECTED: [[1], [0.367], [0.135], [0.050]]
return 0;
}
ortfac subpackagecmatrixlq function/************************************************************************* LQ decomposition of a rectangular complex matrix of size MxN COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix A whose indexes range within [0..M-1, 0..N-1] M - number of rows in matrix A. N - number of columns in matrix A. Output parameters: A - matrices Q and L in compact form Tau - array of scalar factors which are used to form matrix Q. Array whose indexes range within [0.. Min(M,N)-1] Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size MxM, L - lower triangular (or lower trapezoid) matrix of size MxN. -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994 *************************************************************************/void alglib::cmatrixlq( complex_2d_array& a, ae_int_t m, ae_int_t n, complex_1d_array& tau); void alglib::smp_cmatrixlq( complex_2d_array& a, ae_int_t m, ae_int_t n, complex_1d_array& tau);
cmatrixlqunpackl function/************************************************************************* Unpacking of matrix L from the LQ decomposition of a matrix A Input parameters: A - matrices Q and L in compact form. Output of CMatrixLQ subroutine. M - number of rows in given matrix A. M>=0. N - number of columns in given matrix A. N>=0. Output parameters: L - matrix L, array[0..M-1, 0..N-1]. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixlqunpackl( complex_2d_array a, ae_int_t m, ae_int_t n, complex_2d_array& l);
cmatrixlqunpackq function/************************************************************************* Partial unpacking of matrix Q from LQ decomposition of a complex matrix A. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrices Q and R in compact form. Output of CMatrixLQ subroutine . M - number of rows in matrix A. M>=0. N - number of columns in matrix A. N>=0. Tau - scalar factors which are used to form Q. Output of CMatrixLQ subroutine . QRows - required number of rows in matrix Q. N>=QColumns>=0. Output parameters: Q - first QRows rows of matrix Q. Array whose index ranges within [0..QRows-1, 0..N-1]. If QRows=0, array isn't changed. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixlqunpackq( complex_2d_array a, ae_int_t m, ae_int_t n, complex_1d_array tau, ae_int_t qrows, complex_2d_array& q); void alglib::smp_cmatrixlqunpackq( complex_2d_array a, ae_int_t m, ae_int_t n, complex_1d_array tau, ae_int_t qrows, complex_2d_array& q);
cmatrixqr function/************************************************************************* QR decomposition of a rectangular complex matrix of size MxN COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix A whose indexes range within [0..M-1, 0..N-1] M - number of rows in matrix A. N - number of columns in matrix A. Output parameters: A - matrices Q and R in compact form Tau - array of scalar factors which are used to form matrix Q. Array whose indexes range within [0.. Min(M,N)-1] Matrix A is represented as A = QR, where Q is an orthogonal matrix of size MxM, R - upper triangular (or upper trapezoid) matrix of size MxN. -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994 *************************************************************************/void alglib::cmatrixqr( complex_2d_array& a, ae_int_t m, ae_int_t n, complex_1d_array& tau); void alglib::smp_cmatrixqr( complex_2d_array& a, ae_int_t m, ae_int_t n, complex_1d_array& tau);
cmatrixqrunpackq function/************************************************************************* Partial unpacking of matrix Q from QR decomposition of a complex matrix A. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrices Q and R in compact form. Output of CMatrixQR subroutine . M - number of rows in matrix A. M>=0. N - number of columns in matrix A. N>=0. Tau - scalar factors which are used to form Q. Output of CMatrixQR subroutine . QColumns - required number of columns in matrix Q. M>=QColumns>=0. Output parameters: Q - first QColumns columns of matrix Q. Array whose index ranges within [0..M-1, 0..QColumns-1]. If QColumns=0, array isn't changed. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixqrunpackq( complex_2d_array a, ae_int_t m, ae_int_t n, complex_1d_array tau, ae_int_t qcolumns, complex_2d_array& q); void alglib::smp_cmatrixqrunpackq( complex_2d_array a, ae_int_t m, ae_int_t n, complex_1d_array tau, ae_int_t qcolumns, complex_2d_array& q);
cmatrixqrunpackr function/************************************************************************* Unpacking of matrix R from the QR decomposition of a matrix A Input parameters: A - matrices Q and R in compact form. Output of CMatrixQR subroutine. M - number of rows in given matrix A. M>=0. N - number of columns in given matrix A. N>=0. Output parameters: R - matrix R, array[0..M-1, 0..N-1]. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixqrunpackr( complex_2d_array a, ae_int_t m, ae_int_t n, complex_2d_array& r);
hmatrixtd function/************************************************************************* Reduction of a Hermitian matrix which is given by its higher or lower triangular part to a real tridiagonal matrix using unitary similarity transformation: Q'*A*Q = T. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix to be transformed array with elements [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - storage format. If IsUpper = True, then matrix A is given by its upper triangle, and the lower triangle is not used and not modified by the algorithm, and vice versa if IsUpper = False. Output parameters: A - matrices T and Q in compact form (see lower) Tau - array of factors which are forming matrices H(i) array with elements [0..N-2]. D - main diagonal of real symmetric matrix T. array with elements [0..N-1]. E - secondary diagonal of real symmetric matrix T. array with elements [0..N-2]. If IsUpper=True, the matrix Q is represented as a product of elementary reflectors Q = H(n-2) . . . H(2) H(0). Each H(i) has the form H(i) = I - tau * v * v' where tau is a complex scalar, and v is a complex vector with v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in A(0:i-1,i+1), and tau in TAU(i). If IsUpper=False, the matrix Q is represented as a product of elementary reflectors Q = H(0) H(2) . . . H(n-2). Each H(i) has the form H(i) = I - tau * v * v' where tau is a complex scalar, and v is a complex vector with v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v1 v2 v3 ) ( d ) ( d e v2 v3 ) ( e d ) ( d e v3 ) ( v0 e d ) ( d e ) ( v0 v1 e d ) ( d ) ( v0 v1 v2 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1992 *************************************************************************/void alglib::hmatrixtd( complex_2d_array& a, ae_int_t n, bool isupper, complex_1d_array& tau, real_1d_array& d, real_1d_array& e);
hmatrixtdunpackq function/************************************************************************* Unpacking matrix Q which reduces a Hermitian matrix to a real tridiagonal form. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - the result of a HMatrixTD subroutine N - size of matrix A. IsUpper - storage format (a parameter of HMatrixTD subroutine) Tau - the result of a HMatrixTD subroutine Output parameters: Q - transformation matrix. array with elements [0..N-1, 0..N-1]. -- ALGLIB -- Copyright 2005-2010 by Bochkanov Sergey *************************************************************************/void alglib::hmatrixtdunpackq( complex_2d_array a, ae_int_t n, bool isupper, complex_1d_array tau, complex_2d_array& q);
rmatrixbd function/************************************************************************* Reduction of a rectangular matrix to bidiagonal form The algorithm reduces the rectangular matrix A to bidiagonal form by orthogonal transformations P and Q: A = Q*B*(P^T). COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Multithreaded acceleration is NOT supported for this function because ! bidiagonal decompostion is inherently sequential in nature. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - source matrix. array[0..M-1, 0..N-1] M - number of rows in matrix A. N - number of columns in matrix A. Output parameters: A - matrices Q, B, P in compact form (see below). TauQ - scalar factors which are used to form matrix Q. TauP - scalar factors which are used to form matrix P. The main diagonal and one of the secondary diagonals of matrix A are replaced with bidiagonal matrix B. Other elements contain elementary reflections which form MxM matrix Q and NxN matrix P, respectively. If M>=N, B is the upper bidiagonal MxN matrix and is stored in the corresponding elements of matrix A. Matrix Q is represented as a product of elementary reflections Q = H(0)*H(1)*...*H(n-1), where H(i) = 1-tau*v*v'. Here tau is a scalar which is stored in TauQ[i], and vector v has the following structure: v(0:i-1)=0, v(i)=1, v(i+1:m-1) is stored in elements A(i+1:m-1,i). Matrix P is as follows: P = G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i], u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1). If M<N, B is the lower bidiagonal MxN matrix and is stored in the corresponding elements of matrix A. Q = H(0)*H(1)*...*H(m-2), where H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1) is stored in elements A(i+2:m-1,i). P = G(0)*G(1)*...*G(m-1), G(i) = 1-tau*u*u', tau is stored in TauP, u(0:i-1)=0, u(i)=1, u(i+1:n-1) is stored in A(i,i+1:n-1). EXAMPLE: m=6, n=5 (m > n): m=5, n=6 (m < n): ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) ( v1 v2 v3 v4 v5 ) Here vi and ui are vectors which form H(i) and G(i), and d and e - are the diagonal and off-diagonal elements of matrix B. -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994. Sergey Bochkanov, ALGLIB project, translation from FORTRAN to pseudocode, 2007-2010. *************************************************************************/void alglib::rmatrixbd( real_2d_array& a, ae_int_t m, ae_int_t n, real_1d_array& tauq, real_1d_array& taup);
rmatrixbdmultiplybyp function/************************************************************************* Multiplication by matrix P which reduces matrix A to bidiagonal form. The algorithm allows pre- or post-multiply by P or P'. Input parameters: QP - matrices Q and P in compact form. Output of RMatrixBD subroutine. M - number of rows in matrix A. N - number of columns in matrix A. TAUP - scalar factors which are used to form P. Output of RMatrixBD subroutine. Z - multiplied matrix. Array whose indexes range within [0..ZRows-1,0..ZColumns-1]. ZRows - number of rows in matrix Z. If FromTheRight=False, ZRows=N, otherwise ZRows can be arbitrary. ZColumns - number of columns in matrix Z. If FromTheRight=True, ZColumns=N, otherwise ZColumns can be arbitrary. FromTheRight - pre- or post-multiply. DoTranspose - multiply by P or P'. Output parameters: Z - product of Z and P. Array whose indexes range within [0..ZRows-1,0..ZColumns-1]. If ZRows=0 or ZColumns=0, the array is not modified. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixbdmultiplybyp( real_2d_array qp, ae_int_t m, ae_int_t n, real_1d_array taup, real_2d_array& z, ae_int_t zrows, ae_int_t zcolumns, bool fromtheright, bool dotranspose);
rmatrixbdmultiplybyq function/************************************************************************* Multiplication by matrix Q which reduces matrix A to bidiagonal form. The algorithm allows pre- or post-multiply by Q or Q'. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: QP - matrices Q and P in compact form. Output of ToBidiagonal subroutine. M - number of rows in matrix A. N - number of columns in matrix A. TAUQ - scalar factors which are used to form Q. Output of ToBidiagonal subroutine. Z - multiplied matrix. array[0..ZRows-1,0..ZColumns-1] ZRows - number of rows in matrix Z. If FromTheRight=False, ZRows=M, otherwise ZRows can be arbitrary. ZColumns - number of columns in matrix Z. If FromTheRight=True, ZColumns=M, otherwise ZColumns can be arbitrary. FromTheRight - pre- or post-multiply. DoTranspose - multiply by Q or Q'. Output parameters: Z - product of Z and Q. Array[0..ZRows-1,0..ZColumns-1] If ZRows=0 or ZColumns=0, the array is not modified. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixbdmultiplybyq( real_2d_array qp, ae_int_t m, ae_int_t n, real_1d_array tauq, real_2d_array& z, ae_int_t zrows, ae_int_t zcolumns, bool fromtheright, bool dotranspose);
rmatrixbdunpackdiagonals function/************************************************************************* Unpacking of the main and secondary diagonals of bidiagonal decomposition of matrix A. Input parameters: B - output of RMatrixBD subroutine. M - number of rows in matrix B. N - number of columns in matrix B. Output parameters: IsUpper - True, if the matrix is upper bidiagonal. otherwise IsUpper is False. D - the main diagonal. Array whose index ranges within [0..Min(M,N)-1]. E - the secondary diagonal (upper or lower, depending on the value of IsUpper). Array index ranges within [0..Min(M,N)-1], the last element is not used. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixbdunpackdiagonals( real_2d_array b, ae_int_t m, ae_int_t n, bool& isupper, real_1d_array& d, real_1d_array& e);
rmatrixbdunpackpt function/************************************************************************* Unpacking matrix P which reduces matrix A to bidiagonal form. The subroutine returns transposed matrix P. Input parameters: QP - matrices Q and P in compact form. Output of ToBidiagonal subroutine. M - number of rows in matrix A. N - number of columns in matrix A. TAUP - scalar factors which are used to form P. Output of ToBidiagonal subroutine. PTRows - required number of rows of matrix P^T. N >= PTRows >= 0. Output parameters: PT - first PTRows columns of matrix P^T Array[0..PTRows-1, 0..N-1] If PTRows=0, the array is not modified. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixbdunpackpt( real_2d_array qp, ae_int_t m, ae_int_t n, real_1d_array taup, ae_int_t ptrows, real_2d_array& pt);
rmatrixbdunpackq function/************************************************************************* Unpacking matrix Q which reduces a matrix to bidiagonal form. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: QP - matrices Q and P in compact form. Output of ToBidiagonal subroutine. M - number of rows in matrix A. N - number of columns in matrix A. TAUQ - scalar factors which are used to form Q. Output of ToBidiagonal subroutine. QColumns - required number of columns in matrix Q. M>=QColumns>=0. Output parameters: Q - first QColumns columns of matrix Q. Array[0..M-1, 0..QColumns-1] If QColumns=0, the array is not modified. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixbdunpackq( real_2d_array qp, ae_int_t m, ae_int_t n, real_1d_array tauq, ae_int_t qcolumns, real_2d_array& q);
rmatrixhessenberg function/************************************************************************* Reduction of a square matrix to upper Hessenberg form: Q'*A*Q = H, where Q is an orthogonal matrix, H - Hessenberg matrix. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix A with elements [0..N-1, 0..N-1] N - size of matrix A. Output parameters: A - matrices Q and P in compact form (see below). Tau - array of scalar factors which are used to form matrix Q. Array whose index ranges within [0..N-2] Matrix H is located on the main diagonal, on the lower secondary diagonal and above the main diagonal of matrix A. The elements which are used to form matrix Q are situated in array Tau and below the lower secondary diagonal of matrix A as follows: Matrix Q is represented as a product of elementary reflections Q = H(0)*H(2)*...*H(n-2), where each H(i) is given by H(i) = 1 - tau * v * (v^T) where tau is a scalar stored in Tau[I]; v - is a real vector, so that v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) stored in A(i+2:n-1,i). -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1992 *************************************************************************/void alglib::rmatrixhessenberg( real_2d_array& a, ae_int_t n, real_1d_array& tau);
rmatrixhessenbergunpackh function/************************************************************************* Unpacking matrix H (the result of matrix A reduction to upper Hessenberg form) Input parameters: A - output of RMatrixHessenberg subroutine. N - size of matrix A. Output parameters: H - matrix H. Array whose indexes range within [0..N-1, 0..N-1]. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixhessenbergunpackh( real_2d_array a, ae_int_t n, real_2d_array& h);
rmatrixhessenbergunpackq function/************************************************************************* Unpacking matrix Q which reduces matrix A to upper Hessenberg form COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - output of RMatrixHessenberg subroutine. N - size of matrix A. Tau - scalar factors which are used to form Q. Output of RMatrixHessenberg subroutine. Output parameters: Q - matrix Q. Array whose indexes range within [0..N-1, 0..N-1]. -- ALGLIB -- 2005-2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixhessenbergunpackq( real_2d_array a, ae_int_t n, real_1d_array tau, real_2d_array& q);
rmatrixlq function/************************************************************************* LQ decomposition of a rectangular matrix of size MxN COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix A whose indexes range within [0..M-1, 0..N-1]. M - number of rows in matrix A. N - number of columns in matrix A. Output parameters: A - matrices L and Q in compact form (see below) Tau - array of scalar factors which are used to form matrix Q. Array whose index ranges within [0..Min(M,N)-1]. Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size MxM, L - lower triangular (or lower trapezoid) matrix of size M x N. The elements of matrix L are located on and below the main diagonal of matrix A. The elements which are located in Tau array and above the main diagonal of matrix A are used to form matrix Q as follows: Matrix Q is represented as a product of elementary reflections Q = H(k-1)*H(k-2)*...*H(1)*H(0), where k = min(m,n), and each H(i) is of the form H(i) = 1 - tau * v * (v^T) where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1)=0, v(i) = 1, v(i+1:n-1) stored in A(i,i+1:n-1). -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixlq( real_2d_array& a, ae_int_t m, ae_int_t n, real_1d_array& tau); void alglib::smp_rmatrixlq( real_2d_array& a, ae_int_t m, ae_int_t n, real_1d_array& tau);
rmatrixlqunpackl function/************************************************************************* Unpacking of matrix L from the LQ decomposition of a matrix A Input parameters: A - matrices Q and L in compact form. Output of RMatrixLQ subroutine. M - number of rows in given matrix A. M>=0. N - number of columns in given matrix A. N>=0. Output parameters: L - matrix L, array[0..M-1, 0..N-1]. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixlqunpackl( real_2d_array a, ae_int_t m, ae_int_t n, real_2d_array& l);
rmatrixlqunpackq function/************************************************************************* Partial unpacking of matrix Q from the LQ decomposition of a matrix A COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrices L and Q in compact form. Output of RMatrixLQ subroutine. M - number of rows in given matrix A. M>=0. N - number of columns in given matrix A. N>=0. Tau - scalar factors which are used to form Q. Output of the RMatrixLQ subroutine. QRows - required number of rows in matrix Q. N>=QRows>=0. Output parameters: Q - first QRows rows of matrix Q. Array whose indexes range within [0..QRows-1, 0..N-1]. If QRows=0, the array remains unchanged. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixlqunpackq( real_2d_array a, ae_int_t m, ae_int_t n, real_1d_array tau, ae_int_t qrows, real_2d_array& q); void alglib::smp_rmatrixlqunpackq( real_2d_array a, ae_int_t m, ae_int_t n, real_1d_array tau, ae_int_t qrows, real_2d_array& q);
rmatrixqr function/************************************************************************* QR decomposition of a rectangular matrix of size MxN COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix A whose indexes range within [0..M-1, 0..N-1]. M - number of rows in matrix A. N - number of columns in matrix A. Output parameters: A - matrices Q and R in compact form (see below). Tau - array of scalar factors which are used to form matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)]. Matrix A is represented as A = QR, where Q is an orthogonal matrix of size MxM, R - upper triangular (or upper trapezoid) matrix of size M x N. The elements of matrix R are located on and above the main diagonal of matrix A. The elements which are located in Tau array and below the main diagonal of matrix A are used to form matrix Q as follows: Matrix Q is represented as a product of elementary reflections Q = H(0)*H(2)*...*H(k-1), where k = min(m,n), and each H(i) is in the form H(i) = 1 - tau * v * (v^T) where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i). -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixqr( real_2d_array& a, ae_int_t m, ae_int_t n, real_1d_array& tau); void alglib::smp_rmatrixqr( real_2d_array& a, ae_int_t m, ae_int_t n, real_1d_array& tau);
rmatrixqrunpackq function/************************************************************************* Partial unpacking of matrix Q from the QR decomposition of a matrix A COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that QP decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=512, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrices Q and R in compact form. Output of RMatrixQR subroutine. M - number of rows in given matrix A. M>=0. N - number of columns in given matrix A. N>=0. Tau - scalar factors which are used to form Q. Output of the RMatrixQR subroutine. QColumns - required number of columns of matrix Q. M>=QColumns>=0. Output parameters: Q - first QColumns columns of matrix Q. Array whose indexes range within [0..M-1, 0..QColumns-1]. If QColumns=0, the array remains unchanged. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixqrunpackq( real_2d_array a, ae_int_t m, ae_int_t n, real_1d_array tau, ae_int_t qcolumns, real_2d_array& q); void alglib::smp_rmatrixqrunpackq( real_2d_array a, ae_int_t m, ae_int_t n, real_1d_array tau, ae_int_t qcolumns, real_2d_array& q);
rmatrixqrunpackr function/************************************************************************* Unpacking of matrix R from the QR decomposition of a matrix A Input parameters: A - matrices Q and R in compact form. Output of RMatrixQR subroutine. M - number of rows in given matrix A. M>=0. N - number of columns in given matrix A. N>=0. Output parameters: R - matrix R, array[0..M-1, 0..N-1]. -- ALGLIB routine -- 17.02.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixqrunpackr( real_2d_array a, ae_int_t m, ae_int_t n, real_2d_array& r);
smatrixtd function/************************************************************************* Reduction of a symmetric matrix which is given by its higher or lower triangular part to a tridiagonal matrix using orthogonal similarity transformation: Q'*A*Q=T. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - matrix to be transformed array with elements [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - storage format. If IsUpper = True, then matrix A is given by its upper triangle, and the lower triangle is not used and not modified by the algorithm, and vice versa if IsUpper = False. Output parameters: A - matrices T and Q in compact form (see lower) Tau - array of factors which are forming matrices H(i) array with elements [0..N-2]. D - main diagonal of symmetric matrix T. array with elements [0..N-1]. E - secondary diagonal of symmetric matrix T. array with elements [0..N-2]. If IsUpper=True, the matrix Q is represented as a product of elementary reflectors Q = H(n-2) . . . H(2) H(0). Each H(i) has the form H(i) = I - tau * v * v' where tau is a real scalar, and v is a real vector with v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in A(0:i-1,i+1), and tau in TAU(i). If IsUpper=False, the matrix Q is represented as a product of elementary reflectors Q = H(0) H(2) . . . H(n-2). Each H(i) has the form H(i) = I - tau * v * v' where tau is a real scalar, and v is a real vector with v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i), and tau in TAU(i). The contents of A on exit are illustrated by the following examples with n = 5: if UPLO = 'U': if UPLO = 'L': ( d e v1 v2 v3 ) ( d ) ( d e v2 v3 ) ( e d ) ( d e v3 ) ( v0 e d ) ( d e ) ( v0 v1 e d ) ( d ) ( v0 v1 v2 e d ) where d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector defining H(i). -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University October 31, 1992 *************************************************************************/void alglib::smatrixtd( real_2d_array& a, ae_int_t n, bool isupper, real_1d_array& tau, real_1d_array& d, real_1d_array& e);
smatrixtdunpackq function/************************************************************************* Unpacking matrix Q which reduces symmetric matrix to a tridiagonal form. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. Input parameters: A - the result of a SMatrixTD subroutine N - size of matrix A. IsUpper - storage format (a parameter of SMatrixTD subroutine) Tau - the result of a SMatrixTD subroutine Output parameters: Q - transformation matrix. array with elements [0..N-1, 0..N-1]. -- ALGLIB -- Copyright 2005-2010 by Bochkanov Sergey *************************************************************************/void alglib::smatrixtdunpackq( real_2d_array a, ae_int_t n, bool isupper, real_1d_array tau, real_2d_array& q);
parametric subpackage| parametric_rdp | Parametric Ramer-Douglas-Peucker approximation |
pspline2interpolant class/************************************************************************* Parametric spline inteprolant: 2-dimensional curve. You should not try to access its members directly - use PSpline2XXXXXXXX() functions instead. *************************************************************************/class pspline2interpolant { };
pspline3interpolant class/************************************************************************* Parametric spline inteprolant: 3-dimensional curve. You should not try to access its members directly - use PSpline3XXXXXXXX() functions instead. *************************************************************************/class pspline3interpolant { };
parametricrdpfixed function/************************************************************************* This subroutine fits piecewise linear curve to points with Ramer-Douglas- Peucker algorithm. This function performs PARAMETRIC fit, i.e. it can be used to fit curves like circles. On input it accepts dataset which describes parametric multidimensional curve X(t), with X being vector, and t taking values in [0,N), where N is a number of points in dataset. As result, it returns reduced dataset X2, which can be used to build parametric curve X2(t), which approximates X(t) with desired precision (or has specified number of sections). INPUT PARAMETERS: X - array of multidimensional points: * at least N elements, leading N elements are used if more than N elements were specified * order of points is IMPORTANT because it is parametric fit * each row of array is one point which has D coordinates N - number of elements in X D - number of dimensions (elements per row of X) StopM - stopping condition - desired number of sections: * at most M sections are generated by this function * less than M sections can be generated if we have N<M (or some X are non-distinct). * zero StopM means that algorithm does not stop after achieving some pre-specified section count StopEps - stopping condition - desired precision: * algorithm stops after error in each section is at most Eps * zero Eps means that algorithm does not stop after achieving some pre-specified precision OUTPUT PARAMETERS: X2 - array of corner points for piecewise approximation, has length NSections+1 or zero (for NSections=0). Idx2 - array of indexes (parameter values): * has length NSections+1 or zero (for NSections=0). * each element of Idx2 corresponds to same-numbered element of X2 * each element of Idx2 is index of corresponding element of X2 at original array X, i.e. I-th row of X2 is Idx2[I]-th row of X. * elements of Idx2 can be treated as parameter values which should be used when building new parametric curve * Idx2[0]=0, Idx2[NSections]=N-1 NSections- number of sections found by algorithm, NSections<=M, NSections can be zero for degenerate datasets (N<=1 or all X[] are non-distinct). NOTE: algorithm stops after: a) dividing curve into StopM sections b) achieving required precision StopEps c) dividing curve into N-1 sections If both StopM and StopEps are non-zero, algorithm is stopped by the FIRST criterion which is satisfied. In case both StopM and StopEps are zero, algorithm stops because of (c). -- ALGLIB -- Copyright 02.10.2014 by Bochkanov Sergey *************************************************************************/void alglib::parametricrdpfixed( real_2d_array x, ae_int_t n, ae_int_t d, ae_int_t stopm, double stopeps, real_2d_array& x2, integer_1d_array& idx2, ae_int_t& nsections);
Examples: [1]
pspline2arclength function/************************************************************************* This function calculates arc length, i.e. length of curve between t=a and t=b. INPUT PARAMETERS: P - parametric spline interpolant A,B - parameter values corresponding to arc ends: * B>A will result in positive length returned * B<A will result in negative length returned RESULT: length of arc starting at T=A and ending at T=B. -- ALGLIB PROJECT -- Copyright 30.05.2010 by Bochkanov Sergey *************************************************************************/double alglib::pspline2arclength( pspline2interpolant p, double a, double b);
pspline2build function/************************************************************************* This function builds non-periodic 2-dimensional parametric spline which starts at (X[0],Y[0]) and ends at (X[N-1],Y[N-1]). INPUT PARAMETERS: XY - points, array[0..N-1,0..1]. XY[I,0:1] corresponds to the Ith point. Order of points is important! N - points count, N>=5 for Akima splines, N>=2 for other types of splines. ST - spline type: * 0 Akima spline * 1 parabolically terminated Catmull-Rom spline (Tension=0) * 2 parabolically terminated cubic spline PT - parameterization type: * 0 uniform * 1 chord length * 2 centripetal OUTPUT PARAMETERS: P - parametric spline interpolant NOTES: * this function assumes that there all consequent points are distinct. I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on. However, non-consequent points may coincide, i.e. we can have (x0,y0)= =(x2,y2). -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2build( real_2d_array xy, ae_int_t n, ae_int_t st, ae_int_t pt, pspline2interpolant& p);
pspline2buildperiodic function/************************************************************************* This function builds periodic 2-dimensional parametric spline which starts at (X[0],Y[0]), goes through all points to (X[N-1],Y[N-1]) and then back to (X[0],Y[0]). INPUT PARAMETERS: XY - points, array[0..N-1,0..1]. XY[I,0:1] corresponds to the Ith point. XY[N-1,0:1] must be different from XY[0,0:1]. Order of points is important! N - points count, N>=3 for other types of splines. ST - spline type: * 1 Catmull-Rom spline (Tension=0) with cyclic boundary conditions * 2 cubic spline with cyclic boundary conditions PT - parameterization type: * 0 uniform * 1 chord length * 2 centripetal OUTPUT PARAMETERS: P - parametric spline interpolant NOTES: * this function assumes that there all consequent points are distinct. I.e. (x0,y0)<>(x1,y1), (x1,y1)<>(x2,y2), (x2,y2)<>(x3,y3) and so on. However, non-consequent points may coincide, i.e. we can have (x0,y0)= =(x2,y2). * last point of sequence is NOT equal to the first point. You shouldn't make curve "explicitly periodic" by making them equal. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2buildperiodic( real_2d_array xy, ae_int_t n, ae_int_t st, ae_int_t pt, pspline2interpolant& p);
pspline2calc function/************************************************************************* This function calculates the value of the parametric spline for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-position Y - Y-position -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2calc( pspline2interpolant p, double t, double& x, double& y);
pspline2diff function/************************************************************************* This function calculates derivative, i.e. it returns (dX/dT,dY/dT). INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - X-derivative Y - Y-value DY - Y-derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2diff( pspline2interpolant p, double t, double& x, double& dx, double& y, double& dy);
pspline2diff2 function/************************************************************************* This function calculates first and second derivative with respect to T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - derivative D2X - second derivative Y - Y-value DY - derivative D2Y - second derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2diff2( pspline2interpolant p, double t, double& x, double& dx, double& d2x, double& y, double& dy, double& d2y);
pspline2parametervalues function/************************************************************************* This function returns vector of parameter values correspoding to points. I.e. for P created from (X[0],Y[0])...(X[N-1],Y[N-1]) and U=TValues(P) we have (X[0],Y[0]) = PSpline2Calc(P,U[0]), (X[1],Y[1]) = PSpline2Calc(P,U[1]), (X[2],Y[2]) = PSpline2Calc(P,U[2]), ... INPUT PARAMETERS: P - parametric spline interpolant OUTPUT PARAMETERS: N - array size T - array[0..N-1] NOTES: * for non-periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]=1 * for periodic splines U[0]=0, U[0]<U[1]<...<U[N-1], U[N-1]<1 -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2parametervalues( pspline2interpolant p, ae_int_t& n, real_1d_array& t);
pspline2tangent function/************************************************************************* This function calculates tangent vector for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-component of tangent vector (normalized) Y - Y-component of tangent vector (normalized) NOTE: X^2+Y^2 is either 1 (for non-zero tangent vector) or 0. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline2tangent( pspline2interpolant p, double t, double& x, double& y);
pspline3arclength function/************************************************************************* This function calculates arc length, i.e. length of curve between t=a and t=b. INPUT PARAMETERS: P - parametric spline interpolant A,B - parameter values corresponding to arc ends: * B>A will result in positive length returned * B<A will result in negative length returned RESULT: length of arc starting at T=A and ending at T=B. -- ALGLIB PROJECT -- Copyright 30.05.2010 by Bochkanov Sergey *************************************************************************/double alglib::pspline3arclength( pspline3interpolant p, double a, double b);
pspline3build function/************************************************************************* This function builds non-periodic 3-dimensional parametric spline which starts at (X[0],Y[0],Z[0]) and ends at (X[N-1],Y[N-1],Z[N-1]). Same as PSpline2Build() function, but for 3D, so we won't duplicate its description here. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3build( real_2d_array xy, ae_int_t n, ae_int_t st, ae_int_t pt, pspline3interpolant& p);
pspline3buildperiodic function/************************************************************************* This function builds periodic 3-dimensional parametric spline which starts at (X[0],Y[0],Z[0]), goes through all points to (X[N-1],Y[N-1],Z[N-1]) and then back to (X[0],Y[0],Z[0]). Same as PSpline2Build() function, but for 3D, so we won't duplicate its description here. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3buildperiodic( real_2d_array xy, ae_int_t n, ae_int_t st, ae_int_t pt, pspline3interpolant& p);
pspline3calc function/************************************************************************* This function calculates the value of the parametric spline for a given value of parameter T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-position Y - Y-position Z - Z-position -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3calc( pspline3interpolant p, double t, double& x, double& y, double& z);
pspline3diff function/************************************************************************* This function calculates derivative, i.e. it returns (dX/dT,dY/dT,dZ/dT). INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - X-derivative Y - Y-value DY - Y-derivative Z - Z-value DZ - Z-derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3diff( pspline3interpolant p, double t, double& x, double& dx, double& y, double& dy, double& z, double& dz);
pspline3diff2 function/************************************************************************* This function calculates first and second derivative with respect to T. INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-value DX - derivative D2X - second derivative Y - Y-value DY - derivative D2Y - second derivative Z - Z-value DZ - derivative D2Z - second derivative -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3diff2( pspline3interpolant p, double t, double& x, double& dx, double& d2x, double& y, double& dy, double& d2y, double& z, double& dz, double& d2z);
pspline3parametervalues function/************************************************************************* This function returns vector of parameter values correspoding to points. Same as PSpline2ParameterValues(), but for 3D. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3parametervalues( pspline3interpolant p, ae_int_t& n, real_1d_array& t);
pspline3tangent function/************************************************************************* This function calculates tangent vector for a given value of parameter T INPUT PARAMETERS: P - parametric spline interpolant T - point: * T in [0,1] corresponds to interval spanned by points * for non-periodic splines T<0 (or T>1) correspond to parts of the curve before the first (after the last) point * for periodic splines T<0 (or T>1) are projected into [0,1] by making T=T-floor(T). OUTPUT PARAMETERS: X - X-component of tangent vector (normalized) Y - Y-component of tangent vector (normalized) Z - Z-component of tangent vector (normalized) NOTE: X^2+Y^2+Z^2 is either 1 (for non-zero tangent vector) or 0. -- ALGLIB PROJECT -- Copyright 28.05.2010 by Bochkanov Sergey *************************************************************************/void alglib::pspline3tangent( pspline3interpolant p, double t, double& x, double& y, double& z);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use RDP algorithm to approximate parametric 2D curve given by
// locations in t=0,1,2,3 (see below), which form piecewise linear
// trajectory through D-dimensional space (2-dimensional in our example).
//
// |
// |
// - * * X2................X3
// | .
// | .
// - * * . * * * *
// | .
// | .
// - * X1 * * * *
// | .....
// | ....
// X0----|-----|-----|-----|-----|-----|---
//
ae_int_t npoints = 4;
ae_int_t ndimensions = 2;
real_2d_array x = "[[0,0],[2,1],[3,3],[6,3]]";
//
// Approximation of parametric curve is performed by another parametric curve
// with lesser amount of points. It allows to work with "compressed"
// representation, which needs smaller amount of memory. Say, in our example
// (we allow points with error smaller than 0.8) approximation will have
// just two sequential sections connecting X0 with X2, and X2 with X3.
//
// |
// |
// - * * X2................X3
// | .
// | .
// - * . * * * *
// | .
// | .
// - . X1 * * * *
// | .
// | .
// X0----|-----|-----|-----|-----|-----|---
//
//
real_2d_array y;
integer_1d_array idxy;
ae_int_t nsections;
ae_int_t limitcnt = 0;
double limiteps = 0.8;
parametricrdpfixed(x, npoints, ndimensions, limitcnt, limiteps, y, idxy, nsections);
printf("%d\n", int(nsections)); // EXPECTED: 2
printf("%s\n", idxy.tostring().c_str()); // EXPECTED: [0,2,3]
return 0;
}
pca subpackagepcabuildbasis function/************************************************************************* Principal components analysis This function builds orthogonal basis where first axis corresponds to direction with maximum variance, second axis maximizes variance in the subspace orthogonal to first axis and so on. This function builds FULL basis, i.e. returns N vectors corresponding to ALL directions, no matter how informative. If you need just a few (say, 10 or 50) of the most important directions, you may find it faster to use one of the reduced versions: * pcatruncatedsubspace() - for subspace iteration based method It should be noted that, unlike LDA, PCA does not use class labels. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * multithreading support ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Multithreading typically gives sublinear (wrt to cores count) speedup, ! because only some parts of the algorithm can be parallelized. ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. Best results are achieved for high-dimensional problems ! (NVars is at least 256). ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - dataset, array[0..NPoints-1,0..NVars-1]. matrix contains ONLY INDEPENDENT VARIABLES. NPoints - dataset size, NPoints>=0 NVars - number of independent variables, NVars>=1 OUTPUT PARAMETERS: Info - return code: * -4, if SVD subroutine haven't converged * -1, if wrong parameters has been passed (NPoints<0, NVars<1) * 1, if task is solved S2 - array[0..NVars-1]. variance values corresponding to basis vectors. V - array[0..NVars-1,0..NVars-1] matrix, whose columns store basis vectors. -- ALGLIB -- Copyright 25.08.2008 by Bochkanov Sergey *************************************************************************/void alglib::pcabuildbasis( real_2d_array x, ae_int_t npoints, ae_int_t nvars, ae_int_t& info, real_1d_array& s2, real_2d_array& v); void alglib::smp_pcabuildbasis( real_2d_array x, ae_int_t npoints, ae_int_t nvars, ae_int_t& info, real_1d_array& s2, real_2d_array& v);
pcatruncatedsubspace function/************************************************************************* Principal components analysis This function performs truncated PCA, i.e. returns just a few most important directions. Internally it uses iterative eigensolver which is very efficient when only a minor fraction of full basis is required. Thus, if you need full basis, it is better to use pcabuildbasis() function. It should be noted that, unlike LDA, PCA does not use class labels. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * multithreading support ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! For a situation when you need just a few eigenvectors (~1-10), ! multithreading typically gives sublinear (wrt to cores count) speedup. ! For larger problems it may give you nearly linear increase in ! performance. ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. Best results are achieved for high-dimensional problems ! (NVars is at least 256). ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: X - dataset, array[0..NPoints-1,0..NVars-1]. matrix contains ONLY INDEPENDENT VARIABLES. NPoints - dataset size, NPoints>=0 NVars - number of independent variables, NVars>=1 NNeeded - number of requested directions, in [1,NVars] range; this function is efficient only for NNeeded<<NVars. Eps - desired precision of vectors returned; underlying solver will stop iterations as soon as absolute error in corresponding singular values reduces to roughly eps*MAX(lambda[]), with lambda[] being array of eigen values. Zero value means that algorithm performs number of iterations specified by maxits parameter, without paying attention to precision. MaxIts - number of iterations performed by subspace iteration method. Zero value means that no limit on iteration count is placed (eps-based stopping condition is used). OUTPUT PARAMETERS: S2 - array[NNeeded]. Variance values corresponding to basis vectors. V - array[NVars,NNeeded] matrix, whose columns store basis vectors. NOTE: passing eps=0 and maxits=0 results in small eps being selected as stopping condition. Exact value of automatically selected eps is version- -dependent. -- ALGLIB -- Copyright 10.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::pcatruncatedsubspace( real_2d_array x, ae_int_t npoints, ae_int_t nvars, ae_int_t nneeded, double eps, ae_int_t maxits, real_1d_array& s2, real_2d_array& v); void alglib::smp_pcatruncatedsubspace( real_2d_array x, ae_int_t npoints, ae_int_t nvars, ae_int_t nneeded, double eps, ae_int_t maxits, real_1d_array& s2, real_2d_array& v);
poissondistr subpackageinvpoissondistribution function/************************************************************************* Inverse Poisson distribution Finds the Poisson variable x such that the integral from 0 to x of the Poisson density is equal to the given probability y. This is accomplished using the inverse gamma integral function and the relation m = igami( k+1, y ). ACCURACY: See inverse incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invpoissondistribution(ae_int_t k, double y);
poissoncdistribution function/************************************************************************* Complemented Poisson distribution Returns the sum of the terms k+1 to infinity of the Poisson distribution: inf. j -- -m m > e -- -- j! j=k+1 The terms are not summed directly; instead the incomplete gamma integral is employed, according to the formula y = pdtrc( k, m ) = igam( k+1, m ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::poissoncdistribution(ae_int_t k, double m);
poissondistribution function/************************************************************************* Poisson distribution Returns the sum of the first k+1 terms of the Poisson distribution: k j -- -m m > e -- -- j! j=0 The terms are not summed directly; instead the incomplete gamma integral is employed, according to the relation y = pdtr( k, m ) = igamc( k+1, m ). The arguments must both be positive. ACCURACY: See incomplete gamma function Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::poissondistribution(ae_int_t k, double m);
polint subpackage| polint_d_calcdiff | Interpolation and differentiation using barycentric representation | |
| polint_d_conv | Conversion between power basis and barycentric representation | |
| polint_d_spec | Polynomial interpolation on special grids (equidistant, Chebyshev I/II) |
polynomialbar2cheb function/************************************************************************* Conversion from barycentric representation to Chebyshev basis. This function has O(N^2) complexity. INPUT PARAMETERS: P - polynomial in barycentric form A,B - base interval for Chebyshev polynomials (see below) A<>B OUTPUT PARAMETERS T - coefficients of Chebyshev representation; P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N-1 }, where Ti - I-th Chebyshev polynomial. NOTES: barycentric interpolant passed as P may be either polynomial obtained from polynomial interpolation/ fitting or rational function which is NOT polynomial. We can't distinguish between these two cases, and this algorithm just tries to work assuming that P IS a polynomial. If not, algorithm will return results, but they won't have any meaning. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::polynomialbar2cheb( barycentricinterpolant p, double a, double b, real_1d_array& t);
polynomialbar2pow function/************************************************************************* Conversion from barycentric representation to power basis. This function has O(N^2) complexity. INPUT PARAMETERS: P - polynomial in barycentric form C - offset (see below); 0.0 is used as default value. S - scale (see below); 1.0 is used as default value. S<>0. OUTPUT PARAMETERS A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 } N - number of coefficients (polynomial degree plus 1) NOTES: 1. this function accepts offset and scale, which can be set to improve numerical properties of polynomial. For example, if P was obtained as result of interpolation on [-1,+1], you can set C=0 and S=1 and represent P as sum of 1, x, x^2, x^3 and so on. In most cases you it is exactly what you need. However, if your interpolation model was built on [999,1001], you will see significant growth of numerical errors when using {1, x, x^2, x^3} as basis. Representing P as sum of 1, (x-1000), (x-1000)^2, (x-1000)^3 will be better option. Such representation can be obtained by using 1000.0 as offset C and 1.0 as scale S. 2. power basis is ill-conditioned and tricks described above can't solve this problem completely. This function will return coefficients in any case, but for N>8 they will become unreliable. However, N's less than 5 are pretty safe. 3. barycentric interpolant passed as P may be either polynomial obtained from polynomial interpolation/ fitting or rational function which is NOT polynomial. We can't distinguish between these two cases, and this algorithm just tries to work assuming that P IS a polynomial. If not, algorithm will return results, but they won't have any meaning. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::polynomialbar2pow( barycentricinterpolant p, real_1d_array& a); void alglib::polynomialbar2pow( barycentricinterpolant p, double c, double s, real_1d_array& a);
Examples: [1]
polynomialbuild function/************************************************************************* Lagrange intepolant: generation of the model on the general grid. This function has O(N^2) complexity. INPUT PARAMETERS: X - abscissas, array[0..N-1] Y - function values, array[0..N-1] N - number of points, N>=1 OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::polynomialbuild( real_1d_array x, real_1d_array y, barycentricinterpolant& p); void alglib::polynomialbuild( real_1d_array x, real_1d_array y, ae_int_t n, barycentricinterpolant& p);
Examples: [1]
polynomialbuildcheb1 function/************************************************************************* Lagrange intepolant on Chebyshev grid (first kind). This function has O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] Y - function values at the nodes, array[0..N-1], Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n))) N - number of points, N>=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::polynomialbuildcheb1( double a, double b, real_1d_array y, barycentricinterpolant& p); void alglib::polynomialbuildcheb1( double a, double b, real_1d_array y, ae_int_t n, barycentricinterpolant& p);
Examples: [1]
polynomialbuildcheb2 function/************************************************************************* Lagrange intepolant on Chebyshev grid (second kind). This function has O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] Y - function values at the nodes, array[0..N-1], Y[I] = Y(0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1))) N - number of points, N>=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::polynomialbuildcheb2( double a, double b, real_1d_array y, barycentricinterpolant& p); void alglib::polynomialbuildcheb2( double a, double b, real_1d_array y, ae_int_t n, barycentricinterpolant& p);
Examples: [1]
polynomialbuildeqdist function/************************************************************************* Lagrange intepolant: generation of the model on equidistant grid. This function has O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] Y - function values at the nodes, array[0..N-1] N - number of points, N>=1 for N=1 a constant model is constructed. OUTPUT PARAMETERS P - barycentric model which represents Lagrange interpolant (see ratint unit info and BarycentricCalc() description for more information). -- ALGLIB -- Copyright 03.12.2009 by Bochkanov Sergey *************************************************************************/void alglib::polynomialbuildeqdist( double a, double b, real_1d_array y, barycentricinterpolant& p); void alglib::polynomialbuildeqdist( double a, double b, real_1d_array y, ae_int_t n, barycentricinterpolant& p);
Examples: [1]
polynomialcalccheb1 function/************************************************************************* Fast polynomial interpolation function on Chebyshev points (first kind) with O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on Chebyshev grid (first kind), X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*(2*i+1)/(2*n)) for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolIntBuildCheb1()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/double alglib::polynomialcalccheb1( double a, double b, real_1d_array f, double t); double alglib::polynomialcalccheb1( double a, double b, real_1d_array f, ae_int_t n, double t);
Examples: [1]
polynomialcalccheb2 function/************************************************************************* Fast polynomial interpolation function on Chebyshev points (second kind) with O(N) complexity. INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on Chebyshev grid (second kind), X[i] = 0.5*(B+A) + 0.5*(B-A)*Cos(PI*i/(n-1)) for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolIntBuildCheb2()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/double alglib::polynomialcalccheb2( double a, double b, real_1d_array f, double t); double alglib::polynomialcalccheb2( double a, double b, real_1d_array f, ae_int_t n, double t);
Examples: [1]
polynomialcalceqdist function/************************************************************************* Fast equidistant polynomial interpolation function with O(N) complexity INPUT PARAMETERS: A - left boundary of [A,B] B - right boundary of [A,B] F - function values, array[0..N-1] N - number of points on equidistant grid, N>=1 for N=1 a constant model is constructed. T - position where P(x) is calculated RESULT value of the Lagrange interpolant at T IMPORTANT this function provides fast interface which is not overflow-safe nor it is very precise. the best option is to use PolynomialBuildEqDist()/BarycentricCalc() subroutines unless you are pretty sure that your data will not result in overflow. -- ALGLIB -- Copyright 02.12.2009 by Bochkanov Sergey *************************************************************************/double alglib::polynomialcalceqdist( double a, double b, real_1d_array f, double t); double alglib::polynomialcalceqdist( double a, double b, real_1d_array f, ae_int_t n, double t);
Examples: [1]
polynomialcheb2bar function/************************************************************************* Conversion from Chebyshev basis to barycentric representation. This function has O(N^2) complexity. INPUT PARAMETERS: T - coefficients of Chebyshev representation; P(x) = sum { T[i]*Ti(2*(x-A)/(B-A)-1), i=0..N }, where Ti - I-th Chebyshev polynomial. N - number of coefficients: * if given, only leading N elements of T are used * if not given, automatically determined from size of T A,B - base interval for Chebyshev polynomials (see above) A<B OUTPUT PARAMETERS P - polynomial in barycentric form -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::polynomialcheb2bar( real_1d_array t, double a, double b, barycentricinterpolant& p); void alglib::polynomialcheb2bar( real_1d_array t, ae_int_t n, double a, double b, barycentricinterpolant& p);
polynomialpow2bar function/************************************************************************* Conversion from power basis to barycentric representation. This function has O(N^2) complexity. INPUT PARAMETERS: A - coefficients, P(x) = sum { A[i]*((X-C)/S)^i, i=0..N-1 } N - number of coefficients (polynomial degree plus 1) * if given, only leading N elements of A are used * if not given, automatically determined from size of A C - offset (see below); 0.0 is used as default value. S - scale (see below); 1.0 is used as default value. S<>0. OUTPUT PARAMETERS P - polynomial in barycentric form NOTES: 1. this function accepts offset and scale, which can be set to improve numerical properties of polynomial. For example, if you interpolate on [-1,+1], you can set C=0 and S=1 and convert from sum of 1, x, x^2, x^3 and so on. In most cases you it is exactly what you need. However, if your interpolation model was built on [999,1001], you will see significant growth of numerical errors when using {1, x, x^2, x^3} as input basis. Converting from sum of 1, (x-1000), (x-1000)^2, (x-1000)^3 will be better option (you have to specify 1000.0 as offset C and 1.0 as scale S). 2. power basis is ill-conditioned and tricks described above can't solve this problem completely. This function will return barycentric model in any case, but for N>8 accuracy well degrade. However, N's less than 5 are pretty safe. -- ALGLIB -- Copyright 30.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::polynomialpow2bar( real_1d_array a, barycentricinterpolant& p); void alglib::polynomialpow2bar( real_1d_array a, ae_int_t n, double c, double s, barycentricinterpolant& p);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Here we demonstrate polynomial interpolation and differentiation
// of y=x^2-x sampled at [0,1,2]. Barycentric representation of polynomial is used.
//
real_1d_array x = "[0,1,2]";
real_1d_array y = "[0,0,2]";
double t = -1;
double v;
double dv;
double d2v;
barycentricinterpolant p;
// barycentric model is created
polynomialbuild(x, y, p);
// barycentric interpolation is demonstrated
v = barycentriccalc(p, t);
printf("%.4f\n", double(v)); // EXPECTED: 2.0
// barycentric differentation is demonstrated
barycentricdiff1(p, t, v, dv);
printf("%.4f\n", double(v)); // EXPECTED: 2.0
printf("%.4f\n", double(dv)); // EXPECTED: -3.0
// second derivatives with barycentric representation
barycentricdiff1(p, t, v, dv);
printf("%.4f\n", double(v)); // EXPECTED: 2.0
printf("%.4f\n", double(dv)); // EXPECTED: -3.0
barycentricdiff2(p, t, v, dv, d2v);
printf("%.4f\n", double(v)); // EXPECTED: 2.0
printf("%.4f\n", double(dv)); // EXPECTED: -3.0
printf("%.4f\n", double(d2v)); // EXPECTED: 2.0
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Here we demonstrate conversion of y=x^2-x
// between power basis and barycentric representation.
//
real_1d_array a = "[0,-1,+1]";
double t = 2;
real_1d_array a2;
double v;
barycentricinterpolant p;
//
// a=[0,-1,+1] is decomposition of y=x^2-x in the power basis:
//
// y = 0 - 1*x + 1*x^2
//
// We convert it to the barycentric form.
//
polynomialpow2bar(a, p);
// now we have barycentric interpolation; we can use it for interpolation
v = barycentriccalc(p, t);
printf("%.2f\n", double(v)); // EXPECTED: 2.0
// we can also convert back from barycentric representation to power basis
polynomialbar2pow(p, a2);
printf("%s\n", a2.tostring(2).c_str()); // EXPECTED: [0,-1,+1]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Temporaries:
// * values of y=x^2-x sampled at three special grids:
// * equdistant grid spanning [0,2], x[i] = 2*i/(N-1), i=0..N-1
// * Chebyshev-I grid spanning [-1,+1], x[i] = 1 + Cos(PI*(2*i+1)/(2*n)), i=0..N-1
// * Chebyshev-II grid spanning [-1,+1], x[i] = 1 + Cos(PI*i/(n-1)), i=0..N-1
// * barycentric interpolants for these three grids
// * vectors to store coefficients of quadratic representation
//
real_1d_array y_eqdist = "[0,0,2]";
real_1d_array y_cheb1 = "[-0.116025,0.000000,1.616025]";
real_1d_array y_cheb2 = "[0,0,2]";
barycentricinterpolant p_eqdist;
barycentricinterpolant p_cheb1;
barycentricinterpolant p_cheb2;
real_1d_array a_eqdist;
real_1d_array a_cheb1;
real_1d_array a_cheb2;
//
// First, we demonstrate construction of barycentric interpolants on
// special grids. We unpack power representation to ensure that
// interpolant was built correctly.
//
// In all three cases we should get same quadratic function.
//
polynomialbuildeqdist(0.0, 2.0, y_eqdist, p_eqdist);
polynomialbar2pow(p_eqdist, a_eqdist);
printf("%s\n", a_eqdist.tostring(4).c_str()); // EXPECTED: [0,-1,+1]
polynomialbuildcheb1(-1, +1, y_cheb1, p_cheb1);
polynomialbar2pow(p_cheb1, a_cheb1);
printf("%s\n", a_cheb1.tostring(4).c_str()); // EXPECTED: [0,-1,+1]
polynomialbuildcheb2(-1, +1, y_cheb2, p_cheb2);
polynomialbar2pow(p_cheb2, a_cheb2);
printf("%s\n", a_cheb2.tostring(4).c_str()); // EXPECTED: [0,-1,+1]
//
// Now we demonstrate polynomial interpolation without construction
// of the barycentricinterpolant structure.
//
// We calculate interpolant value at x=-2.
// In all three cases we should get same f=6
//
double t = -2;
double v;
v = polynomialcalceqdist(0.0, 2.0, y_eqdist, t);
printf("%.4f\n", double(v)); // EXPECTED: 6.0
v = polynomialcalccheb1(-1, +1, y_cheb1, t);
printf("%.4f\n", double(v)); // EXPECTED: 6.0
v = polynomialcalccheb2(-1, +1, y_cheb2, t);
printf("%.4f\n", double(v)); // EXPECTED: 6.0
return 0;
}
polynomialsolver subpackagepolynomialsolverreport class/************************************************************************* *************************************************************************/class polynomialsolverreport { double maxerr; };
polynomialsolve function/************************************************************************* Polynomial root finding. This function returns all roots of the polynomial P(x) = a0 + a1*x + a2*x^2 + ... + an*x^n Both real and complex roots are returned (see below). INPUT PARAMETERS: A - array[N+1], polynomial coefficients: * A[0] is constant term * A[N] is a coefficient of X^N N - polynomial degree OUTPUT PARAMETERS: X - array of complex roots: * for isolated real root, X[I] is strictly real: IMAGE(X[I])=0 * complex roots are always returned in pairs - roots occupy positions I and I+1, with: * X[I+1]=Conj(X[I]) * IMAGE(X[I]) > 0 * IMAGE(X[I+1]) = -IMAGE(X[I]) < 0 * multiple real roots may have non-zero imaginary part due to roundoff errors. There is no reliable way to distinguish real root of multiplicity 2 from two complex roots in the presence of roundoff errors. Rep - report, additional information, following fields are set: * Rep.MaxErr - max( |P(xi)| ) for i=0..N-1. This field allows to quickly estimate "quality" of the roots being returned. NOTE: this function uses companion matrix method to find roots. In case internal EVD solver fails do find eigenvalues, exception is generated. NOTE: roots are not "polished" and no matrix balancing is performed for them. -- ALGLIB -- Copyright 24.02.2014 by Bochkanov Sergey *************************************************************************/void alglib::polynomialsolve( real_1d_array a, ae_int_t n, complex_1d_array& x, polynomialsolverreport& rep);
psif subpackagepsi function/************************************************************************* Psi (digamma) function d - psi(x) = -- ln | (x) dx is the logarithmic derivative of the gamma function. For integer x, n-1 - psi(n) = -EUL + > 1/k. - k=1 This formula is used for 0 < n <= 10. If x is negative, it is transformed to a positive argument by the reflection formula psi(1-x) = psi(x) + pi cot(pi x). For general positive x, the argument is made greater than 10 using the recurrence psi(x+1) = psi(x) + 1/x. Then the following asymptotic expansion is applied: inf. B - 2k psi(x) = log(x) - 1/2x - > ------- - 2k k=1 2k x where the B2k are Bernoulli numbers. ACCURACY: Relative error (except absolute when |psi| < 1): arithmetic domain # trials peak rms IEEE 0,30 30000 1.3e-15 1.4e-16 IEEE -30,0 40000 1.5e-15 2.2e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1992, 2000 by Stephen L. Moshier *************************************************************************/double alglib::psi(double x);
ratint subpackagebarycentricinterpolant class/************************************************************************* Barycentric interpolant. *************************************************************************/class barycentricinterpolant { };
barycentricbuildfloaterhormann function/************************************************************************* Rational interpolant without poles The subroutine constructs the rational interpolating function without real poles (see 'Barycentric rational interpolation with no poles and high rates of approximation', Michael S. Floater. and Kai Hormann, for more information on this subject). Input parameters: X - interpolation nodes, array[0..N-1]. Y - function values, array[0..N-1]. N - number of nodes, N>0. D - order of the interpolation scheme, 0 <= D <= N-1. D<0 will cause an error. D>=N it will be replaced with D=N-1. if you don't know what D to choose, use small value about 3-5. Output parameters: B - barycentric interpolant. Note: this algorithm always succeeds and calculates the weights with close to machine precision. -- ALGLIB PROJECT -- Copyright 17.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::barycentricbuildfloaterhormann( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t d, barycentricinterpolant& b);
barycentricbuildxyw function/************************************************************************* Rational interpolant from X/Y/W arrays F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i])) INPUT PARAMETERS: X - interpolation nodes, array[0..N-1] F - function values, array[0..N-1] W - barycentric weights, array[0..N-1] N - nodes count, N>0 OUTPUT PARAMETERS: B - barycentric interpolant built from (X, Y, W) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentricbuildxyw( real_1d_array x, real_1d_array y, real_1d_array w, ae_int_t n, barycentricinterpolant& b);
barycentriccalc function/************************************************************************* Rational interpolation using barycentric formula F(t) = SUM(i=0,n-1,w[i]*f[i]/(t-x[i])) / SUM(i=0,n-1,w[i]/(t-x[i])) Input parameters: B - barycentric interpolant built with one of model building subroutines. T - interpolation point Result: barycentric interpolant F(t) -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/double alglib::barycentriccalc(barycentricinterpolant b, double t);
barycentricdiff1 function/************************************************************************* Differentiation of barycentric interpolant: first derivative. Algorithm used in this subroutine is very robust and should not fail until provided with values too close to MaxRealNumber (usually MaxRealNumber/N or greater will overflow). INPUT PARAMETERS: B - barycentric interpolant built with one of model building subroutines. T - interpolation point OUTPUT PARAMETERS: F - barycentric interpolant at T DF - first derivative NOTE -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentricdiff1( barycentricinterpolant b, double t, double& f, double& df);
barycentricdiff2 function/************************************************************************* Differentiation of barycentric interpolant: first/second derivatives. INPUT PARAMETERS: B - barycentric interpolant built with one of model building subroutines. T - interpolation point OUTPUT PARAMETERS: F - barycentric interpolant at T DF - first derivative D2F - second derivative NOTE: this algorithm may fail due to overflow/underflor if used on data whose values are close to MaxRealNumber or MinRealNumber. Use more robust BarycentricDiff1() subroutine in such cases. -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentricdiff2( barycentricinterpolant b, double t, double& f, double& df, double& d2f);
barycentriclintransx function/************************************************************************* This subroutine performs linear transformation of the argument. INPUT PARAMETERS: B - rational interpolant in barycentric form CA, CB - transformation coefficients: x = CA*t + CB OUTPUT PARAMETERS: B - transformed interpolant with X replaced by T -- ALGLIB PROJECT -- Copyright 19.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentriclintransx( barycentricinterpolant b, double ca, double cb);
barycentriclintransy function/************************************************************************* This subroutine performs linear transformation of the barycentric interpolant. INPUT PARAMETERS: B - rational interpolant in barycentric form CA, CB - transformation coefficients: B2(x) = CA*B(x) + CB OUTPUT PARAMETERS: B - transformed interpolant -- ALGLIB PROJECT -- Copyright 19.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentriclintransy( barycentricinterpolant b, double ca, double cb);
barycentricunpack function/************************************************************************* Extracts X/Y/W arrays from rational interpolant INPUT PARAMETERS: B - barycentric interpolant OUTPUT PARAMETERS: N - nodes count, N>0 X - interpolation nodes, array[0..N-1] F - function values, array[0..N-1] W - barycentric weights, array[0..N-1] -- ALGLIB -- Copyright 17.08.2009 by Bochkanov Sergey *************************************************************************/void alglib::barycentricunpack( barycentricinterpolant b, ae_int_t& n, real_1d_array& x, real_1d_array& y, real_1d_array& w);
rbf subpackage| rbf_d_hrbf | Simple model built with HRBF algorithm | |
| rbf_d_polterm | RBF models - working with polynomial term | |
| rbf_d_serialize | Serialization/unserialization | |
| rbf_d_vector | Working with vector functions |
rbfcalcbuffer class/************************************************************************* Buffer object which is used to perform nearest neighbor requests in the multithreaded mode (multiple threads working with same KD-tree object). This object should be created with KDTreeCreateBuffer(). *************************************************************************/class rbfcalcbuffer { };
rbfmodel class/************************************************************************* RBF model. Never try to directly work with fields of this object - always use ALGLIB functions to use this object. *************************************************************************/class rbfmodel { };
rbfreport class/************************************************************************* RBF solution report: * TerminationType - termination type, positive values - success, non-positive - failure. Fields which are set by modern RBF solvers (hierarchical): * RMSError - root-mean-square error; NAN for old solvers (ML, QNN) * MaxError - maximum error; NAN for old solvers (ML, QNN) *************************************************************************/class rbfreport { double rmserror; double maxerror; ae_int_t arows; ae_int_t acols; ae_int_t annz; ae_int_t iterationscount; ae_int_t nmv; ae_int_t terminationtype; };
rbfbuildmodel function/************************************************************************* This function builds RBF model and returns report (contains some information which can be used for evaluation of the algorithm properties). Call to this function modifies RBF model by calculating its centers/radii/ weights and saving them into RBFModel structure. Initially RBFModel contain zero coefficients, but after call to this function we will have coefficients which were calculated in order to fit our dataset. After you called this function you can call RBFCalc(), RBFGridCalc() and other model calculation functions. INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call Rep - report: * Rep.TerminationType: * -5 - non-distinct basis function centers were detected, interpolation aborted; only QNN returns this error code, other algorithms can handle non- distinct nodes. * -4 - nonconvergence of the internal SVD solver * -3 incorrect model construction algorithm was chosen: QNN or RBF-ML, combined with one of the incompatible features - NX=1 or NX>3; points with per-dimension scales. * 1 - successful termination Fields which are set only by modern RBF solvers (hierarchical or nonnegative; older solvers like QNN and ML initialize these fields by NANs): * rep.rmserror - root-mean-square error at nodes * rep.maxerror - maximum error at nodes Fields are used for debugging purposes: * Rep.IterationsCount - iterations count of the LSQR solver * Rep.NMV - number of matrix-vector products * Rep.ARows - rows count for the system matrix * Rep.ACols - columns count for the system matrix * Rep.ANNZ - number of significantly non-zero elements (elements above some algorithm-determined threshold) NOTE: failure to build model will leave current state of the structure unchanged. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfbuildmodel(rbfmodel s, rbfreport& rep);
rbfcalc function/************************************************************************* This function calculates values of the RBF model at the given point. This is general function which can be used for arbitrary NX (dimension of the space of arguments) and NY (dimension of the function itself). However when you have NY=1 you may find more convenient to use rbfcalc2() or rbfcalc3(). If you want to perform parallel model evaluation from multiple threads, use rbftscalcbuf() with per-thread buffer object. This function returns 0.0 when model is not initialized. INPUT PARAMETERS: S - RBF model X - coordinates, array[NX]. X may have more than NX elements, in this case only leading NX will be used. OUTPUT PARAMETERS: Y - function value, array[NY]. Y is out-parameter and reallocated after call to this function. In case you want to reuse previously allocated Y, you may use RBFCalcBuf(), which reallocates Y only when it is too small. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfcalc(rbfmodel s, real_1d_array x, real_1d_array& y);
Examples: [1]
rbfcalc1 function/************************************************************************* This function calculates values of the RBF model in the given point. IMPORTANT: this function works only with modern (hierarchical) RBFs. It can not be used with legacy (version 1) RBFs because older RBF code does not support 1-dimensional models. This function should be used when we have NY=1 (scalar function) and NX=1 (1-dimensional space). If you have 3-dimensional space, use rbfcalc3(). If you have 2-dimensional space, use rbfcalc3(). If you have general situation (NX-dimensional space, NY-dimensional function) you should use generic rbfcalc(). If you want to perform parallel model evaluation from multiple threads, use rbftscalcbuf() with per-thread buffer object. This function returns 0.0 when: * model is not initialized * NX<>1 * NY<>1 INPUT PARAMETERS: S - RBF model X0 - X-coordinate, finite number RESULT: value of the model or 0.0 (as defined above) -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/double alglib::rbfcalc1(rbfmodel s, double x0);
rbfcalc2 function/************************************************************************* This function calculates values of the RBF model in the given point. This function should be used when we have NY=1 (scalar function) and NX=2 (2-dimensional space). If you have 3-dimensional space, use rbfcalc3(). If you have general situation (NX-dimensional space, NY-dimensional function) you should use generic rbfcalc(). If you want to calculate function values many times, consider using rbfgridcalc2v(), which is far more efficient than many subsequent calls to rbfcalc2(). If you want to perform parallel model evaluation from multiple threads, use rbftscalcbuf() with per-thread buffer object. This function returns 0.0 when: * model is not initialized * NX<>2 *NY<>1 INPUT PARAMETERS: S - RBF model X0 - first coordinate, finite number X1 - second coordinate, finite number RESULT: value of the model or 0.0 (as defined above) -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/double alglib::rbfcalc2(rbfmodel s, double x0, double x1);
rbfcalc3 function/************************************************************************* This function calculates value of the RBF model in the given point. This function should be used when we have NY=1 (scalar function) and NX=3 (3-dimensional space). If you have 2-dimensional space, use rbfcalc2(). If you have general situation (NX-dimensional space, NY-dimensional function) you should use generic rbfcalc(). If you want to calculate function values many times, consider using rbfgridcalc3v(), which is far more efficient than many subsequent calls to rbfcalc3(). If you want to perform parallel model evaluation from multiple threads, use rbftscalcbuf() with per-thread buffer object. This function returns 0.0 when: * model is not initialized * NX<>3 *NY<>1 INPUT PARAMETERS: S - RBF model X0 - first coordinate, finite number X1 - second coordinate, finite number X2 - third coordinate, finite number RESULT: value of the model or 0.0 (as defined above) -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/double alglib::rbfcalc3(rbfmodel s, double x0, double x1, double x2);
rbfcalcbuf function/************************************************************************* This function calculates values of the RBF model at the given point. Same as rbfcalc(), but does not reallocate Y when in is large enough to store function values. If you want to perform parallel model evaluation from multiple threads, use rbftscalcbuf() with per-thread buffer object. INPUT PARAMETERS: S - RBF model X - coordinates, array[NX]. X may have more than NX elements, in this case only leading NX will be used. Y - possibly preallocated array OUTPUT PARAMETERS: Y - function value, array[NY]. Y is not reallocated when it is larger than NY. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfcalcbuf(rbfmodel s, real_1d_array x, real_1d_array& y);
rbfcreate function/************************************************************************* This function creates RBF model for a scalar (NY=1) or vector (NY>1) function in a NX-dimensional space (NX>=1). Newly created model is empty. It can be used for interpolation right after creation, but it just returns zeros. You have to add points to the model, tune interpolation settings, and then call model construction function rbfbuildmodel() which will update model according to your specification. USAGE: 1. User creates model with rbfcreate() 2. User adds dataset with rbfsetpoints() (points do NOT have to be on a regular grid) or rbfsetpointsandscales(). 3. (OPTIONAL) User chooses polynomial term by calling: * rbflinterm() to set linear term * rbfconstterm() to set constant term * rbfzeroterm() to set zero term By default, linear term is used. 4. User tweaks algorithm properties with rbfsetalgohierarchical() method (or chooses one of the legacy algorithms - QNN (rbfsetalgoqnn) or ML (rbfsetalgomultilayer)). 5. User calls rbfbuildmodel() function which rebuilds model according to the specification 6. User may call rbfcalc() to calculate model value at the specified point, rbfgridcalc() to calculate model values at the points of the regular grid. User may extract model coefficients with rbfunpack() call. IMPORTANT: we recommend you to use latest model construction algorithm - hierarchical RBFs, which is activated by rbfsetalgohierarchical() function. This algorithm is the fastest one, and most memory- efficient. However, it is incompatible with older versions of ALGLIB (pre-3.11). So, if you serialize hierarchical model, you will be unable to load it in pre-3.11 ALGLIB. Other model types (QNN and RBF-ML) are still backward-compatible. INPUT PARAMETERS: NX - dimension of the space, NX>=1 NY - function dimension, NY>=1 OUTPUT PARAMETERS: S - RBF model (initially equals to zero) NOTE 1: memory requirements. RBF models require amount of memory which is proportional to the number of data points. Some additional memory is allocated during model construction, but most of this memory is freed after model coefficients are calculated. Amount of this additional memory depends on model construction algorithm being used. NOTE 2: prior to ALGLIB version 3.11, RBF models supported only NX=2 or NX=3. Any attempt to create single-dimensional or more than 3-dimensional RBF model resulted in exception. ALGLIB 3.11 supports any NX>0, but models created with NX!=2 and NX!=3 are incompatible with (a) older versions of ALGLIB, (b) old model construction algorithms (QNN or RBF-ML). So, if you create a model with NX=2 or NX=3, then, depending on specific model construction algorithm being chosen, you will (QNN and RBF-ML) or will not (HierarchicalRBF) get backward compatibility with older versions of ALGLIB. You have a choice here. However, if you create a model with NX neither 2 nor 3, you have no backward compatibility from the start, and you are forced to use hierarchical RBFs and ALGLIB 3.11 or later. -- ALGLIB -- Copyright 13.12.2011, 20.06.2016 by Bochkanov Sergey *************************************************************************/void alglib::rbfcreate(ae_int_t nx, ae_int_t ny, rbfmodel& s);
rbfcreatecalcbuffer function/************************************************************************* This function creates buffer structure which can be used to perform parallel RBF model evaluations (with one RBF model instance being used from multiple threads, as long as different threads use different instances of buffer). This buffer object can be used with rbftscalcbuf() function (here "ts" stands for "thread-safe", "buf" is a suffix which denotes function which reuses previously allocated output space). How to use it: * create RBF model structure with rbfcreate() * load data, tune parameters * call rbfbuildmodel() * call rbfcreatecalcbuffer(), once per thread working with RBF model (you should call this function only AFTER call to rbfbuildmodel(), see below for more information) * call rbftscalcbuf() from different threads, with each thread working with its own copy of buffer object. INPUT PARAMETERS S - RBF model OUTPUT PARAMETERS Buf - external buffer. IMPORTANT: buffer object should be used only with RBF model object which was used to initialize buffer. Any attempt to use buffer with different object is dangerous - you may get memory violation error because sizes of internal arrays do not fit to dimensions of RBF structure. IMPORTANT: you should call this function only for model which was built with rbfbuildmodel() function, after successful invocation of rbfbuildmodel(). Sizes of some internal structures are determined only after model is built, so buffer object created before model construction stage will be useless (and any attempt to use it will result in exception). -- ALGLIB -- Copyright 02.04.2016 by Sergey Bochkanov *************************************************************************/void alglib::rbfcreatecalcbuffer(rbfmodel s, rbfcalcbuffer& buf);
rbfgetmodelversion function/************************************************************************* This function returns model version. INPUT PARAMETERS: S - RBF model RESULT: * 1 - for models created by QNN and RBF-ML algorithms, compatible with ALGLIB 3.10 or earlier. * 2 - for models created by HierarchicalRBF, requires ALGLIB 3.11 or later -- ALGLIB -- Copyright 06.07.2016 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::rbfgetmodelversion(rbfmodel s);
rbfgridcalc2 function/************************************************************************* This is legacy function for gridded calculation of RBF model. It is superseded by rbfgridcalc2v() and rbfgridcalc2vsubset() functions. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfgridcalc2( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_2d_array& y);
rbfgridcalc2v function/************************************************************************* This function calculates values of the RBF model at the regular grid, which has N0*N1 points, with Point[I,J] = (X0[I], X1[J]). Vector-valued RBF models are supported. This function returns 0.0 when: * model is not initialized * NX<>2 FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes multicore support (C++ and C# ! computational cores) for this function. ! ! Parallel processing gives close-to-linear speedup on multicore systems, ! assuming that problem is large enough to be divided between cores. ! You should solve at least 200x200 problem to get nearly-linear speed- ! up (high efficiency). ! ! Parallel processing is implemented only for modern (hierarchical) ! RBFs. Legacy version 1 RBFs (created by QNN or RBF-ML) are still ! processed serially. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - RBF model, used in read-only mode, can be shared between multiple invocations of this function from multiple threads. X0 - array of grid nodes, first coordinates, array[N0]. Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N0 - grid size (number of nodes) in the first dimension X1 - array of grid nodes, second coordinates, array[N1] Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N1 - grid size (number of nodes) in the second dimension OUTPUT PARAMETERS: Y - function values, array[NY*N0*N1], where NY is a number of "output" vector values (this function supports vector- valued RBF models). Y is out-variable and is reallocated by this function. Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]), for: * K=0...NY-1 * I0=0...N0-1 * I1=0...N1-1 NOTE: this function supports weakly ordered grid nodes, i.e. you may have X[i]=X[i+1] for some i. It does not provide you any performance benefits due to duplication of points, just convenience and flexibility. NOTE: this function is re-entrant, i.e. you may use same rbfmodel structure in multiple threads calling this function for different grids. NOTE: if you need function values on some subset of regular grid, which may be described as "several compact and dense islands", you may use rbfgridcalc2vsubset(). -- ALGLIB -- Copyright 27.01.2017 by Bochkanov Sergey *************************************************************************/void alglib::rbfgridcalc2v( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_1d_array& y); void alglib::smp_rbfgridcalc2v( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_1d_array& y);
rbfgridcalc2vsubset function/************************************************************************* This function calculates values of the RBF model at some subset of regular grid: * grid has N0*N1 points, with Point[I,J] = (X0[I], X1[J]) * only values at some subset of this grid are required Vector-valued RBF models are supported. This function returns 0.0 when: * model is not initialized * NX<>2 FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes multicore support (C++ and C# ! computational cores) for this function. ! ! Parallel processing gives close-to-linear speedup on multicore systems, ! assuming that problem is large enough to be divided between cores. ! You should solve at least 200x200 problem to get nearly-linear speed- ! up (high efficiency). ! ! Parallel processing is implemented only for modern (hierarchical) ! RBFs. Legacy version 1 RBFs (created by QNN or RBF-ML) are still ! processed serially. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - RBF model, used in read-only mode, can be shared between multiple invocations of this function from multiple threads. X0 - array of grid nodes, first coordinates, array[N0]. Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N0 - grid size (number of nodes) in the first dimension X1 - array of grid nodes, second coordinates, array[N1] Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N1 - grid size (number of nodes) in the second dimension FlagY - array[N0*N1]: * Y[I0+I1*N0] corresponds to node (X0[I0],X1[I1]) * it is a "bitmap" array which contains False for nodes which are NOT calculated, and True for nodes which are required. OUTPUT PARAMETERS: Y - function values, array[NY*N0*N1*N2], where NY is a number of "output" vector values (this function supports vector- valued RBF models): * Y[K+NY*(I0+I1*N0)]=F_k(X0[I0],X1[I1]), for K=0...NY-1, I0=0...N0-1, I1=0...N1-1. * elements of Y[] which correspond to FlagY[]=True are loaded by model values (which may be exactly zero for some nodes). * elements of Y[] which correspond to FlagY[]=False MAY be initialized by zeros OR may be calculated. This function processes grid as a hierarchy of nested blocks and micro-rows. If just one element of micro-row is required, entire micro-row (up to 8 nodes in the current version, but no promises) is calculated. NOTE: this function supports weakly ordered grid nodes, i.e. you may have X[i]=X[i+1] for some i. It does not provide you any performance benefits due to duplication of points, just convenience and flexibility. NOTE: this function is re-entrant, i.e. you may use same rbfmodel structure in multiple threads calling this function for different grids. -- ALGLIB -- Copyright 04.03.2016 by Bochkanov Sergey *************************************************************************/void alglib::rbfgridcalc2vsubset( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, boolean_1d_array flagy, real_1d_array& y); void alglib::smp_rbfgridcalc2vsubset( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, boolean_1d_array flagy, real_1d_array& y);
rbfgridcalc3v function/************************************************************************* This function calculates values of the RBF model at the regular grid, which has N0*N1*N2 points, with Point[I,J,K] = (X0[I], X1[J], X2[K]). Vector-valued RBF models are supported. This function returns 0.0 when: * model is not initialized * NX<>3 FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes multicore support (C++ and C# ! computational cores) for this function. ! ! Parallel processing gives close-to-linear speedup on multicore systems, ! assuming that problem is large enough to be divided between cores. ! You should solve at least 50x50x50 problem to get nearly-linear speed- ! up (high efficiency). ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - RBF model, used in read-only mode, can be shared between multiple invocations of this function from multiple threads. X0 - array of grid nodes, first coordinates, array[N0]. Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N0 - grid size (number of nodes) in the first dimension X1 - array of grid nodes, second coordinates, array[N1] Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N1 - grid size (number of nodes) in the second dimension X2 - array of grid nodes, third coordinates, array[N2] Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N2 - grid size (number of nodes) in the third dimension OUTPUT PARAMETERS: Y - function values, array[NY*N0*N1*N2], where NY is a number of "output" vector values (this function supports vector- valued RBF models). Y is out-variable and is reallocated by this function. Y[K+NY*(I0+I1*N0+I2*N0*N1)]=F_k(X0[I0],X1[I1],X2[I2]), for: * K=0...NY-1 * I0=0...N0-1 * I1=0...N1-1 * I2=0...N2-1 NOTE: this function supports weakly ordered grid nodes, i.e. you may have X[i]=X[i+1] for some i. It does not provide you any performance benefits due to duplication of points, just convenience and flexibility. NOTE: this function is re-entrant, i.e. you may use same rbfmodel structure in multiple threads calling this function for different grids. NOTE: if you need function values on some subset of regular grid, which may be described as "several compact and dense islands", you may use rbfgridcalc3vsubset(). -- ALGLIB -- Copyright 04.03.2016 by Bochkanov Sergey *************************************************************************/void alglib::rbfgridcalc3v( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_1d_array x2, ae_int_t n2, real_1d_array& y); void alglib::smp_rbfgridcalc3v( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_1d_array x2, ae_int_t n2, real_1d_array& y);
rbfgridcalc3vsubset function/************************************************************************* This function calculates values of the RBF model at some subset of regular grid: * grid has N0*N1*N2 points, with Point[I,J,K] = (X0[I], X1[J], X2[K]) * only values at some subset of this grid are required Vector-valued RBF models are supported. This function returns 0.0 when: * model is not initialized * NX<>3 FOR USERS OF COMMERCIAL EDITION: ! Commercial version of ALGLIB includes multicore support (C++ and C# ! computational cores) for this function. ! ! Parallel processing gives close-to-linear speedup on multicore systems, ! assuming that problem is large enough to be divided between cores. ! You should solve at least 50x50x50 problem to get nearly-linear speed- ! up (high efficiency). ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! This note is given for users of commercial edition; if you use GPL ! edition, you still will be able to call smp-version of this function, ! but all computations will be done serially. ! ! We recommend you to carefully read ALGLIB Reference Manual, section ! called 'SMP support', before using parallel version of this function. INPUT PARAMETERS: S - RBF model, used in read-only mode, can be shared between multiple invocations of this function from multiple threads. X0 - array of grid nodes, first coordinates, array[N0]. Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N0 - grid size (number of nodes) in the first dimension X1 - array of grid nodes, second coordinates, array[N1] Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N1 - grid size (number of nodes) in the second dimension X2 - array of grid nodes, third coordinates, array[N2] Must be ordered by ascending. Exception is generated if the array is not correctly ordered. N2 - grid size (number of nodes) in the third dimension FlagY - array[N0*N1*N2]: * Y[I0+I1*N0+I2*N0*N1] corresponds to node (X0[I0],X1[I1],X2[I2]) * it is a "bitmap" array which contains False for nodes which are NOT calculated, and True for nodes which are required. OUTPUT PARAMETERS: Y - function values, array[NY*N0*N1*N2], where NY is a number of "output" vector values (this function supports vector- valued RBF models): * Y[K+NY*(I0+I1*N0+I2*N0*N1)]=F_k(X0[I0],X1[I1],X2[I2]), for K=0...NY-1, I0=0...N0-1, I1=0...N1-1, I2=0...N2-1. * elements of Y[] which correspond to FlagY[]=True are loaded by model values (which may be exactly zero for some nodes). * elements of Y[] which correspond to FlagY[]=False MAY be initialized by zeros OR may be calculated. This function processes grid as a hierarchy of nested blocks and micro-rows. If just one element of micro-row is required, entire micro-row (up to 8 nodes in the current version, but no promises) is calculated. NOTE: this function supports weakly ordered grid nodes, i.e. you may have X[i]=X[i+1] for some i. It does not provide you any performance benefits due to duplication of points, just convenience and flexibility. NOTE: this function is re-entrant, i.e. you may use same rbfmodel structure in multiple threads calling this function for different grids. -- ALGLIB -- Copyright 04.03.2016 by Bochkanov Sergey *************************************************************************/void alglib::rbfgridcalc3vsubset( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_1d_array x2, ae_int_t n2, boolean_1d_array flagy, real_1d_array& y); void alglib::smp_rbfgridcalc3vsubset( rbfmodel s, real_1d_array x0, ae_int_t n0, real_1d_array x1, ae_int_t n1, real_1d_array x2, ae_int_t n2, boolean_1d_array flagy, real_1d_array& y);
rbfserialize function/************************************************************************* This function serializes data structure to string. Important properties of s_out: * it contains alphanumeric characters, dots, underscores, minus signs * these symbols are grouped into words, which are separated by spaces and Windows-style (CR+LF) newlines * although serializer uses spaces and CR+LF as separators, you can replace any separator character by arbitrary combination of spaces, tabs, Windows or Unix newlines. It allows flexible reformatting of the string in case you want to include it into text or XML file. But you should not insert separators into the middle of the "words" nor you should change case of letters. * s_out can be freely moved between 32-bit and 64-bit systems, little and big endian machines, and so on. You can serialize structure on 32-bit machine and unserialize it on 64-bit one (or vice versa), or serialize it on SPARC and unserialize on x86. You can also serialize it in C++ version of ALGLIB and unserialize in C# one, and vice versa. *************************************************************************/void rbfserialize(rbfmodel &obj, std::string &s_out); void rbfserialize(rbfmodel &obj, std::ostream &s_out);
rbfsetalgohierarchical function/************************************************************************* This function sets RBF interpolation algorithm. ALGLIB supports several RBF algorithms with different properties. This algorithm is called Hierarchical RBF. It similar to its previous incarnation, RBF-ML, i.e. it also builds a sequence of models with decreasing radii. However, it uses more economical way of building upper layers (ones with large radii), which results in faster model construction and evaluation, as well as smaller memory footprint during construction. This algorithm has following important features: * ability to handle millions of points * controllable smoothing via nonlinearity penalization * support for NX-dimensional models with NX=1 or NX>3 (unlike QNN or RBF-ML) * support for specification of per-dimensional radii via scale vector, which is set by means of rbfsetpointsandscales() function. This feature is useful if you solve spatio-temporal interpolation problems, where different radii are required for spatial and temporal dimensions. Running times are roughly proportional to: * N*log(N)*NLayers - for model construction * N*NLayers - for model evaluation You may see that running time does not depend on search radius or points density, just on number of layers in the hierarchy. IMPORTANT: this model construction algorithm was introduced in ALGLIB 3.11 and produces models which are INCOMPATIBLE with previous versions of ALGLIB. You can not unserialize models produced with this function in ALGLIB 3.10 or earlier. INPUT PARAMETERS: S - RBF model, initialized by rbfcreate() call RBase - RBase parameter, RBase>0 NLayers - NLayers parameter, NLayers>0, recommended value to start with - about 5. LambdaNS- >=0, nonlinearity penalty coefficient, negative values are not allowed. This parameter adds controllable smoothing to the problem, which may reduce noise. Specification of non- zero lambda means that in addition to fitting error solver will also minimize LambdaNS*|S''(x)|^2 (appropriately generalized to multiple dimensions. Specification of exactly zero value means that no penalty is added (we do not even evaluate matrix of second derivatives which is necessary for smoothing). Calculation of nonlinearity penalty is costly - it results in several-fold increase of model construction time. Evaluation time remains the same. Optimal lambda is problem-dependent and requires trial and error. Good value to start from is 1e-5...1e-6, which corresponds to slightly noticeable smoothing of the function. Value 1e-2 usually means that quite heavy smoothing is applied. TUNING ALGORITHM In order to use this algorithm you have to choose three parameters: * initial radius RBase * number of layers in the model NLayers * penalty coefficient LambdaNS Initial radius is easy to choose - you can pick any number several times larger than the average distance between points. Algorithm won't break down if you choose radius which is too large (model construction time will increase, but model will be built correctly). Choose such number of layers that RLast=RBase/2^(NLayers-1) (radius used by the last layer) will be smaller than the typical distance between points. In case model error is too large, you can increase number of layers. Having more layers will make model construction and evaluation proportionally slower, but it will allow you to have model which precisely fits your data. From the other side, if you want to suppress noise, you can DECREASE number of layers to make your model less flexible (or specify non-zero LambdaNS). TYPICAL ERRORS 1. Using too small number of layers - RBF models with large radius are not flexible enough to reproduce small variations in the target function. You need many layers with different radii, from large to small, in order to have good model. 2. Using initial radius which is too small. You will get model with "holes" in the areas which are too far away from interpolation centers. However, algorithm will work correctly (and quickly) in this case. -- ALGLIB -- Copyright 20.06.2016 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetalgohierarchical( rbfmodel s, double rbase, ae_int_t nlayers, double lambdans);
rbfsetalgomultilayer function/************************************************************************* DEPRECATED:since version 3.11 ALGLIB includes new RBF model construction algorithm, Hierarchical RBF. This algorithm is faster and requires less memory than QNN and RBF-ML. It is especially good for large-scale interpolation problems. So, we recommend you to consider Hierarchical RBF as default option. ========================================================================== This function sets RBF interpolation algorithm. ALGLIB supports several RBF algorithms with different properties. This algorithm is called RBF-ML. It builds multilayer RBF model, i.e. model with subsequently decreasing radii, which allows us to combine smoothness (due to large radii of the first layers) with exactness (due to small radii of the last layers) and fast convergence. Internally RBF-ML uses many different means of acceleration, from sparse matrices to KD-trees, which results in algorithm whose working time is roughly proportional to N*log(N)*Density*RBase^2*NLayers, where N is a number of points, Density is an average density if points per unit of the interpolation space, RBase is an initial radius, NLayers is a number of layers. RBF-ML is good for following kinds of interpolation problems: 1. "exact" problems (perfect fit) with well separated points 2. least squares problems with arbitrary distribution of points (algorithm gives perfect fit where it is possible, and resorts to least squares fit in the hard areas). 3. noisy problems where we want to apply some controlled amount of smoothing. INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call RBase - RBase parameter, RBase>0 NLayers - NLayers parameter, NLayers>0, recommended value to start with - about 5. LambdaV - regularization value, can be useful when solving problem in the least squares sense. Optimal lambda is problem- dependent and require trial and error. In our experience, good lambda can be as large as 0.1, and you can use 0.001 as initial guess. Default value - 0.01, which is used when LambdaV is not given. You can specify zero value, but it is not recommended to do so. TUNING ALGORITHM In order to use this algorithm you have to choose three parameters: * initial radius RBase * number of layers in the model NLayers * regularization coefficient LambdaV Initial radius is easy to choose - you can pick any number several times larger than the average distance between points. Algorithm won't break down if you choose radius which is too large (model construction time will increase, but model will be built correctly). Choose such number of layers that RLast=RBase/2^(NLayers-1) (radius used by the last layer) will be smaller than the typical distance between points. In case model error is too large, you can increase number of layers. Having more layers will make model construction and evaluation proportionally slower, but it will allow you to have model which precisely fits your data. From the other side, if you want to suppress noise, you can DECREASE number of layers to make your model less flexible. Regularization coefficient LambdaV controls smoothness of the individual models built for each layer. We recommend you to use default value in case you don't want to tune this parameter, because having non-zero LambdaV accelerates and stabilizes internal iterative algorithm. In case you want to suppress noise you can use LambdaV as additional parameter (larger value = more smoothness) to tune. TYPICAL ERRORS 1. Using initial radius which is too large. Memory requirements of the RBF-ML are roughly proportional to N*Density*RBase^2 (where Density is an average density of points per unit of the interpolation space). In the extreme case of the very large RBase we will need O(N^2) units of memory - and many layers in order to decrease radius to some reasonably small value. 2. Using too small number of layers - RBF models with large radius are not flexible enough to reproduce small variations in the target function. You need many layers with different radii, from large to small, in order to have good model. 3. Using initial radius which is too small. You will get model with "holes" in the areas which are too far away from interpolation centers. However, algorithm will work correctly (and quickly) in this case. 4. Using too many layers - you will get too large and too slow model. This model will perfectly reproduce your function, but maybe you will be able to achieve similar results with less layers (and less memory). -- ALGLIB -- Copyright 02.03.2012 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetalgomultilayer( rbfmodel s, double rbase, ae_int_t nlayers); void alglib::rbfsetalgomultilayer( rbfmodel s, double rbase, ae_int_t nlayers, double lambdav);
rbfsetalgoqnn function/************************************************************************* DEPRECATED:since version 3.11 ALGLIB includes new RBF model construction algorithm, Hierarchical RBF. This algorithm is faster and requires less memory than QNN and RBF-ML. It is especially good for large-scale interpolation problems. So, we recommend you to consider Hierarchical RBF as default option. ========================================================================== This function sets RBF interpolation algorithm. ALGLIB supports several RBF algorithms with different properties. This algorithm is called RBF-QNN and it is good for point sets with following properties: a) all points are distinct b) all points are well separated. c) points distribution is approximately uniform. There is no "contour lines", clusters of points, or other small-scale structures. Algorithm description: 1) interpolation centers are allocated to data points 2) interpolation radii are calculated as distances to the nearest centers times Q coefficient (where Q is a value from [0.75,1.50]). 3) after performing (2) radii are transformed in order to avoid situation when single outlier has very large radius and influences many points across all dataset. Transformation has following form: new_r[i] = min(r[i],Z*median(r[])) where r[i] is I-th radius, median() is a median radius across entire dataset, Z is user-specified value which controls amount of deviation from median radius. When (a) is violated, we will be unable to build RBF model. When (b) or (c) are violated, model will be built, but interpolation quality will be low. See http://www.alglib.net/interpolation/ for more information on this subject. This algorithm is used by default. Additional Q parameter controls smoothness properties of the RBF basis: * Q<0.75 will give perfectly conditioned basis, but terrible smoothness properties (RBF interpolant will have sharp peaks around function values) * Q around 1.0 gives good balance between smoothness and condition number * Q>1.5 will lead to badly conditioned systems and slow convergence of the underlying linear solver (although smoothness will be very good) * Q>2.0 will effectively make optimizer useless because it won't converge within reasonable amount of iterations. It is possible to set such large Q, but it is advised not to do so. INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call Q - Q parameter, Q>0, recommended value - 1.0 Z - Z parameter, Z>0, recommended value - 5.0 NOTE: this function has some serialization-related subtleties. We recommend you to study serialization examples from ALGLIB Reference Manual if you want to perform serialization of your models. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetalgoqnn(rbfmodel s); void alglib::rbfsetalgoqnn(rbfmodel s, double q, double z);
rbfsetconstterm function/************************************************************************* This function sets constant term (model is a sum of radial basis functions plus constant). This function won't have effect until next call to RBFBuildModel(). INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call NOTE: this function has some serialization-related subtleties. We recommend you to study serialization examples from ALGLIB Reference Manual if you want to perform serialization of your models. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetconstterm(rbfmodel s);
Examples: [1]
rbfsetlinterm function/************************************************************************* This function sets linear term (model is a sum of radial basis functions plus linear polynomial). This function won't have effect until next call to RBFBuildModel(). INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call NOTE: this function has some serialization-related subtleties. We recommend you to study serialization examples from ALGLIB Reference Manual if you want to perform serialization of your models. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetlinterm(rbfmodel s);
Examples: [1]
rbfsetpoints function/************************************************************************* This function adds dataset. This function overrides results of the previous calls, i.e. multiple calls of this function will result in only the last set being added. IMPORTANT: ALGLIB version 3.11 and later allows you to specify a set of per-dimension scales. Interpolation radii are multiplied by the scale vector. It may be useful if you have mixed spatio-temporal data (say, a set of 3D slices recorded at different times). You should call rbfsetpointsandscales() function to use this feature. INPUT PARAMETERS: S - RBF model, initialized by rbfcreate() call. XY - points, array[N,NX+NY]. One row corresponds to one point in the dataset. First NX elements are coordinates, next NY elements are function values. Array may be larger than specified, in this case only leading [N,NX+NY] elements will be used. N - number of points in the dataset After you've added dataset and (optionally) tuned algorithm settings you should call rbfbuildmodel() in order to build a model for you. NOTE: dataset added by this function is not saved during model serialization. MODEL ITSELF is serialized, but data used to build it are not. So, if you 1) add dataset to empty RBF model, 2) serialize and unserialize it, then you will get an empty RBF model with no dataset being attached. From the other side, if you call rbfbuildmodel() between (1) and (2), then after (2) you will get your fully constructed RBF model - but again with no dataset attached, so subsequent calls to rbfbuildmodel() will produce empty model. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetpoints(rbfmodel s, real_2d_array xy); void alglib::rbfsetpoints(rbfmodel s, real_2d_array xy, ae_int_t n);
rbfsetpointsandscales function/************************************************************************* This function adds dataset and a vector of per-dimension scales. It may be useful if you have mixed spatio-temporal data - say, a set of 3D slices recorded at different times. Such data typically require different RBF radii for spatial and temporal dimensions. ALGLIB solves this problem by specifying single RBF radius, which is (optionally) multiplied by the scale vector. This function overrides results of the previous calls, i.e. multiple calls of this function will result in only the last set being added. IMPORTANT: only HierarchicalRBF algorithm can work with scaled points. So, using this function results in RBF models which can be used in ALGLIB 3.11 or later. Previous versions of the library will be unable to unserialize models produced by HierarchicalRBF algo. Any attempt to use this function with RBF-ML or QNN algorithms will result in -3 error code being returned (incorrect algorithm). INPUT PARAMETERS: R - RBF model, initialized by rbfcreate() call. XY - points, array[N,NX+NY]. One row corresponds to one point in the dataset. First NX elements are coordinates, next NY elements are function values. Array may be larger than specified, in this case only leading [N,NX+NY] elements will be used. N - number of points in the dataset S - array[NX], scale vector, S[i]>0. After you've added dataset and (optionally) tuned algorithm settings you should call rbfbuildmodel() in order to build a model for you. NOTE: dataset added by this function is not saved during model serialization. MODEL ITSELF is serialized, but data used to build it are not. So, if you 1) add dataset to empty RBF model, 2) serialize and unserialize it, then you will get an empty RBF model with no dataset being attached. From the other side, if you call rbfbuildmodel() between (1) and (2), then after (2) you will get your fully constructed RBF model - but again with no dataset attached, so subsequent calls to rbfbuildmodel() will produce empty model. -- ALGLIB -- Copyright 20.06.2016 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetpointsandscales( rbfmodel r, real_2d_array xy, real_1d_array s); void alglib::rbfsetpointsandscales( rbfmodel r, real_2d_array xy, ae_int_t n, real_1d_array s);
rbfsetv2bf function/************************************************************************* This function sets basis function type, which can be: * 0 for classic Gaussian * 1 for fast and compact bell-like basis function, which becomes exactly zero at distance equal to 3*R (default option). INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call BF - basis function type: * 0 - classic Gaussian * 1 - fast and compact one -- ALGLIB -- Copyright 01.02.2017 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetv2bf(rbfmodel s, ae_int_t bf);
rbfsetv2its function/************************************************************************* This function sets stopping criteria of the underlying linear solver for hierarchical (version 2) RBF constructor. INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call MaxIts - this criterion will stop algorithm after MaxIts iterations. Typically a few hundreds iterations is required, with 400 being a good default value to start experimentation. Zero value means that default value will be selected. -- ALGLIB -- Copyright 01.02.2017 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetv2its(rbfmodel s, ae_int_t maxits);
rbfsetv2supportr function/************************************************************************* This function sets support radius parameter of hierarchical (version 2) RBF constructor. Hierarchical RBF model achieves great speed-up by removing from the model excessive (too dense) nodes. Say, if you have RBF radius equal to 1 meter, and two nodes are just 1 millimeter apart, you may remove one of them without reducing model quality. Support radius parameter is used to justify which points need removal, and which do not. If two points are less than SUPPORT_R*CUR_RADIUS units of distance apart, one of them is removed from the model. The larger support radius is, the faster model construction AND evaluation are. However, too large values result in "bumpy" models. INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call R - support radius coefficient, >=0. Recommended values are [0.1,0.4] range, with 0.1 being default value. -- ALGLIB -- Copyright 01.02.2017 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetv2supportr(rbfmodel s, double r);
rbfsetzeroterm function/************************************************************************* This function sets zero term (model is a sum of radial basis functions without polynomial term). This function won't have effect until next call to RBFBuildModel(). INPUT PARAMETERS: S - RBF model, initialized by RBFCreate() call NOTE: this function has some serialization-related subtleties. We recommend you to study serialization examples from ALGLIB Reference Manual if you want to perform serialization of your models. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfsetzeroterm(rbfmodel s);
Examples: [1]
rbftscalcbuf function/************************************************************************* This function calculates values of the RBF model at the given point, using external buffer object (internal temporaries of RBF model are not modified). This function allows to use same RBF model object in different threads, assuming that different threads use different instances of buffer structure. INPUT PARAMETERS: S - RBF model, may be shared between different threads Buf - buffer object created for this particular instance of RBF model with rbfcreatecalcbuffer(). X - coordinates, array[NX]. X may have more than NX elements, in this case only leading NX will be used. Y - possibly preallocated array OUTPUT PARAMETERS: Y - function value, array[NY]. Y is not reallocated when it is larger than NY. -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbftscalcbuf( rbfmodel s, rbfcalcbuffer buf, real_1d_array x, real_1d_array& y);
rbfunpack function/************************************************************************* This function "unpacks" RBF model by extracting its coefficients. INPUT PARAMETERS: S - RBF model OUTPUT PARAMETERS: NX - dimensionality of argument NY - dimensionality of the target function XWR - model information, array[NC,NX+NY+1]. One row of the array corresponds to one basis function: * first NX columns - coordinates of the center * next NY columns - weights, one per dimension of the function being modelled For ModelVersion=1: * last column - radius, same for all dimensions of the function being modelled For ModelVersion=2: * last NX columns - radii, one per dimension NC - number of the centers V - polynomial term , array[NY,NX+1]. One row per one dimension of the function being modelled. First NX elements are linear coefficients, V[NX] is equal to the constant part. ModelVersion-version of the RBF model: * 1 - for models created by QNN and RBF-ML algorithms, compatible with ALGLIB 3.10 or earlier. * 2 - for models created by HierarchicalRBF, requires ALGLIB 3.11 or later -- ALGLIB -- Copyright 13.12.2011 by Bochkanov Sergey *************************************************************************/void alglib::rbfunpack( rbfmodel s, ae_int_t& nx, ae_int_t& ny, real_2d_array& xwr, ae_int_t& nc, real_2d_array& v, ae_int_t& modelversion);
Examples: [1]
rbfunserialize function/************************************************************************* This function unserializes data structure from string. *************************************************************************/void rbfunserialize(const std::string &s_in, rbfmodel &obj); void rbfunserialize(const std::istream &s_in, rbfmodel &obj);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example illustrates basic concepts of the RBF models: creation, modification,
// evaluation.
//
// Suppose that we have set of 2-dimensional points with associated
// scalar function values, and we want to build a RBF model using
// our data.
//
// NOTE: we can work with 3D models too :)
//
// Typical sequence of steps is given below:
// 1. we create RBF model object
// 2. we attach our dataset to the RBF model and tune algorithm settings
// 3. we rebuild RBF model using QNN algorithm on new data
// 4. we use RBF model (evaluate, serialize, etc.)
//
double v;
//
// Step 1: RBF model creation.
//
// We have to specify dimensionality of the space (2 or 3) and
// dimensionality of the function (scalar or vector).
//
// New model is empty - it can be evaluated,
// but we just get zero value at any point.
//
rbfmodel model;
rbfcreate(2, 1, model);
v = rbfcalc2(model, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 0.000
//
// Step 2: we add dataset.
//
// XY contains two points - x0=(-1,0) and x1=(+1,0) -
// and two function values f(x0)=2, f(x1)=3.
//
// We added points, but model was not rebuild yet.
// If we call rbfcalc2(), we still will get 0.0 as result.
//
real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
rbfsetpoints(model, xy);
v = rbfcalc2(model, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 0.000
//
// Step 3: rebuild model
//
// After we've configured model, we should rebuild it -
// it will change coefficients stored internally in the
// rbfmodel structure.
//
// We use hierarchical RBF algorithm with following parameters:
// * RBase - set to 1.0
// * NLayers - three layers are used (although such simple problem
// does not need more than 1 layer)
// * LambdaReg - is set to zero value, no smoothing is required
//
rbfreport rep;
rbfsetalgohierarchical(model, 1.0, 3, 0.0);
rbfbuildmodel(model, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
//
// Step 4: model was built
//
// After call of rbfbuildmodel(), rbfcalc2() will return
// value of the new model.
//
v = rbfcalc2(model, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 2.500
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example show how to work with polynomial term
//
// Suppose that we have set of 2-dimensional points with associated
// scalar function values, and we want to build a RBF model using
// our data.
//
// We use hierarchical RBF algorithm with following parameters:
// * RBase - set to 1.0
// * NLayers - three layers are used (although such simple problem
// does not need more than 1 layer)
// * LambdaReg - is set to zero value, no smoothing is required
//
double v;
rbfmodel model;
real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
rbfreport rep;
rbfcreate(2, 1, model);
rbfsetpoints(model, xy);
rbfsetalgohierarchical(model, 1.0, 3, 0.0);
//
// By default, RBF model uses linear term. It means that model
// looks like
// f(x,y) = SUM(RBF[i]) + a*x + b*y + c
// where RBF[i] is I-th radial basis function and a*x+by+c is a
// linear term. Having linear terms in a model gives us:
// (1) improved extrapolation properties
// (2) linearity of the model when data can be perfectly fitted
// by the linear function
// (3) linear asymptotic behavior
//
// Our simple dataset can be modelled by the linear function
// f(x,y) = 0.5*x + 2.5
// and rbfbuildmodel() with default settings should preserve this
// linearity.
//
ae_int_t nx;
ae_int_t ny;
ae_int_t nc;
ae_int_t modelversion;
real_2d_array xwr;
real_2d_array c;
rbfbuildmodel(model, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.500,0.000,2.500]]
// asymptotic behavior of our function is linear
v = rbfcalc2(model, 1000.0, 0.0);
printf("%.1f\n", double(v)); // EXPECTED: 502.50
//
// Instead of linear term we can use constant term. In this case
// we will get model which has form
// f(x,y) = SUM(RBF[i]) + c
// where RBF[i] is I-th radial basis function and c is a constant,
// which is equal to the average function value on the dataset.
//
// Because we've already attached dataset to the model the only
// thing we have to do is to call rbfsetconstterm() and then
// rebuild model with rbfbuildmodel().
//
rbfsetconstterm(model);
rbfbuildmodel(model, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.000,0.000,2.500]]
// asymptotic behavior of our function is constant
v = rbfcalc2(model, 1000.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 2.500
//
// Finally, we can use zero term. Just plain RBF without polynomial
// part:
// f(x,y) = SUM(RBF[i])
// where RBF[i] is I-th radial basis function.
//
rbfsetzeroterm(model);
rbfbuildmodel(model, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
rbfunpack(model, nx, ny, xwr, nc, c, modelversion);
printf("%s\n", c.tostring(2).c_str()); // EXPECTED: [[0.000,0.000,0.000]]
// asymptotic behavior of our function is just zero constant
v = rbfcalc2(model, 1000.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 0.000
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example show how to serialize and unserialize RBF model
//
// Suppose that we have set of 2-dimensional points with associated
// scalar function values, and we want to build a RBF model using
// our data. Then we want to serialize it to string and to unserialize
// from string, loading to another instance of RBF model.
//
// Here we assume that you already know how to create RBF models.
//
std::string s;
double v;
rbfmodel model0;
rbfmodel model1;
real_2d_array xy = "[[-1,0,2],[+1,0,3]]";
rbfreport rep;
// model initialization
rbfcreate(2, 1, model0);
rbfsetpoints(model0, xy);
rbfsetalgohierarchical(model0, 1.0, 3, 0.0);
rbfbuildmodel(model0, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
//
// Serialization - it looks easy,
// but you should carefully read next section.
//
alglib::rbfserialize(model0, s);
alglib::rbfunserialize(s, model1);
// both models return same value
v = rbfcalc2(model0, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 2.500
v = rbfcalc2(model1, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 2.500
//
// Previous section shows that model state is saved/restored during
// serialization. However, some properties are NOT serialized.
//
// Serialization saves/restores RBF model, but it does NOT saves/restores
// settings which were used to build current model. In particular, dataset
// which was used to build model, is not preserved.
//
// What does it mean in for us?
//
// Do you remember this sequence: rbfcreate-rbfsetpoints-rbfbuildmodel?
// First step creates model, second step adds dataset and tunes model
// settings, third step builds model using current dataset and model
// construction settings.
//
// If you call rbfbuildmodel() without calling rbfsetpoints() first, you
// will get empty (zero) RBF model. In our example, model0 contains
// dataset which was added by rbfsetpoints() call. However, model1 does
// NOT contain dataset - because dataset is NOT serialized.
//
// This, if we call rbfbuildmodel(model0,rep), we will get same model,
// which returns 2.5 at (x,y)=(0,0). However, after same call model1 will
// return zero - because it contains RBF model (coefficients), but does NOT
// contain dataset which was used to build this model.
//
// Basically, it means that:
// * serialization of the RBF model preserves anything related to the model
// EVALUATION
// * but it does NOT creates perfect copy of the original object.
//
rbfbuildmodel(model0, rep);
v = rbfcalc2(model0, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 2.500
rbfbuildmodel(model1, rep);
v = rbfcalc2(model1, 0.0, 0.0);
printf("%.2f\n", double(v)); // EXPECTED: 0.000
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Suppose that we have set of 2-dimensional points with associated VECTOR
// function values, and we want to build a RBF model using our data.
//
// Typical sequence of steps is given below:
// 1. we create RBF model object
// 2. we attach our dataset to the RBF model and tune algorithm settings
// 3. we rebuild RBF model using new data
// 4. we use RBF model (evaluate, serialize, etc.)
//
real_1d_array x;
real_1d_array y;
//
// Step 1: RBF model creation.
//
// We have to specify dimensionality of the space (equal to 2) and
// dimensionality of the function (2-dimensional vector function).
//
// New model is empty - it can be evaluated,
// but we just get zero value at any point.
//
rbfmodel model;
rbfcreate(2, 2, model);
x = "[+1,+1]";
rbfcalc(model, x, y);
printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,0.000]
//
// Step 2: we add dataset.
//
// XY arrays containt four points:
// * (x0,y0) = (+1,+1), f(x0,y0)=(0,-1)
// * (x1,y1) = (+1,-1), f(x1,y1)=(-1,0)
// * (x2,y2) = (-1,-1), f(x2,y2)=(0,+1)
// * (x3,y3) = (-1,+1), f(x3,y3)=(+1,0)
//
real_2d_array xy = "[[+1,+1,0,-1],[+1,-1,-1,0],[-1,-1,0,+1],[-1,+1,+1,0]]";
rbfsetpoints(model, xy);
// We added points, but model was not rebuild yet.
// If we call rbfcalc(), we still will get 0.0 as result.
rbfcalc(model, x, y);
printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,0.000]
//
// Step 3: rebuild model
//
// We use hierarchical RBF algorithm with following parameters:
// * RBase - set to 1.0
// * NLayers - three layers are used (although such simple problem
// does not need more than 1 layer)
// * LambdaReg - is set to zero value, no smoothing is required
//
// After we've configured model, we should rebuild it -
// it will change coefficients stored internally in the
// rbfmodel structure.
//
rbfreport rep;
rbfsetalgohierarchical(model, 1.0, 3, 0.0);
rbfbuildmodel(model, rep);
printf("%d\n", int(rep.terminationtype)); // EXPECTED: 1
//
// Step 4: model was built
//
// After call of rbfbuildmodel(), rbfcalc() will return
// value of the new model.
//
rbfcalc(model, x, y);
printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [0.000,-1.000]
return 0;
}
rcond subpackagecmatrixlurcond1 function/************************************************************************* Estimate of the condition number of a matrix given by its LU decomposition (1-norm) The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: LUA - LU decomposition of a matrix in compact form. Output of the CMatrixLU subroutine. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::cmatrixlurcond1(complex_2d_array lua, ae_int_t n);
cmatrixlurcondinf function/************************************************************************* Estimate of the condition number of a matrix given by its LU decomposition (infinity norm). The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: LUA - LU decomposition of a matrix in compact form. Output of the CMatrixLU subroutine. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::cmatrixlurcondinf(complex_2d_array lua, ae_int_t n);
cmatrixrcond1 function/************************************************************************* Estimate of a matrix condition number (1-norm) The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::cmatrixrcond1(complex_2d_array a, ae_int_t n);
cmatrixrcondinf function/************************************************************************* Estimate of a matrix condition number (infinity-norm). The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::cmatrixrcondinf(complex_2d_array a, ae_int_t n);
cmatrixtrrcond1 function/************************************************************************* Triangular matrix: estimate of a condition number (1-norm) The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array[0..N-1, 0..N-1]. N - size of A. IsUpper - True, if the matrix is upper triangular. IsUnit - True, if the matrix has a unit diagonal. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::cmatrixtrrcond1( complex_2d_array a, ae_int_t n, bool isupper, bool isunit);
cmatrixtrrcondinf function/************************************************************************* Triangular matrix: estimate of a matrix condition number (infinity-norm). The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - True, if the matrix is upper triangular. IsUnit - True, if the matrix has a unit diagonal. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::cmatrixtrrcondinf( complex_2d_array a, ae_int_t n, bool isupper, bool isunit);
hpdmatrixcholeskyrcond function/************************************************************************* Condition number estimate of a Hermitian positive definite matrix given by Cholesky decomposition. The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). It should be noted that 1-norm and inf-norm condition numbers of symmetric matrices are equal, so the algorithm doesn't take into account the differences between these types of norms. Input parameters: CD - Cholesky decomposition of matrix A, output of SMatrixCholesky subroutine. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::hpdmatrixcholeskyrcond( complex_2d_array a, ae_int_t n, bool isupper);
hpdmatrixrcond function/************************************************************************* Condition number estimate of a Hermitian positive definite matrix. The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). It should be noted that 1-norm and inf-norm of condition numbers of symmetric matrices are equal, so the algorithm doesn't take into account the differences between these types of norms. Input parameters: A - Hermitian positive definite matrix which is given by its upper or lower triangle depending on the value of IsUpper. Array with elements [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - storage format. Result: 1/LowerBound(cond(A)), if matrix A is positive definite, -1, if matrix A is not positive definite, and its condition number could not be found by this algorithm. NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::hpdmatrixrcond( complex_2d_array a, ae_int_t n, bool isupper);
rmatrixlurcond1 function/************************************************************************* Estimate of the condition number of a matrix given by its LU decomposition (1-norm) The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: LUA - LU decomposition of a matrix in compact form. Output of the RMatrixLU subroutine. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::rmatrixlurcond1(real_2d_array lua, ae_int_t n);
rmatrixlurcondinf function/************************************************************************* Estimate of the condition number of a matrix given by its LU decomposition (infinity norm). The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: LUA - LU decomposition of a matrix in compact form. Output of the RMatrixLU subroutine. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::rmatrixlurcondinf(real_2d_array lua, ae_int_t n);
rmatrixrcond1 function/************************************************************************* Estimate of a matrix condition number (1-norm) The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::rmatrixrcond1(real_2d_array a, ae_int_t n);
rmatrixrcondinf function/************************************************************************* Estimate of a matrix condition number (infinity-norm). The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::rmatrixrcondinf(real_2d_array a, ae_int_t n);
rmatrixtrrcond1 function/************************************************************************* Triangular matrix: estimate of a condition number (1-norm) The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array[0..N-1, 0..N-1]. N - size of A. IsUpper - True, if the matrix is upper triangular. IsUnit - True, if the matrix has a unit diagonal. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::rmatrixtrrcond1( real_2d_array a, ae_int_t n, bool isupper, bool isunit);
rmatrixtrrcondinf function/************************************************************************* Triangular matrix: estimate of a matrix condition number (infinity-norm). The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). Input parameters: A - matrix. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - True, if the matrix is upper triangular. IsUnit - True, if the matrix has a unit diagonal. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::rmatrixtrrcondinf( real_2d_array a, ae_int_t n, bool isupper, bool isunit);
spdmatrixcholeskyrcond function/************************************************************************* Condition number estimate of a symmetric positive definite matrix given by Cholesky decomposition. The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). It should be noted that 1-norm and inf-norm condition numbers of symmetric matrices are equal, so the algorithm doesn't take into account the differences between these types of norms. Input parameters: CD - Cholesky decomposition of matrix A, output of SMatrixCholesky subroutine. N - size of matrix A. Result: 1/LowerBound(cond(A)) NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::spdmatrixcholeskyrcond( real_2d_array a, ae_int_t n, bool isupper);
spdmatrixrcond function/************************************************************************* Condition number estimate of a symmetric positive definite matrix. The algorithm calculates a lower bound of the condition number. In this case, the algorithm does not return a lower bound of the condition number, but an inverse number (to avoid an overflow in case of a singular matrix). It should be noted that 1-norm and inf-norm of condition numbers of symmetric matrices are equal, so the algorithm doesn't take into account the differences between these types of norms. Input parameters: A - symmetric positive definite matrix which is given by its upper or lower triangle depending on the value of IsUpper. Array with elements [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - storage format. Result: 1/LowerBound(cond(A)), if matrix A is positive definite, -1, if matrix A is not positive definite, and its condition number could not be found by this algorithm. NOTE: if k(A) is very large, then matrix is assumed degenerate, k(A)=INF, 0.0 is returned in such cases. *************************************************************************/double alglib::spdmatrixrcond(real_2d_array a, ae_int_t n, bool isupper);
schur subpackagermatrixschur function/************************************************************************* Subroutine performing the Schur decomposition of a general matrix by using the QR algorithm with multiple shifts. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Multithreaded acceleration is NOT supported for this function. ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. The source matrix A is represented as S'*A*S = T, where S is an orthogonal matrix (Schur vectors), T - upper quasi-triangular matrix (with blocks of sizes 1x1 and 2x2 on the main diagonal). Input parameters: A - matrix to be decomposed. Array whose indexes range within [0..N-1, 0..N-1]. N - size of A, N>=0. Output parameters: A - contains matrix T. Array whose indexes range within [0..N-1, 0..N-1]. S - contains Schur vectors. Array whose indexes range within [0..N-1, 0..N-1]. Note 1: The block structure of matrix T can be easily recognized: since all the elements below the blocks are zeros, the elements a[i+1,i] which are equal to 0 show the block border. Note 2: The algorithm performance depends on the value of the internal parameter NS of the InternalSchurDecomposition subroutine which defines the number of shifts in the QR algorithm (similarly to the block width in block-matrix algorithms in linear algebra). If you require maximum performance on your machine, it is recommended to adjust this parameter manually. Result: True, if the algorithm has converged and parameters A and S contain the result. False, if the algorithm has not converged. Algorithm implemented on the basis of the DHSEQR subroutine (LAPACK 3.0 library). *************************************************************************/bool alglib::rmatrixschur(real_2d_array& a, ae_int_t n, real_2d_array& s);
sparse subpackage| sparse_d_1 | Basic operations with sparse matrices | |
| sparse_d_crs | Advanced topic: creation in the CRS format. |
sparsebuffers class/************************************************************************* Temporary buffers for sparse matrix operations. You should pass an instance of this structure to factorization functions. It allows to reuse memory during repeated sparse factorizations. You do not have to call some initialization function - simply passing an instance to factorization function is enough. *************************************************************************/class sparsebuffers { };
sparsematrix class/************************************************************************* Sparse matrix structure. You should use ALGLIB functions to work with sparse matrix. Never try to access its fields directly! NOTES ON THE SPARSE STORAGE FORMATS Sparse matrices can be stored using several formats: * Hash-Table representation * Compressed Row Storage (CRS) * Skyline matrix storage (SKS) Each of the formats has benefits and drawbacks: * Hash-table is good for dynamic operations (insertion of new elements), but does not support linear algebra operations * CRS is good for operations like matrix-vector or matrix-matrix products, but its initialization is less convenient - you have to tell row sizes at the initialization, and you have to fill matrix only row by row, from left to right. * SKS is a special format which is used to store triangular factors from Cholesky factorization. It does not support dynamic modification, and support for linear algebra operations is very limited. Tables below outline information about these two formats: OPERATIONS WITH MATRIX HASH CRS SKS creation + + + SparseGet + + + SparseRewriteExisting + + + SparseSet + SparseAdd + SparseGetRow + + SparseGetCompressedRow + + sparse-dense linear algebra + + *************************************************************************/class sparsematrix { };
sparseadd function/************************************************************************* This function adds value to S[i,j] - element of the sparse matrix. Matrix must be in a Hash-Table mode. In case S[i,j] already exists in the table, V i added to its value. In case S[i,j] is non-existent, it is inserted in the table. Table automatically grows when necessary. INPUT PARAMETERS S - sparse M*N matrix in Hash-Table representation. Exception will be thrown for CRS matrix. I - row index of the element to modify, 0<=I<M J - column index of the element to modify, 0<=J<N V - value to add, must be finite number OUTPUT PARAMETERS S - modified matrix NOTE 1: when S[i,j] is exactly zero after modification, it is deleted from the table. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparseadd(sparsematrix s, ae_int_t i, ae_int_t j, double v);
Examples: [1]
sparseconvertto function/************************************************************************* This function performs in-place conversion to desired sparse storage format. INPUT PARAMETERS S0 - sparse matrix in any format. Fmt - desired storage format of the output, as returned by SparseGetMatrixType() function: * 0 for hash-based storage * 1 for CRS * 2 for SKS OUTPUT PARAMETERS S0 - sparse matrix in requested format. NOTE: in-place conversion wastes a lot of memory which is used to store temporaries. If you perform a lot of repeated conversions, we recommend to use out-of-place buffered conversion functions, like SparseCopyToBuf(), which can reuse already allocated memory. -- ALGLIB PROJECT -- Copyright 16.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparseconvertto(sparsematrix s0, ae_int_t fmt);
sparseconverttocrs function/************************************************************************* This function converts matrix to CRS format. Some algorithms (linear algebra ones, for example) require matrices in CRS format. This function allows to perform in-place conversion. INPUT PARAMETERS S - sparse M*N matrix in any format OUTPUT PARAMETERS S - matrix in CRS format NOTE: this function has no effect when called with matrix which is already in CRS mode. NOTE: this function allocates temporary memory to store a copy of the matrix. If you perform a lot of repeated conversions, we recommend you to use SparseCopyToCRSBuf() function, which can reuse previously allocated memory. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparseconverttocrs(sparsematrix s);
Examples: [1]
sparseconverttohash function/************************************************************************* This function performs in-place conversion to Hash table storage. INPUT PARAMETERS S - sparse matrix in CRS format. OUTPUT PARAMETERS S - sparse matrix in Hash table format. NOTE: this function has no effect when called with matrix which is already in Hash table mode. NOTE: in-place conversion involves allocation of temporary arrays. If you perform a lot of repeated in- place conversions, it may lead to memory fragmentation. Consider using out-of-place SparseCopyToHashBuf() function in this case. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparseconverttohash(sparsematrix s);
sparseconverttosks function/************************************************************************* This function performs in-place conversion to SKS format. INPUT PARAMETERS S - sparse matrix in any format. OUTPUT PARAMETERS S - sparse matrix in SKS format. NOTE: this function has no effect when called with matrix which is already in SKS mode. NOTE: in-place conversion involves allocation of temporary arrays. If you perform a lot of repeated in- place conversions, it may lead to memory fragmentation. Consider using out-of-place SparseCopyToSKSBuf() function in this case. -- ALGLIB PROJECT -- Copyright 15.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparseconverttosks(sparsematrix s);
sparsecopy function/************************************************************************* This function copies S0 to S1. This function completely deallocates memory owned by S1 before creating a copy of S0. If you want to reuse memory, use SparseCopyBuf. NOTE: this function does not verify its arguments, it just copies all fields of the structure. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopy(sparsematrix s0, sparsematrix& s1);
sparsecopybuf function/************************************************************************* This function copies S0 to S1. Memory already allocated in S1 is reused as much as possible. NOTE: this function does not verify its arguments, it just copies all fields of the structure. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopybuf(sparsematrix s0, sparsematrix s1);
sparsecopytobuf function/************************************************************************* This function performs out-of-place conversion to desired sparse storage format. S0 is copied to S1 and converted on-the-fly. Memory allocated in S1 is reused to maximum extent possible. INPUT PARAMETERS S0 - sparse matrix in any format. Fmt - desired storage format of the output, as returned by SparseGetMatrixType() function: * 0 for hash-based storage * 1 for CRS * 2 for SKS OUTPUT PARAMETERS S1 - sparse matrix in requested format. -- ALGLIB PROJECT -- Copyright 16.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytobuf( sparsematrix s0, ae_int_t fmt, sparsematrix s1);
sparsecopytocrs function/************************************************************************* This function performs out-of-place conversion to CRS format. S0 is copied to S1 and converted on-the-fly. INPUT PARAMETERS S0 - sparse matrix in any format. OUTPUT PARAMETERS S1 - sparse matrix in CRS format. NOTE: if S0 is stored as CRS, it is just copied without conversion. NOTE: this function de-allocates memory occupied by S1 before starting CRS conversion. If you perform a lot of repeated CRS conversions, it may lead to memory fragmentation. In this case we recommend you to use SparseCopyToCRSBuf() function which re-uses memory in S1 as much as possible. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytocrs(sparsematrix s0, sparsematrix& s1);
sparsecopytocrsbuf function/************************************************************************* This function performs out-of-place conversion to CRS format. S0 is copied to S1 and converted on-the-fly. Memory allocated in S1 is reused to maximum extent possible. INPUT PARAMETERS S0 - sparse matrix in any format. S1 - matrix which may contain some pre-allocated memory, or can be just uninitialized structure. OUTPUT PARAMETERS S1 - sparse matrix in CRS format. NOTE: if S0 is stored as CRS, it is just copied without conversion. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytocrsbuf(sparsematrix s0, sparsematrix s1);
sparsecopytohash function/************************************************************************* This function performs out-of-place conversion to Hash table storage format. S0 is copied to S1 and converted on-the-fly. INPUT PARAMETERS S0 - sparse matrix in any format. OUTPUT PARAMETERS S1 - sparse matrix in Hash table format. NOTE: if S0 is stored as Hash-table, it is just copied without conversion. NOTE: this function de-allocates memory occupied by S1 before starting conversion. If you perform a lot of repeated conversions, it may lead to memory fragmentation. In this case we recommend you to use SparseCopyToHashBuf() function which re-uses memory in S1 as much as possible. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytohash(sparsematrix s0, sparsematrix& s1);
sparsecopytohashbuf function/************************************************************************* This function performs out-of-place conversion to Hash table storage format. S0 is copied to S1 and converted on-the-fly. Memory allocated in S1 is reused to maximum extent possible. INPUT PARAMETERS S0 - sparse matrix in any format. OUTPUT PARAMETERS S1 - sparse matrix in Hash table format. NOTE: if S0 is stored as Hash-table, it is just copied without conversion. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytohashbuf(sparsematrix s0, sparsematrix s1);
sparsecopytosks function/************************************************************************* This function performs out-of-place conversion to SKS storage format. S0 is copied to S1 and converted on-the-fly. INPUT PARAMETERS S0 - sparse matrix in any format. OUTPUT PARAMETERS S1 - sparse matrix in SKS format. NOTE: if S0 is stored as SKS, it is just copied without conversion. NOTE: this function de-allocates memory occupied by S1 before starting conversion. If you perform a lot of repeated conversions, it may lead to memory fragmentation. In this case we recommend you to use SparseCopyToSKSBuf() function which re-uses memory in S1 as much as possible. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytosks(sparsematrix s0, sparsematrix& s1);
sparsecopytosksbuf function/************************************************************************* This function performs out-of-place conversion to SKS format. S0 is copied to S1 and converted on-the-fly. Memory allocated in S1 is reused to maximum extent possible. INPUT PARAMETERS S0 - sparse matrix in any format. OUTPUT PARAMETERS S1 - sparse matrix in SKS format. NOTE: if S0 is stored as SKS, it is just copied without conversion. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsecopytosksbuf(sparsematrix s0, sparsematrix s1);
sparsecreate function/************************************************************************* This function creates sparse matrix in a Hash-Table format. This function creates Hast-Table matrix, which can be converted to CRS format after its initialization is over. Typical usage scenario for a sparse matrix is: 1. creation in a Hash-Table format 2. insertion of the matrix elements 3. conversion to the CRS representation 4. matrix is passed to some linear algebra algorithm Some information about different matrix formats can be found below, in the "NOTES" section. INPUT PARAMETERS M - number of rows in a matrix, M>=1 N - number of columns in a matrix, N>=1 K - K>=0, expected number of non-zero elements in a matrix. K can be inexact approximation, can be less than actual number of elements (table will grow when needed) or even zero). It is important to understand that although hash-table may grow automatically, it is better to provide good estimate of data size. OUTPUT PARAMETERS S - sparse M*N matrix in Hash-Table representation. All elements of the matrix are zero. NOTE 1 Hash-tables use memory inefficiently, and they have to keep some amount of the "spare memory" in order to have good performance. Hash table for matrix with K non-zero elements will need C*K*(8+2*sizeof(int)) bytes, where C is a small constant, about 1.5-2 in magnitude. CRS storage, from the other side, is more memory-efficient, and needs just K*(8+sizeof(int))+M*sizeof(int) bytes, where M is a number of rows in a matrix. When you convert from the Hash-Table to CRS representation, all unneeded memory will be freed. NOTE 2 Comments of SparseMatrix structure outline information about different sparse storage formats. We recommend you to read them before starting to use ALGLIB sparse matrices. NOTE 3 This function completely overwrites S with new sparse matrix. Previously allocated storage is NOT reused. If you want to reuse already allocated memory, call SparseCreateBuf function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsecreate(ae_int_t m, ae_int_t n, sparsematrix& s); void alglib::sparsecreate( ae_int_t m, ae_int_t n, ae_int_t k, sparsematrix& s);
Examples: [1]
sparsecreatebuf function/************************************************************************* This version of SparseCreate function creates sparse matrix in Hash-Table format, reusing previously allocated storage as much as possible. Read comments for SparseCreate() for more information. INPUT PARAMETERS M - number of rows in a matrix, M>=1 N - number of columns in a matrix, N>=1 K - K>=0, expected number of non-zero elements in a matrix. K can be inexact approximation, can be less than actual number of elements (table will grow when needed) or even zero). It is important to understand that although hash-table may grow automatically, it is better to provide good estimate of data size. S - SparseMatrix structure which MAY contain some already allocated storage. OUTPUT PARAMETERS S - sparse M*N matrix in Hash-Table representation. All elements of the matrix are zero. Previously allocated storage is reused, if its size is compatible with expected number of non-zeros K. -- ALGLIB PROJECT -- Copyright 14.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsecreatebuf(ae_int_t m, ae_int_t n, sparsematrix s); void alglib::sparsecreatebuf( ae_int_t m, ae_int_t n, ae_int_t k, sparsematrix s);
sparsecreatecrs function/************************************************************************* This function creates sparse matrix in a CRS format (expert function for situations when you are running out of memory). This function creates CRS matrix. Typical usage scenario for a CRS matrix is: 1. creation (you have to tell number of non-zero elements at each row at this moment) 2. insertion of the matrix elements (row by row, from left to right) 3. matrix is passed to some linear algebra algorithm This function is a memory-efficient alternative to SparseCreate(), but it is more complex because it requires you to know in advance how large your matrix is. Some information about different matrix formats can be found in comments on SparseMatrix structure. We recommend you to read them before starting to use ALGLIB sparse matrices.. INPUT PARAMETERS M - number of rows in a matrix, M>=1 N - number of columns in a matrix, N>=1 NER - number of elements at each row, array[M], NER[I]>=0 OUTPUT PARAMETERS S - sparse M*N matrix in CRS representation. You have to fill ALL non-zero elements by calling SparseSet() BEFORE you try to use this matrix. NOTE: this function completely overwrites S with new sparse matrix. Previously allocated storage is NOT reused. If you want to reuse already allocated memory, call SparseCreateCRSBuf function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsecreatecrs( ae_int_t m, ae_int_t n, integer_1d_array ner, sparsematrix& s);
Examples: [1]
sparsecreatecrsbuf function/************************************************************************* This function creates sparse matrix in a CRS format (expert function for situations when you are running out of memory). This version of CRS matrix creation function may reuse memory already allocated in S. This function creates CRS matrix. Typical usage scenario for a CRS matrix is: 1. creation (you have to tell number of non-zero elements at each row at this moment) 2. insertion of the matrix elements (row by row, from left to right) 3. matrix is passed to some linear algebra algorithm This function is a memory-efficient alternative to SparseCreate(), but it is more complex because it requires you to know in advance how large your matrix is. Some information about different matrix formats can be found in comments on SparseMatrix structure. We recommend you to read them before starting to use ALGLIB sparse matrices.. INPUT PARAMETERS M - number of rows in a matrix, M>=1 N - number of columns in a matrix, N>=1 NER - number of elements at each row, array[M], NER[I]>=0 S - sparse matrix structure with possibly preallocated memory. OUTPUT PARAMETERS S - sparse M*N matrix in CRS representation. You have to fill ALL non-zero elements by calling SparseSet() BEFORE you try to use this matrix. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsecreatecrsbuf( ae_int_t m, ae_int_t n, integer_1d_array ner, sparsematrix s);
sparsecreatesks function/************************************************************************* This function creates sparse matrix in a SKS format (skyline storage format). In most cases you do not need this function - CRS format better suits most use cases. INPUT PARAMETERS M, N - number of rows(M) and columns (N) in a matrix: * M=N (as for now, ALGLIB supports only square SKS) * N>=1 * M>=1 D - "bottom" bandwidths, array[M], D[I]>=0. I-th element stores number of non-zeros at I-th row, below the diagonal (diagonal itself is not included) U - "top" bandwidths, array[N], U[I]>=0. I-th element stores number of non-zeros at I-th row, above the diagonal (diagonal itself is not included) OUTPUT PARAMETERS S - sparse M*N matrix in SKS representation. All elements are filled by zeros. You may use SparseRewriteExisting() to change their values. NOTE: this function completely overwrites S with new sparse matrix. Previously allocated storage is NOT reused. If you want to reuse already allocated memory, call SparseCreateSKSBuf function. -- ALGLIB PROJECT -- Copyright 13.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsecreatesks( ae_int_t m, ae_int_t n, integer_1d_array d, integer_1d_array u, sparsematrix& s);
sparsecreatesksbuf function/************************************************************************* This is "buffered" version of SparseCreateSKS() which reuses memory previously allocated in S (of course, memory is reallocated if needed). This function creates sparse matrix in a SKS format (skyline storage format). In most cases you do not need this function - CRS format better suits most use cases. INPUT PARAMETERS M, N - number of rows(M) and columns (N) in a matrix: * M=N (as for now, ALGLIB supports only square SKS) * N>=1 * M>=1 D - "bottom" bandwidths, array[M], 0<=D[I]<=I. I-th element stores number of non-zeros at I-th row, below the diagonal (diagonal itself is not included) U - "top" bandwidths, array[N], 0<=U[I]<=I. I-th element stores number of non-zeros at I-th row, above the diagonal (diagonal itself is not included) OUTPUT PARAMETERS S - sparse M*N matrix in SKS representation. All elements are filled by zeros. You may use SparseSet()/SparseAdd() to change their values. -- ALGLIB PROJECT -- Copyright 13.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsecreatesksbuf( ae_int_t m, ae_int_t n, integer_1d_array d, integer_1d_array u, sparsematrix s);
sparseenumerate function/************************************************************************* This function is used to enumerate all elements of the sparse matrix. Before first call user initializes T0 and T1 counters by zero. These counters are used to remember current position in a matrix; after each call they are updated by the function. Subsequent calls to this function return non-zero elements of the sparse matrix, one by one. If you enumerate CRS matrix, matrix is traversed from left to right, from top to bottom. In case you enumerate matrix stored as Hash table, elements are returned in random order. EXAMPLE > T0=0 > T1=0 > while SparseEnumerate(S,T0,T1,I,J,V) do > ....do something with I,J,V INPUT PARAMETERS S - sparse M*N matrix in Hash-Table or CRS representation. T0 - internal counter T1 - internal counter OUTPUT PARAMETERS T0 - new value of the internal counter T1 - new value of the internal counter I - row index of non-zero element, 0<=I<M. J - column index of non-zero element, 0<=J<N V - value of the T-th element RESULT True in case of success (next non-zero element was retrieved) False in case all non-zero elements were enumerated NOTE: you may call SparseRewriteExisting() during enumeration, but it is THE ONLY matrix modification function you can call!!! Other matrix modification functions should not be called during enumeration! -- ALGLIB PROJECT -- Copyright 14.03.2012 by Bochkanov Sergey *************************************************************************/bool alglib::sparseenumerate( sparsematrix s, ae_int_t& t0, ae_int_t& t1, ae_int_t& i, ae_int_t& j, double& v);
sparsefree function/************************************************************************* The function frees all memory occupied by sparse matrix. Sparse matrix structure becomes unusable after this call. OUTPUT PARAMETERS S - sparse matrix to delete -- ALGLIB PROJECT -- Copyright 24.07.2012 by Bochkanov Sergey *************************************************************************/void alglib::sparsefree(sparsematrix& s);
sparseget function/************************************************************************* This function returns S[i,j] - element of the sparse matrix. Matrix can be in any mode (Hash-Table, CRS, SKS), but this function is less efficient for CRS matrices. Hash-Table and SKS matrices can find element in O(1) time, while CRS matrices need O(log(RS)) time, where RS is an number of non-zero elements in a row. INPUT PARAMETERS S - sparse M*N matrix in Hash-Table representation. Exception will be thrown for CRS matrix. I - row index of the element to modify, 0<=I<M J - column index of the element to modify, 0<=J<N RESULT value of S[I,J] or zero (in case no element with such index is found) -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/double alglib::sparseget(sparsematrix s, ae_int_t i, ae_int_t j);
sparsegetcompressedrow function/************************************************************************* This function returns I-th row of the sparse matrix IN COMPRESSED FORMAT - only non-zero elements are returned (with their indexes). Matrix must be stored in CRS or SKS format. INPUT PARAMETERS: S - sparse M*N matrix in CRS format I - row index, 0<=I<M ColIdx - output buffer for column indexes, can be preallocated. In case buffer size is too small to store I-th row, it is automatically reallocated. Vals - output buffer for values, can be preallocated. In case buffer size is too small to store I-th row, it is automatically reallocated. OUTPUT PARAMETERS: ColIdx - column indexes of non-zero elements, sorted by ascending. Symbolically non-zero elements are counted (i.e. if you allocated place for element, but it has zero numerical value - it is counted). Vals - values. Vals[K] stores value of matrix element with indexes (I,ColIdx[K]). Symbolically non-zero elements are counted (i.e. if you allocated place for element, but it has zero numerical value - it is counted). NZCnt - number of symbolically non-zero elements per row. NOTE: when incorrect I (outside of [0,M-1]) or matrix (non CRS/SKS) is passed, this function throws exception. NOTE: this function may allocate additional, unnecessary place for ColIdx and Vals arrays. It is dictated by performance reasons - on SKS matrices it is faster to allocate space at the beginning with some "extra"-space, than performing two passes over matrix - first time to calculate exact space required for data, second time - to store data itself. -- ALGLIB PROJECT -- Copyright 10.12.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsegetcompressedrow( sparsematrix s, ae_int_t i, integer_1d_array& colidx, real_1d_array& vals, ae_int_t& nzcnt);
sparsegetdiagonal function/************************************************************************* This function returns I-th diagonal element of the sparse matrix. Matrix can be in any mode (Hash-Table or CRS storage), but this function is most efficient for CRS matrices - it requires less than 50 CPU cycles to extract diagonal element. For Hash-Table matrices we still have O(1) query time, but function is many times slower. INPUT PARAMETERS S - sparse M*N matrix in Hash-Table representation. Exception will be thrown for CRS matrix. I - index of the element to modify, 0<=I<min(M,N) RESULT value of S[I,I] or zero (in case no element with such index is found) -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/double alglib::sparsegetdiagonal(sparsematrix s, ae_int_t i);
sparsegetlowercount function/************************************************************************* The function returns number of strictly lower triangular non-zero elements in the matrix. It counts SYMBOLICALLY non-zero elements, i.e. entries in the sparse matrix data structure. If some element has zero numerical value, it is still counted. This function has different cost for different types of matrices: * for hash-based matrices it involves complete pass over entire hash-table with O(NNZ) cost, where NNZ is number of non-zero elements * for CRS and SKS matrix types cost of counting is O(N) (N - matrix size). RESULT: number of non-zero elements strictly below main diagonal -- ALGLIB PROJECT -- Copyright 12.02.2014 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::sparsegetlowercount(sparsematrix s);
sparsegetmatrixtype function/************************************************************************* This function returns type of the matrix storage format. INPUT PARAMETERS: S - sparse matrix. RESULT: sparse storage format used by matrix: 0 - Hash-table 1 - CRS (compressed row storage) 2 - SKS (skyline) NOTE: future versions of ALGLIB may include additional sparse storage formats. -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::sparsegetmatrixtype(sparsematrix s);
sparsegetncols function/************************************************************************* The function returns number of columns of a sparse matrix. RESULT: number of columns of a sparse matrix. -- ALGLIB PROJECT -- Copyright 23.08.2012 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::sparsegetncols(sparsematrix s);
sparsegetnrows function/************************************************************************* The function returns number of rows of a sparse matrix. RESULT: number of rows of a sparse matrix. -- ALGLIB PROJECT -- Copyright 23.08.2012 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::sparsegetnrows(sparsematrix s);
sparsegetrow function/************************************************************************* This function returns I-th row of the sparse matrix. Matrix must be stored in CRS or SKS format. INPUT PARAMETERS: S - sparse M*N matrix in CRS format I - row index, 0<=I<M IRow - output buffer, can be preallocated. In case buffer size is too small to store I-th row, it is automatically reallocated. OUTPUT PARAMETERS: IRow - array[M], I-th row. NOTE: this function has O(N) running time, where N is a column count. It allocates and fills N-element array, even although most of its elemets are zero. NOTE: If you have O(non-zeros-per-row) time and memory requirements, use SparseGetCompressedRow() function. It returns data in compressed format. NOTE: when incorrect I (outside of [0,M-1]) or matrix (non CRS/SKS) is passed, this function throws exception. -- ALGLIB PROJECT -- Copyright 10.12.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsegetrow( sparsematrix s, ae_int_t i, real_1d_array& irow);
sparsegetuppercount function/************************************************************************* The function returns number of strictly upper triangular non-zero elements in the matrix. It counts SYMBOLICALLY non-zero elements, i.e. entries in the sparse matrix data structure. If some element has zero numerical value, it is still counted. This function has different cost for different types of matrices: * for hash-based matrices it involves complete pass over entire hash-table with O(NNZ) cost, where NNZ is number of non-zero elements * for CRS and SKS matrix types cost of counting is O(N) (N - matrix size). RESULT: number of non-zero elements strictly above main diagonal -- ALGLIB PROJECT -- Copyright 12.02.2014 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::sparsegetuppercount(sparsematrix s);
sparseiscrs function/************************************************************************* This function checks matrix storage format and returns True when matrix is stored using CRS representation. INPUT PARAMETERS: S - sparse matrix. RESULT: True if matrix type is CRS False if matrix type is not CRS -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/bool alglib::sparseiscrs(sparsematrix s);
sparseishash function/************************************************************************* This function checks matrix storage format and returns True when matrix is stored using Hash table representation. INPUT PARAMETERS: S - sparse matrix. RESULT: True if matrix type is Hash table False if matrix type is not Hash table -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/bool alglib::sparseishash(sparsematrix s);
sparseissks function/************************************************************************* This function checks matrix storage format and returns True when matrix is stored using SKS representation. INPUT PARAMETERS: S - sparse matrix. RESULT: True if matrix type is SKS False if matrix type is not SKS -- ALGLIB PROJECT -- Copyright 20.07.2012 by Bochkanov Sergey *************************************************************************/bool alglib::sparseissks(sparsematrix s);
sparsemm function/************************************************************************* This function calculates matrix-matrix product S*A. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*N matrix in CRS or SKS format. A - array[N][K], input dense matrix. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. K - number of columns of matrix (A). B - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS B - array[M][K], S*A NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsemm( sparsematrix s, real_2d_array a, ae_int_t k, real_2d_array& b);
sparsemm2 function/************************************************************************* This function simultaneously calculates two matrix-matrix products: S*A and S^T*A. S must be square (non-rectangular) matrix stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse N*N matrix in CRS or SKS format. A - array[N][K], input dense matrix. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. K - number of columns of matrix (A). B0 - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. B1 - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS B0 - array[N][K], S*A B1 - array[N][K], S^T*A NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsemm2( sparsematrix s, real_2d_array a, ae_int_t k, real_2d_array& b0, real_2d_array& b1);
sparsemtm function/************************************************************************* This function calculates matrix-matrix product S^T*A. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*N matrix in CRS or SKS format. A - array[M][K], input dense matrix. For performance reasons we make only quick checks - we check that array size is at least M, but we do not check for NAN's or INF's. K - number of columns of matrix (A). B - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS B - array[N][K], S^T*A NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsemtm( sparsematrix s, real_2d_array a, ae_int_t k, real_2d_array& b);
sparsemtv function/************************************************************************* This function calculates matrix-vector product S^T*x. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*N matrix in CRS or SKS format. X - array[M], input vector. For performance reasons we make only quick checks - we check that array size is at least M, but we do not check for NAN's or INF's. Y - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS Y - array[N], S^T*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsemtv(sparsematrix s, real_1d_array x, real_1d_array& y);
sparsemv function/************************************************************************* This function calculates matrix-vector product S*x. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*N matrix in CRS or SKS format. X - array[N], input vector. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. Y - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS Y - array[M], S*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsemv(sparsematrix s, real_1d_array x, real_1d_array& y);
sparsemv2 function/************************************************************************* This function simultaneously calculates two matrix-vector products: S*x and S^T*x. S must be square (non-rectangular) matrix stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse N*N matrix in CRS or SKS format. X - array[N], input vector. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. Y0 - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. Y1 - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS Y0 - array[N], S*x Y1 - array[N], S^T*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsemv2( sparsematrix s, real_1d_array x, real_1d_array& y0, real_1d_array& y1);
sparseresizematrix function/************************************************************************* This procedure resizes Hash-Table matrix. It can be called when you have deleted too many elements from the matrix, and you want to free unneeded memory. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparseresizematrix(sparsematrix s);
sparserewriteexisting function/************************************************************************* This function rewrites existing (non-zero) element. It returns True if element exists or False, when it is called for non-existing (zero) element. This function works with any kind of the matrix. The purpose of this function is to provide convenient thread-safe way to modify sparse matrix. Such modification (already existing element is rewritten) is guaranteed to be thread-safe without any synchronization, as long as different threads modify different elements. INPUT PARAMETERS S - sparse M*N matrix in any kind of representation (Hash, SKS, CRS). I - row index of non-zero element to modify, 0<=I<M J - column index of non-zero element to modify, 0<=J<N V - value to rewrite, must be finite number OUTPUT PARAMETERS S - modified matrix RESULT True in case when element exists False in case when element doesn't exist or it is zero -- ALGLIB PROJECT -- Copyright 14.03.2012 by Bochkanov Sergey *************************************************************************/bool alglib::sparserewriteexisting( sparsematrix s, ae_int_t i, ae_int_t j, double v);
sparseset function/************************************************************************* This function modifies S[i,j] - element of the sparse matrix. For Hash-based storage format: * this function can be called at any moment - during matrix initialization or later * new value can be zero or non-zero. In case new value of S[i,j] is zero, this element is deleted from the table. * this function has no effect when called with zero V for non-existent element. For CRS-bases storage format: * this function can be called ONLY DURING MATRIX INITIALIZATION * new value MUST be non-zero. Exception will be thrown for zero V. * elements must be initialized in correct order - from top row to bottom, within row - from left to right. For SKS storage: NOT SUPPORTED! Use SparseRewriteExisting() to work with SKS matrices. INPUT PARAMETERS S - sparse M*N matrix in Hash-Table or CRS representation. I - row index of the element to modify, 0<=I<M J - column index of the element to modify, 0<=J<N V - value to set, must be finite number, can be zero OUTPUT PARAMETERS S - modified matrix -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparseset(sparsematrix s, ae_int_t i, ae_int_t j, double v);
sparsesmm function/************************************************************************* This function calculates matrix-matrix product S*A, when S is symmetric matrix. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*M matrix in CRS or SKS format. IsUpper - whether upper or lower triangle of S is given: * if upper triangle is given, only S[i,j] for j>=i are used, and lower triangle is ignored (it can be empty - these elements are not referenced at all). * if lower triangle is given, only S[i,j] for j<=i are used, and upper triangle is ignored. A - array[N][K], input dense matrix. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. K - number of columns of matrix (A). B - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS B - array[M][K], S*A NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsesmm( sparsematrix s, bool isupper, real_2d_array a, ae_int_t k, real_2d_array& b);
sparsesmv function/************************************************************************* This function calculates matrix-vector product S*x, when S is symmetric matrix. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*M matrix in CRS or SKS format. IsUpper - whether upper or lower triangle of S is given: * if upper triangle is given, only S[i,j] for j>=i are used, and lower triangle is ignored (it can be empty - these elements are not referenced at all). * if lower triangle is given, only S[i,j] for j<=i are used, and upper triangle is ignored. X - array[N], input vector. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. Y - output buffer, possibly preallocated. In case buffer size is too small to store result, this buffer is automatically resized. OUTPUT PARAMETERS Y - array[M], S*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 14.10.2011 by Bochkanov Sergey *************************************************************************/void alglib::sparsesmv( sparsematrix s, bool isupper, real_1d_array x, real_1d_array& y);
sparseswap function/************************************************************************* This function efficiently swaps contents of S0 and S1. -- ALGLIB PROJECT -- Copyright 16.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparseswap(sparsematrix s0, sparsematrix s1);
sparsetransposesks function/************************************************************************* This function performs efficient in-place transpose of SKS matrix. No additional memory is allocated during transposition. This function supports only skyline storage format (SKS). INPUT PARAMETERS S - sparse matrix in SKS format. OUTPUT PARAMETERS S - sparse matrix, transposed. -- ALGLIB PROJECT -- Copyright 16.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsetransposesks(sparsematrix s);
sparsetrmv function/************************************************************************* This function calculates matrix-vector product op(S)*x, when x is vector, S is symmetric triangular matrix, op(S) is transposition or no operation. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse square matrix in CRS or SKS format. IsUpper - whether upper or lower triangle of S is used: * if upper triangle is given, only S[i,j] for j>=i are used, and lower triangle is ignored (it can be empty - these elements are not referenced at all). * if lower triangle is given, only S[i,j] for j<=i are used, and upper triangle is ignored. IsUnit - unit or non-unit diagonal: * if True, diagonal elements of triangular matrix are considered equal to 1.0. Actual elements stored in S are not referenced at all. * if False, diagonal stored in S is used OpType - operation type: * if 0, S*x is calculated * if 1, (S^T)*x is calculated (transposition) X - array[N] which stores input vector. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. Y - possibly preallocated input buffer. Automatically resized if its size is too small. OUTPUT PARAMETERS Y - array[N], op(S)*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 20.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsetrmv( sparsematrix s, bool isupper, bool isunit, ae_int_t optype, real_1d_array& x, real_1d_array& y);
sparsetrsv function/************************************************************************* This function solves linear system op(S)*y=x where x is vector, S is symmetric triangular matrix, op(S) is transposition or no operation. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse square matrix in CRS or SKS format. IsUpper - whether upper or lower triangle of S is used: * if upper triangle is given, only S[i,j] for j>=i are used, and lower triangle is ignored (it can be empty - these elements are not referenced at all). * if lower triangle is given, only S[i,j] for j<=i are used, and upper triangle is ignored. IsUnit - unit or non-unit diagonal: * if True, diagonal elements of triangular matrix are considered equal to 1.0. Actual elements stored in S are not referenced at all. * if False, diagonal stored in S is used. It is your responsibility to make sure that diagonal is non-zero. OpType - operation type: * if 0, S*x is calculated * if 1, (S^T)*x is calculated (transposition) X - array[N] which stores input vector. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. OUTPUT PARAMETERS X - array[N], inv(op(S))*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. NOTE: no assertion or tests are done during algorithm operation. It is your responsibility to provide invertible matrix to algorithm. -- ALGLIB PROJECT -- Copyright 20.01.2014 by Bochkanov Sergey *************************************************************************/void alglib::sparsetrsv( sparsematrix s, bool isupper, bool isunit, ae_int_t optype, real_1d_array& x);
sparsevsmv function/************************************************************************* This function calculates vector-matrix-vector product x'*S*x, where S is symmetric matrix. Matrix S must be stored in CRS or SKS format (exception will be thrown otherwise). INPUT PARAMETERS S - sparse M*M matrix in CRS or SKS format. IsUpper - whether upper or lower triangle of S is given: * if upper triangle is given, only S[i,j] for j>=i are used, and lower triangle is ignored (it can be empty - these elements are not referenced at all). * if lower triangle is given, only S[i,j] for j<=i are used, and upper triangle is ignored. X - array[N], input vector. For performance reasons we make only quick checks - we check that array size is at least N, but we do not check for NAN's or INF's. RESULT x'*S*x NOTE: this function throws exception when called for non-CRS/SKS matrix. You must convert your matrix with SparseConvertToCRS/SKS() before using this function. -- ALGLIB PROJECT -- Copyright 27.01.2014 by Bochkanov Sergey *************************************************************************/double alglib::sparsevsmv(sparsematrix s, bool isupper, real_1d_array x);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates creation/initialization of the sparse matrix
// and matrix-vector multiplication.
//
// First, we have to create matrix and initialize it. Matrix is initially created
// in the Hash-Table format, which allows convenient initialization. We can modify
// Hash-Table matrix with sparseset() and sparseadd() functions.
//
// NOTE: Unlike CRS format, Hash-Table representation allows you to initialize
// elements in the arbitrary order. You may see that we initialize a[0][0] first,
// then move to the second row, and then move back to the first row.
//
sparsematrix s;
sparsecreate(2, 2, s);
sparseset(s, 0, 0, 2.0);
sparseset(s, 1, 1, 1.0);
sparseset(s, 0, 1, 1.0);
sparseadd(s, 1, 1, 4.0);
//
// Now S is equal to
// [ 2 1 ]
// [ 5 ]
// Lets check it by reading matrix contents with sparseget().
// You may see that with sparseget() you may read both non-zero
// and zero elements.
//
double v;
v = sparseget(s, 0, 0);
printf("%.2f\n", double(v)); // EXPECTED: 2.0000
v = sparseget(s, 0, 1);
printf("%.2f\n", double(v)); // EXPECTED: 1.0000
v = sparseget(s, 1, 0);
printf("%.2f\n", double(v)); // EXPECTED: 0.0000
v = sparseget(s, 1, 1);
printf("%.2f\n", double(v)); // EXPECTED: 5.0000
//
// After successful creation we can use our matrix for linear operations.
//
// However, there is one more thing we MUST do before using S in linear
// operations: we have to convert it from HashTable representation (used for
// initialization and dynamic operations) to CRS format with sparseconverttocrs()
// call. If you omit this call, ALGLIB will generate exception on the first
// attempt to use S in linear operations.
//
sparseconverttocrs(s);
//
// Now S is in the CRS format and we are ready to do linear operations.
// Lets calculate A*x for some x.
//
real_1d_array x = "[1,-1]";
real_1d_array y = "[]";
sparsemv(s, x, y);
printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [1.000,-5.000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "linalg.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// This example demonstrates creation/initialization of the sparse matrix in the
// CRS format.
//
// Hash-Table format used by default is very convenient (it allows easy
// insertion of elements, automatic memory reallocation), but has
// significant memory and performance overhead. Insertion of one element
// costs hundreds of CPU cycles, and memory consumption is several times
// higher than that of CRS.
//
// When you work with really large matrices and when you can tell in
// advance how many elements EXACTLY you need, it can be beneficial to
// create matrix in the CRS format from the very beginning.
//
// If you want to create matrix in the CRS format, you should:
// * use sparsecreatecrs() function
// * know row sizes in advance (number of non-zero entries in the each row)
// * initialize matrix with sparseset() - another function, sparseadd(), is not allowed
// * initialize elements from left to right, from top to bottom, each
// element is initialized only once.
//
sparsematrix s;
integer_1d_array row_sizes = "[2,2,2,1]";
sparsecreatecrs(4, 4, row_sizes, s);
sparseset(s, 0, 0, 2.0);
sparseset(s, 0, 1, 1.0);
sparseset(s, 1, 1, 4.0);
sparseset(s, 1, 2, 2.0);
sparseset(s, 2, 2, 3.0);
sparseset(s, 2, 3, 1.0);
sparseset(s, 3, 3, 9.0);
//
// Now S is equal to
// [ 2 1 ]
// [ 4 2 ]
// [ 3 1 ]
// [ 9 ]
//
// We should point that we have initialized S elements from left to right,
// from top to bottom. CRS representation does NOT allow you to do so in
// the different order. Try to change order of the sparseset() calls above,
// and you will see that your program generates exception.
//
// We can check it by reading matrix contents with sparseget().
// However, you should remember that sparseget() is inefficient on
// CRS matrices (it may have to pass through all elements of the row
// until it finds element you need).
//
double v;
v = sparseget(s, 0, 0);
printf("%.2f\n", double(v)); // EXPECTED: 2.0000
v = sparseget(s, 2, 3);
printf("%.2f\n", double(v)); // EXPECTED: 1.0000
// you may see that you can read zero elements (which are not stored) with sparseget()
v = sparseget(s, 3, 2);
printf("%.2f\n", double(v)); // EXPECTED: 0.0000
//
// After successful creation we can use our matrix for linear operations.
// Lets calculate A*x for some x.
//
real_1d_array x = "[1,-1,1,-1]";
real_1d_array y = "[]";
sparsemv(s, x, y);
printf("%s\n", y.tostring(2).c_str()); // EXPECTED: [1.000,-2.000,2.000,-9]
return 0;
}
spdgevd subpackagesmatrixgevd function/************************************************************************* Algorithm for solving the following generalized symmetric positive-definite eigenproblem: A*x = lambda*B*x (1) or A*B*x = lambda*x (2) or B*A*x = lambda*x (3). where A is a symmetric matrix, B - symmetric positive-definite matrix. The problem is solved by reducing it to an ordinary symmetric eigenvalue problem. Input parameters: A - symmetric matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrices A and B. IsUpperA - storage format of matrix A. B - symmetric positive-definite matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. IsUpperB - storage format of matrix B. ZNeeded - if ZNeeded is equal to: * 0, the eigenvectors are not returned; * 1, the eigenvectors are returned. ProblemType - if ProblemType is equal to: * 1, the following problem is solved: A*x = lambda*B*x; * 2, the following problem is solved: A*B*x = lambda*x; * 3, the following problem is solved: B*A*x = lambda*x. Output parameters: D - eigenvalues in ascending order. Array whose index ranges within [0..N-1]. Z - if ZNeeded is equal to: * 0, Z hasn't changed; * 1, Z contains eigenvectors. Array whose indexes range within [0..N-1, 0..N-1]. The eigenvectors are stored in matrix columns. It should be noted that the eigenvectors in such problems do not form an orthogonal system. Result: True, if the problem was solved successfully. False, if the error occurred during the Cholesky decomposition of matrix B (the matrix isn't positive-definite) or during the work of the iterative algorithm for solving the symmetric eigenproblem. See also the GeneralizedSymmetricDefiniteEVDReduce subroutine. -- ALGLIB -- Copyright 1.28.2006 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixgevd( real_2d_array a, ae_int_t n, bool isuppera, real_2d_array b, bool isupperb, ae_int_t zneeded, ae_int_t problemtype, real_1d_array& d, real_2d_array& z);
smatrixgevdreduce function/************************************************************************* Algorithm for reduction of the following generalized symmetric positive- definite eigenvalue problem: A*x = lambda*B*x (1) or A*B*x = lambda*x (2) or B*A*x = lambda*x (3) to the symmetric eigenvalues problem C*y = lambda*y (eigenvalues of this and the given problems are the same, and the eigenvectors of the given problem could be obtained by multiplying the obtained eigenvectors by the transformation matrix x = R*y). Here A is a symmetric matrix, B - symmetric positive-definite matrix. Input parameters: A - symmetric matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. N - size of matrices A and B. IsUpperA - storage format of matrix A. B - symmetric positive-definite matrix which is given by its upper or lower triangular part. Array whose indexes range within [0..N-1, 0..N-1]. IsUpperB - storage format of matrix B. ProblemType - if ProblemType is equal to: * 1, the following problem is solved: A*x = lambda*B*x; * 2, the following problem is solved: A*B*x = lambda*x; * 3, the following problem is solved: B*A*x = lambda*x. Output parameters: A - symmetric matrix which is given by its upper or lower triangle depending on IsUpperA. Contains matrix C. Array whose indexes range within [0..N-1, 0..N-1]. R - upper triangular or low triangular transformation matrix which is used to obtain the eigenvectors of a given problem as the product of eigenvectors of C (from the right) and matrix R (from the left). If the matrix is upper triangular, the elements below the main diagonal are equal to 0 (and vice versa). Thus, we can perform the multiplication without taking into account the internal structure (which is an easier though less effective way). Array whose indexes range within [0..N-1, 0..N-1]. IsUpperR - type of matrix R (upper or lower triangular). Result: True, if the problem was reduced successfully. False, if the error occurred during the Cholesky decomposition of matrix B (the matrix is not positive-definite). -- ALGLIB -- Copyright 1.28.2006 by Bochkanov Sergey *************************************************************************/bool alglib::smatrixgevdreduce( real_2d_array& a, ae_int_t n, bool isuppera, real_2d_array b, bool isupperb, ae_int_t problemtype, real_2d_array& r, bool& isupperr);
spline1d subpackage| spline1d_d_convdiff | Resampling using cubic splines | |
| spline1d_d_cubic | Cubic spline interpolation | |
| spline1d_d_griddiff | Differentiation on the grid using cubic splines | |
| spline1d_d_linear | Piecewise linear spline interpolation | |
| spline1d_d_monotone | Monotone interpolation |
spline1dinterpolant class/************************************************************************* 1-dimensional spline interpolant *************************************************************************/class spline1dinterpolant { };
spline1dbuildakima function/************************************************************************* This subroutine builds Akima spline interpolant INPUT PARAMETERS: X - spline nodes, array[0..N-1] Y - function values, array[0..N-1] N - points count (optional): * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dbuildakima( real_1d_array x, real_1d_array y, spline1dinterpolant& c); void alglib::spline1dbuildakima( real_1d_array x, real_1d_array y, ae_int_t n, spline1dinterpolant& c);
spline1dbuildcatmullrom function/************************************************************************* This subroutine builds Catmull-Rom spline interpolant. INPUT PARAMETERS: X - spline nodes, array[0..N-1]. Y - function values, array[0..N-1]. OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundType - boundary condition type: * -1 for periodic boundary condition * 0 for parabolically terminated spline (default) Tension - tension parameter: * tension=0 corresponds to classic Catmull-Rom spline (default) * 0<tension<1 corresponds to more general form - cardinal spline OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dbuildcatmullrom( real_1d_array x, real_1d_array y, spline1dinterpolant& c); void alglib::spline1dbuildcatmullrom( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundtype, double tension, spline1dinterpolant& c);
spline1dbuildcubic function/************************************************************************* This subroutine builds cubic spline interpolant. INPUT PARAMETERS: X - spline nodes, array[0..N-1]. Y - function values, array[0..N-1]. OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dbuildcubic( real_1d_array x, real_1d_array y, spline1dinterpolant& c); void alglib::spline1dbuildcubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundltype, double boundl, ae_int_t boundrtype, double boundr, spline1dinterpolant& c);
spline1dbuildhermite function/************************************************************************* This subroutine builds Hermite spline interpolant. INPUT PARAMETERS: X - spline nodes, array[0..N-1] Y - function values, array[0..N-1] D - derivatives, array[0..N-1] N - points count (optional): * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant. ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dbuildhermite( real_1d_array x, real_1d_array y, real_1d_array d, spline1dinterpolant& c); void alglib::spline1dbuildhermite( real_1d_array x, real_1d_array y, real_1d_array d, ae_int_t n, spline1dinterpolant& c);
spline1dbuildlinear function/************************************************************************* This subroutine builds linear spline interpolant INPUT PARAMETERS: X - spline nodes, array[0..N-1] Y - function values, array[0..N-1] N - points count (optional): * N>=2 * if given, only first N points are used to build spline * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) OUTPUT PARAMETERS: C - spline interpolant ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dbuildlinear( real_1d_array x, real_1d_array y, spline1dinterpolant& c); void alglib::spline1dbuildlinear( real_1d_array x, real_1d_array y, ae_int_t n, spline1dinterpolant& c);
spline1dbuildmonotone function/************************************************************************* This function builds monotone cubic Hermite interpolant. This interpolant is monotonic in [x(0),x(n-1)] and is constant outside of this interval. In case y[] form non-monotonic sequence, interpolant is piecewise monotonic. Say, for x=(0,1,2,3,4) and y=(0,1,2,1,0) interpolant will monotonically grow at [0..2] and monotonically decrease at [2..4]. INPUT PARAMETERS: X - spline nodes, array[0..N-1]. Subroutine automatically sorts points, so caller may pass unsorted array. Y - function values, array[0..N-1] N - the number of points(N>=2). OUTPUT PARAMETERS: C - spline interpolant. -- ALGLIB PROJECT -- Copyright 21.06.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline1dbuildmonotone( real_1d_array x, real_1d_array y, spline1dinterpolant& c); void alglib::spline1dbuildmonotone( real_1d_array x, real_1d_array y, ae_int_t n, spline1dinterpolant& c);
Examples: [1]
spline1dcalc function/************************************************************************* This subroutine calculates the value of the spline at the given point X. INPUT PARAMETERS: C - spline interpolant X - point Result: S(x) -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/double alglib::spline1dcalc(spline1dinterpolant c, double x);
spline1dconvcubic function/************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::spline1dconvcubic( real_1d_array x, real_1d_array y, real_1d_array x2, real_1d_array& y2); void alglib::spline1dconvcubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundltype, double boundl, ae_int_t boundrtype, double boundr, real_1d_array x2, ae_int_t n2, real_1d_array& y2);
Examples: [1]
spline1dconvdiff2cubic function/************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[], first and second derivatives d2[] and dd2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] D2 - first derivatives at X2[] DD2 - second derivatives at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::spline1dconvdiff2cubic( real_1d_array x, real_1d_array y, real_1d_array x2, real_1d_array& y2, real_1d_array& d2, real_1d_array& dd2); void alglib::spline1dconvdiff2cubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundltype, double boundl, ae_int_t boundrtype, double boundr, real_1d_array x2, ae_int_t n2, real_1d_array& y2, real_1d_array& d2, real_1d_array& dd2);
Examples: [1]
spline1dconvdiffcubic function/************************************************************************* This function solves following problem: given table y[] of function values at old nodes x[] and new nodes x2[], it calculates and returns table of function values y2[] and derivatives d2[] (calculated at x2[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - old spline nodes Y - function values X2 - new spline nodes OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points from X/Y are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) N2 - new points count: * N2>=2 * if given, only first N2 points from X2 are used * if not given, automatically detected from X2 size OUTPUT PARAMETERS: F2 - function values at X2[] D2 - first derivatives at X2[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Function values are correctly reordered on return, so F2[I] is always equal to S(X2[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::spline1dconvdiffcubic( real_1d_array x, real_1d_array y, real_1d_array x2, real_1d_array& y2, real_1d_array& d2); void alglib::spline1dconvdiffcubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundltype, double boundl, ae_int_t boundrtype, double boundr, real_1d_array x2, ae_int_t n2, real_1d_array& y2, real_1d_array& d2);
Examples: [1]
spline1ddiff function/************************************************************************* This subroutine differentiates the spline. INPUT PARAMETERS: C - spline interpolant. X - point Result: S - S(x) DS - S'(x) D2S - S''(x) -- ALGLIB PROJECT -- Copyright 24.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1ddiff( spline1dinterpolant c, double x, double& s, double& ds, double& d2s);
spline1dgriddiff2cubic function/************************************************************************* This function solves following problem: given table y[] of function values at nodes x[], it calculates and returns tables of first and second function derivatives d1[] and d2[] (calculated at the same nodes x[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - spline nodes Y - function values OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: D1 - S' values at X[] D2 - S'' values at X[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Derivative values are correctly reordered on return, so D[I] is always equal to S'(X[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::spline1dgriddiff2cubic( real_1d_array x, real_1d_array y, real_1d_array& d1, real_1d_array& d2); void alglib::spline1dgriddiff2cubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundltype, double boundl, ae_int_t boundrtype, double boundr, real_1d_array& d1, real_1d_array& d2);
Examples: [1]
spline1dgriddiffcubic function/************************************************************************* This function solves following problem: given table y[] of function values at nodes x[], it calculates and returns table of function derivatives d[] (calculated at the same nodes x[]). This function yields same result as Spline1DBuildCubic() call followed by sequence of Spline1DDiff() calls, but it can be several times faster when called for ordered X[] and X2[]. INPUT PARAMETERS: X - spline nodes Y - function values OPTIONAL PARAMETERS: N - points count: * N>=2 * if given, only first N points are used * if not given, automatically detected from X/Y sizes (len(X) must be equal to len(Y)) BoundLType - boundary condition type for the left boundary BoundL - left boundary condition (first or second derivative, depending on the BoundLType) BoundRType - boundary condition type for the right boundary BoundR - right boundary condition (first or second derivative, depending on the BoundRType) OUTPUT PARAMETERS: D - derivative values at X[] ORDER OF POINTS Subroutine automatically sorts points, so caller may pass unsorted array. Derivative values are correctly reordered on return, so D[I] is always equal to S'(X[I]) independently of points order. SETTING BOUNDARY VALUES: The BoundLType/BoundRType parameters can have the following values: * -1, which corresonds to the periodic (cyclic) boundary conditions. In this case: * both BoundLType and BoundRType must be equal to -1. * BoundL/BoundR are ignored * Y[last] is ignored (it is assumed to be equal to Y[first]). * 0, which corresponds to the parabolically terminated spline (BoundL and/or BoundR are ignored). * 1, which corresponds to the first derivative boundary condition * 2, which corresponds to the second derivative boundary condition * by default, BoundType=0 is used PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS: Problems with periodic boundary conditions have Y[first_point]=Y[last_point]. However, this subroutine doesn't require you to specify equal values for the first and last points - it automatically forces them to be equal by copying Y[first_point] (corresponds to the leftmost, minimal X[]) to Y[last_point]. However it is recommended to pass consistent values of Y[], i.e. to make Y[first_point]=Y[last_point]. -- ALGLIB PROJECT -- Copyright 03.09.2010 by Bochkanov Sergey *************************************************************************/void alglib::spline1dgriddiffcubic( real_1d_array x, real_1d_array y, real_1d_array& d); void alglib::spline1dgriddiffcubic( real_1d_array x, real_1d_array y, ae_int_t n, ae_int_t boundltype, double boundl, ae_int_t boundrtype, double boundr, real_1d_array& d);
Examples: [1]
spline1dintegrate function/************************************************************************* This subroutine integrates the spline. INPUT PARAMETERS: C - spline interpolant. X - right bound of the integration interval [a, x], here 'a' denotes min(x[]) Result: integral(S(t)dt,a,x) -- ALGLIB PROJECT -- Copyright 23.06.2007 by Bochkanov Sergey *************************************************************************/double alglib::spline1dintegrate(spline1dinterpolant c, double x);
spline1dlintransx function/************************************************************************* This subroutine performs linear transformation of the spline argument. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: x = A*t + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dlintransx(spline1dinterpolant c, double a, double b);
spline1dlintransy function/************************************************************************* This subroutine performs linear transformation of the spline. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: S2(x) = A*S(x) + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dlintransy(spline1dinterpolant c, double a, double b);
spline1dunpack function/************************************************************************* This subroutine unpacks the spline into the coefficients table. INPUT PARAMETERS: C - spline interpolant. X - point OUTPUT PARAMETERS: Tbl - coefficients table, unpacked format, array[0..N-2, 0..5]. For I = 0...N-2: Tbl[I,0] = X[i] Tbl[I,1] = X[i+1] Tbl[I,2] = C0 Tbl[I,3] = C1 Tbl[I,4] = C2 Tbl[I,5] = C3 On [x[i], x[i+1]] spline is equals to: S(x) = C0 + C1*t + C2*t^2 + C3*t^3 t = x-x[i] NOTE: You can rebuild spline with Spline1DBuildHermite() function, which accepts as inputs function values and derivatives at nodes, which are easy to calculate when you have coefficients. -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline1dunpack( spline1dinterpolant c, ae_int_t& n, real_2d_array& tbl);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use cubic spline to do resampling, i.e. having
// values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
// we calculate values/derivatives of cubic spline on
// another grid (equidistant with 9 nodes on [-1,+1])
// WITHOUT CONSTRUCTION OF SPLINE OBJECT.
//
// There are efficient functions spline1dconvcubic(),
// spline1dconvdiffcubic() and spline1dconvdiff2cubic()
// for such calculations.
//
// We use default boundary conditions ("parabolically terminated
// spline") because cubic spline built with such boundary conditions
// will exactly reproduce any quadratic f(x).
//
// Actually, we could use natural conditions, but we feel that
// spline which exactly reproduces f() will show us more
// understandable results.
//
real_1d_array x_old = "[-1.0,-0.5,0.0,+0.5,+1.0]";
real_1d_array y_old = "[+1.0,0.25,0.0,0.25,+1.0]";
real_1d_array x_new = "[-1.00,-0.75,-0.50,-0.25,0.00,+0.25,+0.50,+0.75,+1.00]";
real_1d_array y_new;
real_1d_array d1_new;
real_1d_array d2_new;
//
// First, conversion without differentiation.
//
//
spline1dconvcubic(x_old, y_old, x_new, y_new);
printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
//
// Then, conversion with differentiation (first derivatives only)
//
//
spline1dconvdiffcubic(x_old, y_old, x_new, y_new, d1_new);
printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
printf("%s\n", d1_new.tostring(3).c_str()); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
//
// Finally, conversion with first and second derivatives
//
//
spline1dconvdiff2cubic(x_old, y_old, x_new, y_new, d1_new, d2_new);
printf("%s\n", y_new.tostring(3).c_str()); // EXPECTED: [1.0000, 0.5625, 0.2500, 0.0625, 0.0000, 0.0625, 0.2500, 0.5625, 1.0000]
printf("%s\n", d1_new.tostring(3).c_str()); // EXPECTED: [-2.0, -1.5, -1.0, -0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
printf("%s\n", d2_new.tostring(3).c_str()); // EXPECTED: [2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0, 2.0]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use cubic spline to interpolate f(x)=x^2 sampled
// at 5 equidistant nodes on [-1,+1].
//
// First, we use default boundary conditions ("parabolically terminated
// spline") because cubic spline built with such boundary conditions
// will exactly reproduce any quadratic f(x).
//
// Then we try to use natural boundary conditions
// d2S(-1)/dx^2 = 0.0
// d2S(+1)/dx^2 = 0.0
// and see that such spline interpolated f(x) with small error.
//
real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
double t = 0.25;
double v;
spline1dinterpolant s;
ae_int_t natural_bound_type = 2;
//
// Test exact boundary conditions: build S(x), calculare S(0.25)
// (almost same as original function)
//
spline1dbuildcubic(x, y, s);
v = spline1dcalc(s, t);
printf("%.4f\n", double(v)); // EXPECTED: 0.0625
//
// Test natural boundary conditions: build S(x), calculare S(0.25)
// (small interpolation error)
//
spline1dbuildcubic(x, y, 5, natural_bound_type, 0.0, natural_bound_type, 0.0, s);
v = spline1dcalc(s, t);
printf("%.3f\n", double(v)); // EXPECTED: 0.0580
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use cubic spline to do grid differentiation, i.e. having
// values of f(x)=x^2 sampled at 5 equidistant nodes on [-1,+1]
// we calculate derivatives of cubic spline at nodes WITHOUT
// CONSTRUCTION OF SPLINE OBJECT.
//
// There are efficient functions spline1dgriddiffcubic() and
// spline1dgriddiff2cubic() for such calculations.
//
// We use default boundary conditions ("parabolically terminated
// spline") because cubic spline built with such boundary conditions
// will exactly reproduce any quadratic f(x).
//
// Actually, we could use natural conditions, but we feel that
// spline which exactly reproduces f() will show us more
// understandable results.
//
real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
real_1d_array d1;
real_1d_array d2;
//
// We calculate first derivatives: they must be equal to 2*x
//
spline1dgriddiffcubic(x, y, d1);
printf("%s\n", d1.tostring(3).c_str()); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]
//
// Now test griddiff2, which returns first AND second derivatives.
// First derivative is 2*x, second is equal to 2.0
//
spline1dgriddiff2cubic(x, y, d1, d2);
printf("%s\n", d1.tostring(3).c_str()); // EXPECTED: [-2.0, -1.0, 0.0, +1.0, +2.0]
printf("%s\n", d2.tostring(3).c_str()); // EXPECTED: [ 2.0, 2.0, 2.0, 2.0, 2.0]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use piecewise linear spline to interpolate f(x)=x^2 sampled
// at 5 equidistant nodes on [-1,+1].
//
real_1d_array x = "[-1.0,-0.5,0.0,+0.5,+1.0]";
real_1d_array y = "[+1.0,0.25,0.0,0.25,+1.0]";
double t = 0.25;
double v;
spline1dinterpolant s;
// build spline
spline1dbuildlinear(x, y, s);
// calculate S(0.25) - it is quite different from 0.25^2=0.0625
v = spline1dcalc(s, t);
printf("%.4f\n", double(v)); // EXPECTED: 0.125
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// Spline built witn spline1dbuildcubic() can be non-monotone even when
// Y-values form monotone sequence. Say, for x=[0,1,2] and y=[0,1,1]
// cubic spline will monotonically grow until x=1.5 and then start
// decreasing.
//
// That's why ALGLIB provides special spline construction function
// which builds spline which preserves monotonicity of the original
// dataset.
//
// NOTE: in case original dataset is non-monotonic, ALGLIB splits it
// into monotone subsequences and builds piecewise monotonic spline.
//
real_1d_array x = "[0,1,2]";
real_1d_array y = "[0,1,1]";
spline1dinterpolant s;
// build spline
spline1dbuildmonotone(x, y, s);
// calculate S at x = [-0.5, 0.0, 0.5, 1.0, 1.5, 2.0]
// you may see that spline is really monotonic
double v;
v = spline1dcalc(s, -0.5);
printf("%.4f\n", double(v)); // EXPECTED: 0.0000
v = spline1dcalc(s, 0.0);
printf("%.4f\n", double(v)); // EXPECTED: 0.0000
v = spline1dcalc(s, +0.5);
printf("%.4f\n", double(v)); // EXPECTED: 0.5000
v = spline1dcalc(s, 1.0);
printf("%.4f\n", double(v)); // EXPECTED: 1.0000
v = spline1dcalc(s, 1.5);
printf("%.4f\n", double(v)); // EXPECTED: 1.0000
v = spline1dcalc(s, 2.0);
printf("%.4f\n", double(v)); // EXPECTED: 1.0000
return 0;
}
spline2d subpackage| spline2d_bicubic | Bilinear spline interpolation | |
| spline2d_bilinear | Bilinear spline interpolation | |
| spline2d_copytrans | Copy and transform | |
| spline2d_unpack | Unpacking bilinear spline | |
| spline2d_vector | Copy and transform |
spline2dinterpolant class/************************************************************************* 2-dimensional spline inteprolant *************************************************************************/class spline2dinterpolant { };
spline2dbuildbicubic function/************************************************************************* This subroutine was deprecated in ALGLIB 3.6.0 We recommend you to switch to Spline2DBuildBicubicV(), which is more flexible and accepts its arguments in more convenient order. -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2dbuildbicubic( real_1d_array x, real_1d_array y, real_2d_array f, ae_int_t m, ae_int_t n, spline2dinterpolant& c);
spline2dbuildbicubicv function/************************************************************************* This subroutine builds bicubic vector-valued spline. Input parameters: X - spline abscissas, array[0..N-1] Y - spline ordinates, array[0..M-1] F - function values, array[0..M*N*D-1]: * first D elements store D values at (X[0],Y[0]) * next D elements store D values at (X[1],Y[0]) * general form - D function values at (X[i],Y[j]) are stored at F[D*(J*N+I)...D*(J*N+I)+D-1]. M,N - grid size, M>=2, N>=2 D - vector dimension, D>=1 Output parameters: C - spline interpolant -- ALGLIB PROJECT -- Copyright 16.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline2dbuildbicubicv( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, real_1d_array f, ae_int_t d, spline2dinterpolant& c);
Examples: [1]
spline2dbuildbilinear function/************************************************************************* This subroutine was deprecated in ALGLIB 3.6.0 We recommend you to switch to Spline2DBuildBilinearV(), which is more flexible and accepts its arguments in more convenient order. -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2dbuildbilinear( real_1d_array x, real_1d_array y, real_2d_array f, ae_int_t m, ae_int_t n, spline2dinterpolant& c);
spline2dbuildbilinearv function/************************************************************************* This subroutine builds bilinear vector-valued spline. Input parameters: X - spline abscissas, array[0..N-1] Y - spline ordinates, array[0..M-1] F - function values, array[0..M*N*D-1]: * first D elements store D values at (X[0],Y[0]) * next D elements store D values at (X[1],Y[0]) * general form - D function values at (X[i],Y[j]) are stored at F[D*(J*N+I)...D*(J*N+I)+D-1]. M,N - grid size, M>=2, N>=2 D - vector dimension, D>=1 Output parameters: C - spline interpolant -- ALGLIB PROJECT -- Copyright 16.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline2dbuildbilinearv( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, real_1d_array f, ae_int_t d, spline2dinterpolant& c);
spline2dcalc function/************************************************************************* This subroutine calculates the value of the bilinear or bicubic spline at the given point X. Input parameters: C - coefficients table. Built by BuildBilinearSpline or BuildBicubicSpline. X, Y- point Result: S(x,y) -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/double alglib::spline2dcalc(spline2dinterpolant c, double x, double y);
spline2dcalcv function/************************************************************************* This subroutine calculates bilinear or bicubic vector-valued spline at the given point (X,Y). INPUT PARAMETERS: C - spline interpolant. X, Y- point OUTPUT PARAMETERS: F - array[D] which stores function values. F is out-parameter and it is reallocated after call to this function. In case you want to reuse previously allocated F, you may use Spline2DCalcVBuf(), which reallocates F only when it is too small. -- ALGLIB PROJECT -- Copyright 16.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline2dcalcv( spline2dinterpolant c, double x, double y, real_1d_array& f);
Examples: [1]
spline2dcalcvbuf function/************************************************************************* This subroutine calculates bilinear or bicubic vector-valued spline at the given point (X,Y). INPUT PARAMETERS: C - spline interpolant. X, Y- point F - output buffer, possibly preallocated array. In case array size is large enough to store result, it is not reallocated. Array which is too short will be reallocated OUTPUT PARAMETERS: F - array[D] (or larger) which stores function values -- ALGLIB PROJECT -- Copyright 16.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline2dcalcvbuf( spline2dinterpolant c, double x, double y, real_1d_array& f);
spline2dcopy function/************************************************************************* This subroutine makes the copy of the spline model. Input parameters: C - spline interpolant Output parameters: CC - spline copy -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2dcopy(spline2dinterpolant c, spline2dinterpolant& cc);
Examples: [1]
spline2ddiff function/************************************************************************* This subroutine calculates the value of the bilinear or bicubic spline at the given point X and its derivatives. Input parameters: C - spline interpolant. X, Y- point Output parameters: F - S(x,y) FX - dS(x,y)/dX FY - dS(x,y)/dY FXY - d2S(x,y)/dXdY -- ALGLIB PROJECT -- Copyright 05.07.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2ddiff( spline2dinterpolant c, double x, double y, double& f, double& fx, double& fy, double& fxy);
spline2dlintransf function/************************************************************************* This subroutine performs linear transformation of the spline. Input parameters: C - spline interpolant. A, B- transformation coefficients: S2(x,y) = A*S(x,y) + B Output parameters: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2dlintransf(spline2dinterpolant c, double a, double b);
Examples: [1]
spline2dlintransxy function/************************************************************************* This subroutine performs linear transformation of the spline argument. Input parameters: C - spline interpolant AX, BX - transformation coefficients: x = A*t + B AY, BY - transformation coefficients: y = A*u + B Result: C - transformed spline -- ALGLIB PROJECT -- Copyright 30.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2dlintransxy( spline2dinterpolant c, double ax, double bx, double ay, double by);
Examples: [1]
spline2dresamplebicubic function/************************************************************************* Bicubic spline resampling Input parameters: A - function values at the old grid, array[0..OldHeight-1, 0..OldWidth-1] OldHeight - old grid height, OldHeight>1 OldWidth - old grid width, OldWidth>1 NewHeight - new grid height, NewHeight>1 NewWidth - new grid width, NewWidth>1 Output parameters: B - function values at the new grid, array[0..NewHeight-1, 0..NewWidth-1] -- ALGLIB routine -- 15 May, 2007 Copyright by Bochkanov Sergey *************************************************************************/void alglib::spline2dresamplebicubic( real_2d_array a, ae_int_t oldheight, ae_int_t oldwidth, real_2d_array& b, ae_int_t newheight, ae_int_t newwidth);
spline2dresamplebilinear function/************************************************************************* Bilinear spline resampling Input parameters: A - function values at the old grid, array[0..OldHeight-1, 0..OldWidth-1] OldHeight - old grid height, OldHeight>1 OldWidth - old grid width, OldWidth>1 NewHeight - new grid height, NewHeight>1 NewWidth - new grid width, NewWidth>1 Output parameters: B - function values at the new grid, array[0..NewHeight-1, 0..NewWidth-1] -- ALGLIB routine -- 09.07.2007 Copyright by Bochkanov Sergey *************************************************************************/void alglib::spline2dresamplebilinear( real_2d_array a, ae_int_t oldheight, ae_int_t oldwidth, real_2d_array& b, ae_int_t newheight, ae_int_t newwidth);
spline2dunpack function/************************************************************************* This subroutine was deprecated in ALGLIB 3.6.0 We recommend you to switch to Spline2DUnpackV(), which is more flexible and accepts its arguments in more convenient order. -- ALGLIB PROJECT -- Copyright 29.06.2007 by Bochkanov Sergey *************************************************************************/void alglib::spline2dunpack( spline2dinterpolant c, ae_int_t& m, ae_int_t& n, real_2d_array& tbl);
spline2dunpackv function/************************************************************************* This subroutine unpacks two-dimensional spline into the coefficients table Input parameters: C - spline interpolant. Result: M, N- grid size (x-axis and y-axis) D - number of components Tbl - coefficients table, unpacked format, D - components: [0..(N-1)*(M-1)*D-1, 0..19]. For T=0..D-1 (component index), I = 0...N-2 (x index), J=0..M-2 (y index): K := T + I*D + J*D*(N-1) K-th row stores decomposition for T-th component of the vector-valued function Tbl[K,0] = X[i] Tbl[K,1] = X[i+1] Tbl[K,2] = Y[j] Tbl[K,3] = Y[j+1] Tbl[K,4] = C00 Tbl[K,5] = C01 Tbl[K,6] = C02 Tbl[K,7] = C03 Tbl[K,8] = C10 Tbl[K,9] = C11 ... Tbl[K,19] = C33 On each grid square spline is equals to: S(x) = SUM(c[i,j]*(t^i)*(u^j), i=0..3, j=0..3) t = x-x[j] u = y-y[i] -- ALGLIB PROJECT -- Copyright 16.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline2dunpackv( spline2dinterpolant c, ae_int_t& m, ae_int_t& n, ae_int_t& d, real_2d_array& tbl);
Examples: [1]
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled
// at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
//
real_1d_array x = "[0.0, 0.5, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array f = "[0.00,0.25,1.00,2.00,2.25,3.00]";
double vx = 0.25;
double vy = 0.50;
double v;
double dx;
double dy;
double dxy;
spline2dinterpolant s;
// build spline
spline2dbuildbicubicv(x, 3, y, 2, f, 1, s);
// calculate S(0.25,0.50)
v = spline2dcalc(s, vx, vy);
printf("%.4f\n", double(v)); // EXPECTED: 1.0625
// calculate derivatives
spline2ddiff(s, vx, vy, v, dx, dy, dxy);
printf("%.4f\n", double(v)); // EXPECTED: 1.0625
printf("%.4f\n", double(dx)); // EXPECTED: 0.5000
printf("%.4f\n", double(dy)); // EXPECTED: 2.0000
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use bilinear spline to interpolate f(x,y)=x^2+2*y^2 sampled
// at (x,y) from [0.0, 0.5, 1.0] X [0.0, 1.0].
//
real_1d_array x = "[0.0, 0.5, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array f = "[0.00,0.25,1.00,2.00,2.25,3.00]";
double vx = 0.25;
double vy = 0.50;
double v;
spline2dinterpolant s;
// build spline
spline2dbuildbilinearv(x, 3, y, 2, f, 1, s);
// calculate S(0.25,0.50)
v = spline2dcalc(s, vx, vy);
printf("%.4f\n", double(v)); // EXPECTED: 1.1250
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We build bilinear spline for f(x,y)=x+2*y for (x,y) in [0,1].
// Then we apply several transformations to this spline.
//
real_1d_array x = "[0.0, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array f = "[0.00,1.00,2.00,3.00]";
spline2dinterpolant s;
spline2dinterpolant snew;
double v;
spline2dbuildbilinearv(x, 2, y, 2, f, 1, s);
// copy spline, apply transformation x:=2*xnew, y:=4*ynew
// evaluate at (xnew,ynew) = (0.25,0.25) - should be same as (x,y)=(0.5,1.0)
spline2dcopy(s, snew);
spline2dlintransxy(snew, 2.0, 0.0, 4.0, 0.0);
v = spline2dcalc(snew, 0.25, 0.25);
printf("%.4f\n", double(v)); // EXPECTED: 2.500
// copy spline, apply transformation SNew:=2*S+3
spline2dcopy(s, snew);
spline2dlintransf(snew, 2.0, 3.0);
v = spline2dcalc(snew, 0.5, 1.0);
printf("%.4f\n", double(v)); // EXPECTED: 8.000
//
// Same example, but for vector spline (f0,f1) = {x+2*y, 2*x+y}
//
real_1d_array f2 = "[0.00,0.00, 1.00,2.00, 2.00,1.00, 3.00,3.00]";
real_1d_array vr;
spline2dbuildbilinearv(x, 2, y, 2, f2, 2, s);
// copy spline, apply transformation x:=2*xnew, y:=4*ynew
spline2dcopy(s, snew);
spline2dlintransxy(snew, 2.0, 0.0, 4.0, 0.0);
spline2dcalcv(snew, 0.25, 0.25, vr);
printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [2.500,2.000]
// copy spline, apply transformation SNew:=2*S+3
spline2dcopy(s, snew);
spline2dlintransf(snew, 2.0, 3.0);
spline2dcalcv(snew, 0.5, 1.0, vr);
printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [8.000,7.000]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We build bilinear spline for f(x,y)=x+2*y+3*xy for (x,y) in [0,1].
// Then we demonstrate how to unpack it.
//
real_1d_array x = "[0.0, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array f = "[0.00,1.00,2.00,6.00]";
real_2d_array c;
ae_int_t m;
ae_int_t n;
ae_int_t d;
spline2dinterpolant s;
// build spline
spline2dbuildbilinearv(x, 2, y, 2, f, 1, s);
// unpack and test
spline2dunpackv(s, m, n, d, c);
printf("%s\n", c.tostring(4).c_str()); // EXPECTED: [[0, 1, 0, 1, 0,2,0,0, 1,3,0,0, 0,0,0,0, 0,0,0,0 ]]
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We build bilinear vector-valued spline (f0,f1) = {x+2*y, 2*x+y}
// Spline is built using function values at 2x2 grid: (x,y)=[0,1]*[0,1]
// Then we perform evaluation at (x,y)=(0.1,0.3)
//
real_1d_array x = "[0.0, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array f = "[0.00,0.00, 1.00,2.00, 2.00,1.00, 3.00,3.00]";
spline2dinterpolant s;
real_1d_array vr;
spline2dbuildbilinearv(x, 2, y, 2, f, 2, s);
spline2dcalcv(s, 0.1, 0.3, vr);
printf("%s\n", vr.tostring(4).c_str()); // EXPECTED: [0.700,0.500]
return 0;
}
spline3d subpackage| spline3d_trilinear | Trilinear spline interpolation | |
| spline3d_vector | Vector-valued trilinear spline interpolation |
spline3dinterpolant class/************************************************************************* 3-dimensional spline inteprolant *************************************************************************/class spline3dinterpolant { };
spline3dbuildtrilinearv function/************************************************************************* This subroutine builds trilinear vector-valued spline. INPUT PARAMETERS: X - spline abscissas, array[0..N-1] Y - spline ordinates, array[0..M-1] Z - spline applicates, array[0..L-1] F - function values, array[0..M*N*L*D-1]: * first D elements store D values at (X[0],Y[0],Z[0]) * next D elements store D values at (X[1],Y[0],Z[0]) * next D elements store D values at (X[2],Y[0],Z[0]) * ... * next D elements store D values at (X[0],Y[1],Z[0]) * next D elements store D values at (X[1],Y[1],Z[0]) * next D elements store D values at (X[2],Y[1],Z[0]) * ... * next D elements store D values at (X[0],Y[0],Z[1]) * next D elements store D values at (X[1],Y[0],Z[1]) * next D elements store D values at (X[2],Y[0],Z[1]) * ... * general form - D function values at (X[i],Y[j]) are stored at F[D*(N*(M*K+J)+I)...D*(N*(M*K+J)+I)+D-1]. M,N, L - grid size, M>=2, N>=2, L>=2 D - vector dimension, D>=1 OUTPUT PARAMETERS: C - spline interpolant -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline3dbuildtrilinearv( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, real_1d_array z, ae_int_t l, real_1d_array f, ae_int_t d, spline3dinterpolant& c);
spline3dcalc function/************************************************************************* This subroutine calculates the value of the trilinear or tricubic spline at the given point (X,Y,Z). INPUT PARAMETERS: C - coefficients table. Built by BuildBilinearSpline or BuildBicubicSpline. X, Y, Z - point Result: S(x,y,z) -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/double alglib::spline3dcalc( spline3dinterpolant c, double x, double y, double z);
Examples: [1]
spline3dcalcv function/************************************************************************* This subroutine calculates trilinear or tricubic vector-valued spline at the given point (X,Y,Z). INPUT PARAMETERS: C - spline interpolant. X, Y, Z - point OUTPUT PARAMETERS: F - array[D] which stores function values. F is out-parameter and it is reallocated after call to this function. In case you want to reuse previously allocated F, you may use Spline2DCalcVBuf(), which reallocates F only when it is too small. -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline3dcalcv( spline3dinterpolant c, double x, double y, double z, real_1d_array& f);
Examples: [1]
spline3dcalcvbuf function/************************************************************************* This subroutine calculates bilinear or bicubic vector-valued spline at the given point (X,Y,Z). INPUT PARAMETERS: C - spline interpolant. X, Y, Z - point F - output buffer, possibly preallocated array. In case array size is large enough to store result, it is not reallocated. Array which is too short will be reallocated OUTPUT PARAMETERS: F - array[D] (or larger) which stores function values -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline3dcalcvbuf( spline3dinterpolant c, double x, double y, double z, real_1d_array& f);
spline3dlintransf function/************************************************************************* This subroutine performs linear transformation of the spline. INPUT PARAMETERS: C - spline interpolant. A, B- transformation coefficients: S2(x,y) = A*S(x,y,z) + B OUTPUT PARAMETERS: C - transformed spline -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline3dlintransf(spline3dinterpolant c, double a, double b);
spline3dlintransxyz function/************************************************************************* This subroutine performs linear transformation of the spline argument. INPUT PARAMETERS: C - spline interpolant AX, BX - transformation coefficients: x = A*u + B AY, BY - transformation coefficients: y = A*v + B AZ, BZ - transformation coefficients: z = A*w + B OUTPUT PARAMETERS: C - transformed spline -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline3dlintransxyz( spline3dinterpolant c, double ax, double bx, double ay, double by, double az, double bz);
spline3dresampletrilinear function/************************************************************************* Trilinear spline resampling INPUT PARAMETERS: A - array[0..OldXCount*OldYCount*OldZCount-1], function values at the old grid, : A[0] x=0,y=0,z=0 A[1] x=1,y=0,z=0 A[..] ... A[..] x=oldxcount-1,y=0,z=0 A[..] x=0,y=1,z=0 A[..] ... ... OldZCount - old Z-count, OldZCount>1 OldYCount - old Y-count, OldYCount>1 OldXCount - old X-count, OldXCount>1 NewZCount - new Z-count, NewZCount>1 NewYCount - new Y-count, NewYCount>1 NewXCount - new X-count, NewXCount>1 OUTPUT PARAMETERS: B - array[0..NewXCount*NewYCount*NewZCount-1], function values at the new grid: B[0] x=0,y=0,z=0 B[1] x=1,y=0,z=0 B[..] ... B[..] x=newxcount-1,y=0,z=0 B[..] x=0,y=1,z=0 B[..] ... ... -- ALGLIB routine -- 26.04.2012 Copyright by Bochkanov Sergey *************************************************************************/void alglib::spline3dresampletrilinear( real_1d_array a, ae_int_t oldzcount, ae_int_t oldycount, ae_int_t oldxcount, ae_int_t newzcount, ae_int_t newycount, ae_int_t newxcount, real_1d_array& b);
spline3dunpackv function/************************************************************************* This subroutine unpacks tri-dimensional spline into the coefficients table INPUT PARAMETERS: C - spline interpolant. Result: N - grid size (X) M - grid size (Y) L - grid size (Z) D - number of components SType- spline type. Currently, only one spline type is supported: trilinear spline, as indicated by SType=1. Tbl - spline coefficients: [0..(N-1)*(M-1)*(L-1)*D-1, 0..13]. For T=0..D-1 (component index), I = 0...N-2 (x index), J=0..M-2 (y index), K=0..L-2 (z index): Q := T + I*D + J*D*(N-1) + K*D*(N-1)*(M-1), Q-th row stores decomposition for T-th component of the vector-valued function Tbl[Q,0] = X[i] Tbl[Q,1] = X[i+1] Tbl[Q,2] = Y[j] Tbl[Q,3] = Y[j+1] Tbl[Q,4] = Z[k] Tbl[Q,5] = Z[k+1] Tbl[Q,6] = C000 Tbl[Q,7] = C100 Tbl[Q,8] = C010 Tbl[Q,9] = C110 Tbl[Q,10]= C001 Tbl[Q,11]= C101 Tbl[Q,12]= C011 Tbl[Q,13]= C111 On each grid square spline is equals to: S(x) = SUM(c[i,j,k]*(x^i)*(y^j)*(z^k), i=0..1, j=0..1, k=0..1) t = x-x[j] u = y-y[i] v = z-z[k] NOTE: format of Tbl is given for SType=1. Future versions of ALGLIB can use different formats for different values of SType. -- ALGLIB PROJECT -- Copyright 26.04.2012 by Bochkanov Sergey *************************************************************************/void alglib::spline3dunpackv( spline3dinterpolant c, ae_int_t& n, ae_int_t& m, ae_int_t& l, ae_int_t& d, ae_int_t& stype, real_2d_array& tbl);
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use trilinear spline to interpolate f(x,y,z)=x+xy+z sampled
// at (x,y,z) from [0.0, 1.0] X [0.0, 1.0] X [0.0, 1.0].
//
// We store x, y and z-values at local arrays with same names.
// Function values are stored in the array F as follows:
// f[0] (x,y,z) = (0,0,0)
// f[1] (x,y,z) = (1,0,0)
// f[2] (x,y,z) = (0,1,0)
// f[3] (x,y,z) = (1,1,0)
// f[4] (x,y,z) = (0,0,1)
// f[5] (x,y,z) = (1,0,1)
// f[6] (x,y,z) = (0,1,1)
// f[7] (x,y,z) = (1,1,1)
//
real_1d_array x = "[0.0, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array z = "[0.0, 1.0]";
real_1d_array f = "[0,1,0,2,1,2,1,3]";
double vx = 0.50;
double vy = 0.50;
double vz = 0.50;
double v;
spline3dinterpolant s;
// build spline
spline3dbuildtrilinearv(x, 2, y, 2, z, 2, f, 1, s);
// calculate S(0.5,0.5,0.5)
v = spline3dcalc(s, vx, vy, vz);
printf("%.4f\n", double(v)); // EXPECTED: 1.2500
return 0;
}
#include "stdafx.h"
#include <stdlib.h>
#include <stdio.h>
#include <math.h>
#include "interpolation.h"
using namespace alglib;
int main(int argc, char **argv)
{
//
// We use trilinear vector-valued spline to interpolate {f0,f1}={x+xy+z,x+xy+yz+z}
// sampled at (x,y,z) from [0.0, 1.0] X [0.0, 1.0] X [0.0, 1.0].
//
// We store x, y and z-values at local arrays with same names.
// Function values are stored in the array F as follows:
// f[0] f0, (x,y,z) = (0,0,0)
// f[1] f1, (x,y,z) = (0,0,0)
// f[2] f0, (x,y,z) = (1,0,0)
// f[3] f1, (x,y,z) = (1,0,0)
// f[4] f0, (x,y,z) = (0,1,0)
// f[5] f1, (x,y,z) = (0,1,0)
// f[6] f0, (x,y,z) = (1,1,0)
// f[7] f1, (x,y,z) = (1,1,0)
// f[8] f0, (x,y,z) = (0,0,1)
// f[9] f1, (x,y,z) = (0,0,1)
// f[10] f0, (x,y,z) = (1,0,1)
// f[11] f1, (x,y,z) = (1,0,1)
// f[12] f0, (x,y,z) = (0,1,1)
// f[13] f1, (x,y,z) = (0,1,1)
// f[14] f0, (x,y,z) = (1,1,1)
// f[15] f1, (x,y,z) = (1,1,1)
//
real_1d_array x = "[0.0, 1.0]";
real_1d_array y = "[0.0, 1.0]";
real_1d_array z = "[0.0, 1.0]";
real_1d_array f = "[0,0, 1,1, 0,0, 2,2, 1,1, 2,2, 1,2, 3,4]";
double vx = 0.50;
double vy = 0.50;
double vz = 0.50;
spline3dinterpolant s;
// build spline
spline3dbuildtrilinearv(x, 2, y, 2, z, 2, f, 2, s);
// calculate S(0.5,0.5,0.5) - we have vector of values instead of single value
real_1d_array v;
spline3dcalcv(s, vx, vy, vz, v);
printf("%s\n", v.tostring(4).c_str()); // EXPECTED: [1.2500,1.5000]
return 0;
}
stest subpackageonesamplesigntest function/************************************************************************* Sign test This test checks three hypotheses about the median of the given sample. The following tests are performed: * two-tailed test (null hypothesis - the median is equal to the given value) * left-tailed test (null hypothesis - the median is greater than or equal to the given value) * right-tailed test (null hypothesis - the median is less than or equal to the given value) Requirements: * the scale of measurement should be ordinal, interval or ratio (i.e. the test could not be applied to nominal variables). The test is non-parametric and doesn't require distribution X to be normal Input parameters: X - sample. Array whose index goes from 0 to N-1. N - size of the sample. Median - assumed median value. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. While calculating p-values high-precision binomial distribution approximation is used, so significance levels have about 15 exact digits. -- ALGLIB -- Copyright 08.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::onesamplesigntest( real_1d_array x, ae_int_t n, double median, double& bothtails, double& lefttail, double& righttail);
studenttdistr subpackageinvstudenttdistribution function/************************************************************************* Functional inverse of Student's t distribution Given probability p, finds the argument t such that stdtr(k,t) is equal to p. ACCURACY: Tested at random 1 <= k <= 100. The "domain" refers to p: Relative error: arithmetic domain # trials peak rms IEEE .001,.999 25000 5.7e-15 8.0e-16 IEEE 10^-6,.001 25000 2.0e-12 2.9e-14 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::invstudenttdistribution(ae_int_t k, double p);
studenttdistribution function/************************************************************************* Student's t distribution Computes the integral from minus infinity to t of the Student t distribution with integer k > 0 degrees of freedom: t - | | - | 2 -(k+1)/2 | ( (k+1)/2 ) | ( x ) ---------------------- | ( 1 + --- ) dx - | ( k ) sqrt( k pi ) | ( k/2 ) | | | - -inf. Relation to incomplete beta integral: 1 - stdtr(k,t) = 0.5 * incbet( k/2, 1/2, z ) where z = k/(k + t**2). For t < -2, this is the method of computation. For higher t, a direct method is derived from integration by parts. Since the function is symmetric about t=0, the area under the right tail of the density is found by calling the function with -t instead of t. ACCURACY: Tested at random 1 <= k <= 25. The "domain" refers to t. Relative error: arithmetic domain # trials peak rms IEEE -100,-2 50000 5.9e-15 1.4e-15 IEEE -2,100 500000 2.7e-15 4.9e-17 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 1995, 2000 by Stephen L. Moshier *************************************************************************/double alglib::studenttdistribution(ae_int_t k, double t);
studentttests subpackagestudentttest1 function/************************************************************************* One-sample t-test This test checks three hypotheses about the mean of the given sample. The following tests are performed: * two-tailed test (null hypothesis - the mean is equal to the given value) * left-tailed test (null hypothesis - the mean is greater than or equal to the given value) * right-tailed test (null hypothesis - the mean is less than or equal to the given value). The test is based on the assumption that a given sample has a normal distribution and an unknown dispersion. If the distribution sharply differs from normal, the test will work incorrectly. INPUT PARAMETERS: X - sample. Array whose index goes from 0 to N-1. N - size of sample, N>=0 Mean - assumed value of the mean. OUTPUT PARAMETERS: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. NOTE: this function correctly handles degenerate cases: * when N=0, all p-values are set to 1.0 * when variance of X[] is exactly zero, p-values are set to 1.0 or 0.0, depending on difference between sample mean and value of mean being tested. -- ALGLIB -- Copyright 08.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::studentttest1( real_1d_array x, ae_int_t n, double mean, double& bothtails, double& lefttail, double& righttail);
studentttest2 function/************************************************************************* Two-sample pooled test This test checks three hypotheses about the mean of the given samples. The following tests are performed: * two-tailed test (null hypothesis - the means are equal) * left-tailed test (null hypothesis - the mean of the first sample is greater than or equal to the mean of the second sample) * right-tailed test (null hypothesis - the mean of the first sample is less than or equal to the mean of the second sample). Test is based on the following assumptions: * given samples have normal distributions * dispersions are equal * samples are independent. Input parameters: X - sample 1. Array whose index goes from 0 to N-1. N - size of sample. Y - sample 2. Array whose index goes from 0 to M-1. M - size of sample. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. NOTE: this function correctly handles degenerate cases: * when N=0 or M=0, all p-values are set to 1.0 * when both samples has exactly zero variance, p-values are set to 1.0 or 0.0, depending on difference between means. -- ALGLIB -- Copyright 18.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::studentttest2( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, double& bothtails, double& lefttail, double& righttail);
unequalvariancettest function/************************************************************************* Two-sample unpooled test This test checks three hypotheses about the mean of the given samples. The following tests are performed: * two-tailed test (null hypothesis - the means are equal) * left-tailed test (null hypothesis - the mean of the first sample is greater than or equal to the mean of the second sample) * right-tailed test (null hypothesis - the mean of the first sample is less than or equal to the mean of the second sample). Test is based on the following assumptions: * given samples have normal distributions * samples are independent. Equality of variances is NOT required. Input parameters: X - sample 1. Array whose index goes from 0 to N-1. N - size of the sample. Y - sample 2. Array whose index goes from 0 to M-1. M - size of the sample. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. NOTE: this function correctly handles degenerate cases: * when N=0 or M=0, all p-values are set to 1.0 * when both samples has zero variance, p-values are set to 1.0 or 0.0, depending on difference between means. * when only one sample has zero variance, test reduces to 1-sample version. -- ALGLIB -- Copyright 18.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::unequalvariancettest( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, double& bothtails, double& lefttail, double& righttail);
svd subpackagermatrixsvd function/************************************************************************* Singular value decomposition of a rectangular matrix. COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes one important improvement of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Generally, commercial ALGLIB is several times faster than open-source ! generic C edition, and many times faster than open-source C# edition. ! ! Multithreaded acceleration is only partially supported (some parts are ! optimized, but most - are not). ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. The algorithm calculates the singular value decomposition of a matrix of size MxN: A = U * S * V^T The algorithm finds the singular values and, optionally, matrices U and V^T. The algorithm can find both first min(M,N) columns of matrix U and rows of matrix V^T (singular vectors), and matrices U and V^T wholly (of sizes MxM and NxN respectively). Take into account that the subroutine does not return matrix V but V^T. Input parameters: A - matrix to be decomposed. Array whose indexes range within [0..M-1, 0..N-1]. M - number of rows in matrix A. N - number of columns in matrix A. UNeeded - 0, 1 or 2. See the description of the parameter U. VTNeeded - 0, 1 or 2. See the description of the parameter VT. AdditionalMemory - If the parameter: * equals 0, the algorithm doesn't use additional memory (lower requirements, lower performance). * equals 1, the algorithm uses additional memory of size min(M,N)*min(M,N) of real numbers. It often speeds up the algorithm. * equals 2, the algorithm uses additional memory of size M*min(M,N) of real numbers. It allows to get a maximum performance. The recommended value of the parameter is 2. Output parameters: W - contains singular values in descending order. U - if UNeeded=0, U isn't changed, the left singular vectors are not calculated. if Uneeded=1, U contains left singular vectors (first min(M,N) columns of matrix U). Array whose indexes range within [0..M-1, 0..Min(M,N)-1]. if UNeeded=2, U contains matrix U wholly. Array whose indexes range within [0..M-1, 0..M-1]. VT - if VTNeeded=0, VT isn't changed, the right singular vectors are not calculated. if VTNeeded=1, VT contains right singular vectors (first min(M,N) rows of matrix V^T). Array whose indexes range within [0..min(M,N)-1, 0..N-1]. if VTNeeded=2, VT contains matrix V^T wholly. Array whose indexes range within [0..N-1, 0..N-1]. -- ALGLIB -- Copyright 2005 by Bochkanov Sergey *************************************************************************/bool alglib::rmatrixsvd( real_2d_array a, ae_int_t m, ae_int_t n, ae_int_t uneeded, ae_int_t vtneeded, ae_int_t additionalmemory, real_1d_array& w, real_2d_array& u, real_2d_array& vt); bool alglib::smp_rmatrixsvd( real_2d_array a, ae_int_t m, ae_int_t n, ae_int_t uneeded, ae_int_t vtneeded, ae_int_t additionalmemory, real_1d_array& w, real_2d_array& u, real_2d_array& vt);
trfac subpackagecmatrixlu function/************************************************************************* LU decomposition of a general complex matrix with row pivoting A is represented as A = P*L*U, where: * L is lower unitriangular matrix * U is upper triangular matrix * P = P0*P1*...*PK, K=min(M,N)-1, Pi - permutation matrix for I and Pivots[I] This is cache-oblivous implementation of LU decomposition. It is optimized for square matrices. As for rectangular matrices: * best case - M>>N * worst case - N>>M, small M, large N, matrix does not fit in CPU cache COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - array[0..M-1, 0..N-1]. M - number of rows in matrix A. N - number of columns in matrix A. OUTPUT PARAMETERS: A - matrices L and U in compact form: * L is stored under main diagonal * U is stored on and above main diagonal Pivots - permutation matrix in compact form. array[0..Min(M-1,N-1)]. -- ALGLIB routine -- 10.01.2010 Bochkanov Sergey *************************************************************************/void alglib::cmatrixlu( complex_2d_array& a, ae_int_t m, ae_int_t n, integer_1d_array& pivots); void alglib::smp_cmatrixlu( complex_2d_array& a, ae_int_t m, ae_int_t n, integer_1d_array& pivots);
hpdmatrixcholesky function/************************************************************************* Cache-oblivious Cholesky decomposition The algorithm computes Cholesky decomposition of a Hermitian positive- definite matrix. The result of an algorithm is a representation of A as A=U'*U or A=L*L' (here X' detones conj(X^T)). COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - upper or lower triangle of a factorized matrix. array with elements [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - if IsUpper=True, then A contains an upper triangle of a symmetric matrix, otherwise A contains a lower one. OUTPUT PARAMETERS: A - the result of factorization. If IsUpper=True, then the upper triangle contains matrix U, so that A = U'*U, and the elements below the main diagonal are not modified. Similarly, if IsUpper = False. RESULT: If the matrix is positive-definite, the function returns True. Otherwise, the function returns False. Contents of A is not determined in such case. -- ALGLIB routine -- 15.12.2009 Bochkanov Sergey *************************************************************************/bool alglib::hpdmatrixcholesky( complex_2d_array& a, ae_int_t n, bool isupper); bool alglib::smp_hpdmatrixcholesky( complex_2d_array& a, ae_int_t n, bool isupper);
rmatrixlu function/************************************************************************* LU decomposition of a general real matrix with row pivoting A is represented as A = P*L*U, where: * L is lower unitriangular matrix * U is upper triangular matrix * P = P0*P1*...*PK, K=min(M,N)-1, Pi - permutation matrix for I and Pivots[I] This is cache-oblivous implementation of LU decomposition. It is optimized for square matrices. As for rectangular matrices: * best case - M>>N * worst case - N>>M, small M, large N, matrix does not fit in CPU cache COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that LU decomposition is harder to ! parallelize than, say, matrix-matrix product - this algorithm has ! many internal synchronization points which can not be avoided. However ! parallelism starts to be profitable starting from N=1024, achieving ! near-linear speedup for N=4096 or higher. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - array[0..M-1, 0..N-1]. M - number of rows in matrix A. N - number of columns in matrix A. OUTPUT PARAMETERS: A - matrices L and U in compact form: * L is stored under main diagonal * U is stored on and above main diagonal Pivots - permutation matrix in compact form. array[0..Min(M-1,N-1)]. -- ALGLIB routine -- 10.01.2010 Bochkanov Sergey *************************************************************************/void alglib::rmatrixlu( real_2d_array& a, ae_int_t m, ae_int_t n, integer_1d_array& pivots); void alglib::smp_rmatrixlu( real_2d_array& a, ae_int_t m, ae_int_t n, integer_1d_array& pivots);
sparsecholeskyskyline function/************************************************************************* Sparse Cholesky decomposition for skyline matrixm using in-place algorithm without allocating additional storage. The algorithm computes Cholesky decomposition of a symmetric positive- definite sparse matrix. The result of an algorithm is a representation of A as A=U^T*U or A=L*L^T This function is a more efficient alternative to general, but slower SparseCholeskyX(), because it does not create temporary copies of the target. It performs factorization in-place, which gives best performance on low-profile matrices. Its drawback, however, is that it can not perform profile-reducing permutation of input matrix. INPUT PARAMETERS: A - sparse matrix in skyline storage (SKS) format. N - size of matrix A (can be smaller than actual size of A) IsUpper - if IsUpper=True, then factorization is performed on upper triangle. Another triangle is ignored (it may contant some data, but it is not changed). OUTPUT PARAMETERS: A - the result of factorization, stored in SKS. If IsUpper=True, then the upper triangle contains matrix U, such that A = U^T*U. Lower triangle is not changed. Similarly, if IsUpper = False. In this case L is returned, and we have A = L*(L^T). Note that THIS function does not perform permutation of rows to reduce bandwidth. RESULT: If the matrix is positive-definite, the function returns True. Otherwise, the function returns False. Contents of A is not determined in such case. NOTE: for performance reasons this function does NOT check that input matrix includes only finite values. It is your responsibility to make sure that there are no infinite or NAN values in the matrix. -- ALGLIB routine -- 16.01.2014 Bochkanov Sergey *************************************************************************/bool alglib::sparsecholeskyskyline( sparsematrix a, ae_int_t n, bool isupper);
spdmatrixcholesky function/************************************************************************* Cache-oblivious Cholesky decomposition The algorithm computes Cholesky decomposition of a symmetric positive- definite matrix. The result of an algorithm is a representation of A as A=U^T*U or A=L*L^T COMMERCIAL EDITION OF ALGLIB: ! Commercial version of ALGLIB includes two important improvements of ! this function, which can be used from C++ and C#: ! * Intel MKL support (lightweight Intel MKL is shipped with ALGLIB) ! * multicore support ! ! Intel MKL gives approximately constant (with respect to number of ! worker threads) acceleration factor which depends on CPU being used, ! problem size and "baseline" ALGLIB edition which is used for ! comparison. ! ! Say, on SSE2-capable CPU with N=1024, HPC ALGLIB will be: ! * about 2-3x faster than ALGLIB for C++ without MKL ! * about 7-10x faster than "pure C#" edition of ALGLIB ! Difference in performance will be more striking on newer CPU's with ! support for newer SIMD instructions. Generally, MKL accelerates any ! problem whose size is at least 128, with best efficiency achieved for ! N's larger than 512. ! ! Commercial edition of ALGLIB also supports multithreaded acceleration ! of this function. We should note that Cholesky decomposition is harder ! to parallelize than, say, matrix-matrix product - this algorithm has ! several synchronization points which can not be avoided. However, ! parallelism starts to be profitable starting from N=500. ! ! In order to use multicore features you have to: ! * use commercial version of ALGLIB ! * call this function with "smp_" prefix, which indicates that ! multicore code will be used (for multicore support) ! ! We recommend you to read 'Working with commercial version' section of ! ALGLIB Reference Manual in order to find out how to use performance- ! related features provided by commercial edition of ALGLIB. INPUT PARAMETERS: A - upper or lower triangle of a factorized matrix. array with elements [0..N-1, 0..N-1]. N - size of matrix A. IsUpper - if IsUpper=True, then A contains an upper triangle of a symmetric matrix, otherwise A contains a lower one. OUTPUT PARAMETERS: A - the result of factorization. If IsUpper=True, then the upper triangle contains matrix U, so that A = U^T*U, and the elements below the main diagonal are not modified. Similarly, if IsUpper = False. RESULT: If the matrix is positive-definite, the function returns True. Otherwise, the function returns False. Contents of A is not determined in such case. -- ALGLIB routine -- 15.12.2009 Bochkanov Sergey *************************************************************************/bool alglib::spdmatrixcholesky( real_2d_array& a, ae_int_t n, bool isupper); bool alglib::smp_spdmatrixcholesky( real_2d_array& a, ae_int_t n, bool isupper);
spdmatrixcholeskyupdateadd1 function/************************************************************************* Update of Cholesky decomposition: rank-1 update to original A. "Buffered" version which uses preallocated buffer which is saved between subsequent function calls. This function uses internally allocated buffer which is not saved between subsequent calls. So, if you perform a lot of subsequent updates, we recommend you to use "buffered" version of this function: SPDMatrixCholeskyUpdateAdd1Buf(). INPUT PARAMETERS: A - upper or lower Cholesky factor. array with elements [0..N-1, 0..N-1]. Exception is thrown if array size is too small. N - size of matrix A, N>0 IsUpper - if IsUpper=True, then A contains upper Cholesky factor; otherwise A contains a lower one. U - array[N], rank-1 update to A: A_mod = A + u*u' Exception is thrown if array size is too small. BufR - possibly preallocated buffer; automatically resized if needed. It is recommended to reuse this buffer if you perform a lot of subsequent decompositions. OUTPUT PARAMETERS: A - updated factorization. If IsUpper=True, then the upper triangle contains matrix U, and the elements below the main diagonal are not modified. Similarly, if IsUpper = False. NOTE: this function always succeeds, so it does not return completion code NOTE: this function checks sizes of input arrays, but it does NOT checks for presence of infinities or NAN's. -- ALGLIB -- 03.02.2014 Sergey Bochkanov *************************************************************************/void alglib::spdmatrixcholeskyupdateadd1( real_2d_array& a, ae_int_t n, bool isupper, real_1d_array u);
spdmatrixcholeskyupdateadd1buf function/************************************************************************* Update of Cholesky decomposition: rank-1 update to original A. "Buffered" version which uses preallocated buffer which is saved between subsequent function calls. See comments for SPDMatrixCholeskyUpdateAdd1() for more information. INPUT PARAMETERS: A - upper or lower Cholesky factor. array with elements [0..N-1, 0..N-1]. Exception is thrown if array size is too small. N - size of matrix A, N>0 IsUpper - if IsUpper=True, then A contains upper Cholesky factor; otherwise A contains a lower one. U - array[N], rank-1 update to A: A_mod = A + u*u' Exception is thrown if array size is too small. BufR - possibly preallocated buffer; automatically resized if needed. It is recommended to reuse this buffer if you perform a lot of subsequent decompositions. OUTPUT PARAMETERS: A - updated factorization. If IsUpper=True, then the upper triangle contains matrix U, and the elements below the main diagonal are not modified. Similarly, if IsUpper = False. -- ALGLIB -- 03.02.2014 Sergey Bochkanov *************************************************************************/void alglib::spdmatrixcholeskyupdateadd1buf( real_2d_array& a, ae_int_t n, bool isupper, real_1d_array u, real_1d_array& bufr);
spdmatrixcholeskyupdatefix function/************************************************************************* Update of Cholesky decomposition: "fixing" some variables. This function uses internally allocated buffer which is not saved between subsequent calls. So, if you perform a lot of subsequent updates, we recommend you to use "buffered" version of this function: SPDMatrixCholeskyUpdateFixBuf(). "FIXING" EXPLAINED: Suppose we have N*N positive definite matrix A. "Fixing" some variable means filling corresponding row/column of A by zeros, and setting diagonal element to 1. For example, if we fix 2nd variable in 4*4 matrix A, it becomes Af: ( A00 A01 A02 A03 ) ( Af00 0 Af02 Af03 ) ( A10 A11 A12 A13 ) ( 0 1 0 0 ) ( A20 A21 A22 A23 ) => ( Af20 0 Af22 Af23 ) ( A30 A31 A32 A33 ) ( Af30 0 Af32 Af33 ) If we have Cholesky decomposition of A, it must be recalculated after variables were fixed. However, it is possible to use efficient algorithm, which needs O(K*N^2) time to "fix" K variables, given Cholesky decomposition of original, "unfixed" A. INPUT PARAMETERS: A - upper or lower Cholesky factor. array with elements [0..N-1, 0..N-1]. Exception is thrown if array size is too small. N - size of matrix A, N>0 IsUpper - if IsUpper=True, then A contains upper Cholesky factor; otherwise A contains a lower one. Fix - array[N], I-th element is True if I-th variable must be fixed. Exception is thrown if array size is too small. BufR - possibly preallocated buffer; automatically resized if needed. It is recommended to reuse this buffer if you perform a lot of subsequent decompositions. OUTPUT PARAMETERS: A - updated factorization. If IsUpper=True, then the upper triangle contains matrix U, and the elements below the main diagonal are not modified. Similarly, if IsUpper = False. NOTE: this function always succeeds, so it does not return completion code NOTE: this function checks sizes of input arrays, but it does NOT checks for presence of infinities or NAN's. NOTE: this function is efficient only for moderate amount of updated variables - say, 0.1*N or 0.3*N. For larger amount of variables it will still work, but you may get better performance with straightforward Cholesky. -- ALGLIB -- 03.02.2014 Sergey Bochkanov *************************************************************************/void alglib::spdmatrixcholeskyupdatefix( real_2d_array& a, ae_int_t n, bool isupper, boolean_1d_array fix);
spdmatrixcholeskyupdatefixbuf function/************************************************************************* Update of Cholesky decomposition: "fixing" some variables. "Buffered" version which uses preallocated buffer which is saved between subsequent function calls. See comments for SPDMatrixCholeskyUpdateFix() for more information. INPUT PARAMETERS: A - upper or lower Cholesky factor. array with elements [0..N-1, 0..N-1]. Exception is thrown if array size is too small. N - size of matrix A, N>0 IsUpper - if IsUpper=True, then A contains upper Cholesky factor; otherwise A contains a lower one. Fix - array[N], I-th element is True if I-th variable must be fixed. Exception is thrown if array size is too small. BufR - possibly preallocated buffer; automatically resized if needed. It is recommended to reuse this buffer if you perform a lot of subsequent decompositions. OUTPUT PARAMETERS: A - updated factorization. If IsUpper=True, then the upper triangle contains matrix U, and the elements below the main diagonal are not modified. Similarly, if IsUpper = False. -- ALGLIB -- 03.02.2014 Sergey Bochkanov *************************************************************************/void alglib::spdmatrixcholeskyupdatefixbuf( real_2d_array& a, ae_int_t n, bool isupper, boolean_1d_array fix, real_1d_array& bufr);
trigintegrals subpackagehyperbolicsinecosineintegrals function/************************************************************************* Hyperbolic sine and cosine integrals Approximates the integrals x - | | cosh t - 1 Chi(x) = eul + ln x + | ----------- dt, | | t - 0 x - | | sinh t Shi(x) = | ------ dt | | t - 0 where eul = 0.57721566490153286061 is Euler's constant. The integrals are evaluated by power series for x < 8 and by Chebyshev expansions for x between 8 and 88. For large x, both functions approach exp(x)/2x. Arguments greater than 88 in magnitude return MAXNUM. ACCURACY: Test interval 0 to 88. Relative error: arithmetic function # trials peak rms IEEE Shi 30000 6.9e-16 1.6e-16 Absolute error, except relative when |Chi| > 1: IEEE Chi 30000 8.4e-16 1.4e-16 Cephes Math Library Release 2.8: June, 2000 Copyright 1984, 1987, 2000 by Stephen L. Moshier *************************************************************************/void alglib::hyperbolicsinecosineintegrals( double x, double& shi, double& chi);
sinecosineintegrals function/************************************************************************* Sine and cosine integrals Evaluates the integrals x - | cos t - 1 Ci(x) = eul + ln x + | --------- dt, | t - 0 x - | sin t Si(x) = | ----- dt | t - 0 where eul = 0.57721566490153286061 is Euler's constant. The integrals are approximated by rational functions. For x > 8 auxiliary functions f(x) and g(x) are employed such that Ci(x) = f(x) sin(x) - g(x) cos(x) Si(x) = pi/2 - f(x) cos(x) - g(x) sin(x) ACCURACY: Test interval = [0,50]. Absolute error, except relative when > 1: arithmetic function # trials peak rms IEEE Si 30000 4.4e-16 7.3e-17 IEEE Ci 30000 6.9e-16 5.1e-17 Cephes Math Library Release 2.1: January, 1989 Copyright 1984, 1987, 1989 by Stephen L. Moshier *************************************************************************/void alglib::sinecosineintegrals(double x, double& si, double& ci);
variancetests subpackageftest function/************************************************************************* Two-sample F-test This test checks three hypotheses about dispersions of the given samples. The following tests are performed: * two-tailed test (null hypothesis - the dispersions are equal) * left-tailed test (null hypothesis - the dispersion of the first sample is greater than or equal to the dispersion of the second sample). * right-tailed test (null hypothesis - the dispersion of the first sample is less than or equal to the dispersion of the second sample) The test is based on the following assumptions: * the given samples have normal distributions * the samples are independent. Input parameters: X - sample 1. Array whose index goes from 0 to N-1. N - sample size. Y - sample 2. Array whose index goes from 0 to M-1. M - sample size. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. -- ALGLIB -- Copyright 19.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::ftest( real_1d_array x, ae_int_t n, real_1d_array y, ae_int_t m, double& bothtails, double& lefttail, double& righttail);
onesamplevariancetest function/************************************************************************* One-sample chi-square test This test checks three hypotheses about the dispersion of the given sample The following tests are performed: * two-tailed test (null hypothesis - the dispersion equals the given number) * left-tailed test (null hypothesis - the dispersion is greater than or equal to the given number) * right-tailed test (null hypothesis - dispersion is less than or equal to the given number). Test is based on the following assumptions: * the given sample has a normal distribution. Input parameters: X - sample 1. Array whose index goes from 0 to N-1. N - size of the sample. Variance - dispersion value to compare with. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. -- ALGLIB -- Copyright 19.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::onesamplevariancetest( real_1d_array x, ae_int_t n, double variance, double& bothtails, double& lefttail, double& righttail);
wsr subpackagewilcoxonsignedranktest function/************************************************************************* Wilcoxon signed-rank test This test checks three hypotheses about the median of the given sample. The following tests are performed: * two-tailed test (null hypothesis - the median is equal to the given value) * left-tailed test (null hypothesis - the median is greater than or equal to the given value) * right-tailed test (null hypothesis - the median is less than or equal to the given value) Requirements: * the scale of measurement should be ordinal, interval or ratio (i.e. the test could not be applied to nominal variables). * the distribution should be continuous and symmetric relative to its median. * number of distinct values in the X array should be greater than 4 The test is non-parametric and doesn't require distribution X to be normal Input parameters: X - sample. Array whose index goes from 0 to N-1. N - size of the sample. Median - assumed median value. Output parameters: BothTails - p-value for two-tailed test. If BothTails is less than the given significance level the null hypothesis is rejected. LeftTail - p-value for left-tailed test. If LeftTail is less than the given significance level, the null hypothesis is rejected. RightTail - p-value for right-tailed test. If RightTail is less than the given significance level the null hypothesis is rejected. To calculate p-values, special approximation is used. This method lets us calculate p-values with two decimal places in interval [0.0001, 1]. "Two decimal places" does not sound very impressive, but in practice the relative error of less than 1% is enough to make a decision. There is no approximation outside the [0.0001, 1] interval. Therefore, if the significance level outlies this interval, the test returns 0.0001. -- ALGLIB -- Copyright 08.09.2006 by Bochkanov Sergey *************************************************************************/void alglib::wilcoxonsignedranktest( real_1d_array x, ae_int_t n, double e, double& bothtails, double& lefttail, double& righttail);
xdebug subpackagexdebugrecord1 class/************************************************************************* *************************************************************************/class xdebugrecord1 { ae_int_t i; alglib::complex c; real_1d_array a; };
xdebugb1appendcopy function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Appends copy of array to itself. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugb1appendcopy(boolean_1d_array& a);
xdebugb1count function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Counts number of True values in the boolean 1D array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::xdebugb1count(boolean_1d_array a);
xdebugb1not function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by NOT(a[i]). Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugb1not(boolean_1d_array& a);
xdebugb1outeven function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate N-element array with even-numbered elements set to True. Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugb1outeven(ae_int_t n, boolean_1d_array& a);
xdebugb2count function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Counts number of True values in the boolean 2D array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::xdebugb2count(boolean_2d_array a);
xdebugb2not function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by NOT(a[i]). Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugb2not(boolean_2d_array& a);
xdebugb2outsin function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate MxN matrix with elements set to "Sin(3*I+5*J)>0" Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugb2outsin(ae_int_t m, ae_int_t n, boolean_2d_array& a);
xdebugb2transpose function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Transposes array. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugb2transpose(boolean_2d_array& a);
xdebugc1appendcopy function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Appends copy of array to itself. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugc1appendcopy(complex_1d_array& a);
xdebugc1neg function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by -A[I] Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugc1neg(complex_1d_array& a);
xdebugc1outeven function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate N-element array with even-numbered A[K] set to (x,y) = (K*0.25, K*0.125) and odd-numbered ones are set to 0. Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugc1outeven(ae_int_t n, complex_1d_array& a);
xdebugc1sum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of elements in the array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/alglib::complex alglib::xdebugc1sum(complex_1d_array a);
xdebugc2neg function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by -a[i,j] Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugc2neg(complex_2d_array& a);
xdebugc2outsincos function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate MxN matrix with elements set to "Sin(3*I+5*J),Cos(3*I+5*J)" Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugc2outsincos( ae_int_t m, ae_int_t n, complex_2d_array& a);
xdebugc2sum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of elements in the array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/alglib::complex alglib::xdebugc2sum(complex_2d_array a);
xdebugc2transpose function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Transposes array. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugc2transpose(complex_2d_array& a);
xdebugi1appendcopy function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Appends copy of array to itself. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugi1appendcopy(integer_1d_array& a);
xdebugi1neg function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by -A[I] Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugi1neg(integer_1d_array& a);
xdebugi1outeven function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate N-element array with even-numbered A[I] set to I, and odd-numbered ones set to 0. Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugi1outeven(ae_int_t n, integer_1d_array& a);
xdebugi1sum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of elements in the array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::xdebugi1sum(integer_1d_array a);
xdebugi2neg function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by -a[i,j] Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugi2neg(integer_2d_array& a);
xdebugi2outsin function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate MxN matrix with elements set to "Sign(Sin(3*I+5*J))" Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugi2outsin(ae_int_t m, ae_int_t n, integer_2d_array& a);
xdebugi2sum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of elements in the array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/ae_int_t alglib::xdebugi2sum(integer_2d_array a);
xdebugi2transpose function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Transposes array. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugi2transpose(integer_2d_array& a);
xdebuginitrecord1 function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Creates and returns XDebugRecord1 structure: * integer and complex fields of Rec1 are set to 1 and 1+i correspondingly * array field of Rec1 is set to [2,3] -- ALGLIB -- Copyright 27.05.2014 by Bochkanov Sergey *************************************************************************/void alglib::xdebuginitrecord1(xdebugrecord1& rec1);
xdebugmaskedbiasedproductsum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of a[i,j]*(1+b[i,j]) such that c[i,j] is True -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/double alglib::xdebugmaskedbiasedproductsum( ae_int_t m, ae_int_t n, real_2d_array a, real_2d_array b, boolean_2d_array c);
xdebugr1appendcopy function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Appends copy of array to itself. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugr1appendcopy(real_1d_array& a);
xdebugr1neg function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by -A[I] Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugr1neg(real_1d_array& a);
xdebugr1outeven function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate N-element array with even-numbered A[I] set to I*0.25, and odd-numbered ones are set to 0. Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugr1outeven(ae_int_t n, real_1d_array& a);
xdebugr1sum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of elements in the array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/double alglib::xdebugr1sum(real_1d_array a);
xdebugr2neg function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Replace all values in array by -a[i,j] Array is passed using "shared" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugr2neg(real_2d_array& a);
xdebugr2outsin function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Generate MxN matrix with elements set to "Sin(3*I+5*J)" Array is passed using "out" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugr2outsin(ae_int_t m, ae_int_t n, real_2d_array& a);
xdebugr2sum function/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Returns sum of elements in the array. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/double alglib::xdebugr2sum(real_2d_array a);
xdebugr2transpose functioncpp/src/ 0000755 0001750 0001750 00000000000 13105126766 012014 5 ustar sergey sergey cpp/src/solvers.cpp 0000755 0001750 0001750 00001721512 13105126765 014230 0 ustar sergey sergey /************************************************************************* ALGLIB 3.11.0 (source code generated 2017-05-11) Copyright (c) Sergey Bochkanov (ALGLIB project). >>> SOURCE LICENSE >>> This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation (www.fsf.org); either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. A copy of the GNU General Public License is available at http://www.fsf.org/licensing/licenses >>> END OF LICENSE >>> *************************************************************************/ #include "stdafx.h" #include "solvers.h" // disable some irrelevant warnings #if (AE_COMPILER==AE_MSVC) #pragma warning(disable:4100) #pragma warning(disable:4127) #pragma warning(disable:4702) #pragma warning(disable:4996) #endif using namespace std; ///////////////////////////////////////////////////////////////////////// // // THIS SECTION CONTAINS IMPLEMENTATION OF C++ INTERFACE // ///////////////////////////////////////////////////////////////////////// namespace alglib { /************************************************************************* *************************************************************************/ _densesolverreport_owner::_densesolverreport_owner() { p_struct = (alglib_impl::densesolverreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::densesolverreport), NULL); if( p_struct==NULL ) throw ap_error("ALGLIB: malloc error"); alglib_impl::_densesolverreport_init(p_struct, NULL); } _densesolverreport_owner::_densesolverreport_owner(const _densesolverreport_owner &rhs) { p_struct = (alglib_impl::densesolverreport*)alglib_impl::ae_malloc(sizeof(alglib_impl::densesolverreport), NULL); if( p_struct==NULL ) throw ap_error("ALGLIB: malloc error"); alglib_impl::_densesolverreport_init_copy(p_struct, const_cast/************************************************************************* This is debug function intended for testing ALGLIB interface generator. Never use it in any real life project. Transposes array. Array is passed using "var" convention. -- ALGLIB -- Copyright 11.10.2013 by Bochkanov Sergey *************************************************************************/void alglib::xdebugr2transpose(real_2d_array& a);