GSL
***
This file documents the GNU Scientific Library (GSL), a collection of
numerical routines for scientific computing. It corresponds to release
2.3 of the library. Please report any errors in this manual to
.
More information about GSL can be found at the project homepage,
.
Printed copies of this manual can be purchased from Network Theory
Ltd at . The money raised
from sales of the manual helps support the development of GSL.
A Japanese translation of this manual is available from the GSL
project homepage thanks to Daisuke Tominaga.
Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004,
2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016
The GSL Team.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.3 or
any later version published by the Free Software Foundation; with the
Invariant Sections being "GNU General Public License" and "Free Software
Needs Free Documentation", the Front-Cover text being "A GNU Manual",
and with the Back-Cover Text being (a) (see below). A copy of the
license is included in the section entitled "GNU Free Documentation
License".
(a) The Back-Cover Text is: "You have the freedom to copy and modify
this GNU Manual."
1 Introduction
**************
The GNU Scientific Library (GSL) is a collection of routines for
numerical computing. The routines have been written from scratch in C,
and present a modern Applications Programming Interface (API) for C
programmers, allowing wrappers to be written for very high level
languages. The source code is distributed under the GNU General Public
License.
1.1 Routines available in GSL
=============================
The library covers a wide range of topics in numerical computing.
Routines are available for the following areas,
Complex Numbers Roots of Polynomials
Special Functions Vectors and Matrices
Permutations Combinations
Sorting BLAS Support
Linear Algebra CBLAS Library
Fast Fourier Transforms Eigensystems
Random Numbers Quadrature
Random Distributions Quasi-Random Sequences
Histograms Statistics
Monte Carlo Integration N-Tuples
Differential Equations Simulated Annealing
Numerical Differentiation Interpolation
Series Acceleration Chebyshev Approximations
Root-Finding Discrete Hankel Transforms
Least-Squares Fitting Minimization
IEEE Floating-Point Physical Constants
Basis Splines Wavelets
The use of these routines is described in this manual. Each chapter
provides detailed definitions of the functions, followed by example
programs and references to the articles on which the algorithms are
based.
Where possible the routines have been based on reliable public-domain
packages such as FFTPACK and QUADPACK, which the developers of GSL have
reimplemented in C with modern coding conventions.
1.2 GSL is Free Software
========================
The subroutines in the GNU Scientific Library are "free software"; this
means that everyone is free to use them, and to redistribute them in
other free programs. The library is not in the public domain; it is
copyrighted and there are conditions on its distribution. These
conditions are designed to permit everything that a good cooperating
citizen would want to do. What is not allowed is to try to prevent
others from further sharing any version of the software that they might
get from you.
Specifically, we want to make sure that you have the right to share
copies of programs that you are given which use the GNU Scientific
Library, that you receive their source code or else can get it if you
want it, that you can change these programs or use pieces of them in new
free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to
deprive anyone else of these rights. For example, if you distribute
copies of any code which uses the GNU Scientific Library, you must give
the recipients all the rights that you have received. You must make
sure that they, too, receive or can get the source code, both to the
library and the code which uses it. And you must tell them their
rights. This means that the library should not be redistributed in
proprietary programs.
Also, for our own protection, we must make certain that everyone
finds out that there is no warranty for the GNU Scientific Library. If
these programs are modified by someone else and passed on, we want their
recipients to know that what they have is not what we distributed, so
that any problems introduced by others will not reflect on our
reputation.
The precise conditions for the distribution of software related to
the GNU Scientific Library are found in the GNU General Public License
(*note GNU General Public License::). Further information about this
license is available from the GNU Project webpage 'Frequently Asked
Questions about the GNU GPL',
The Free Software Foundation also operates a license consulting service
for commercial users (contact details available from
).
1.3 Obtaining GSL
=================
The source code for the library can be obtained in different ways, by
copying it from a friend, purchasing it on CDROM or downloading it from
the internet. A list of public ftp servers which carry the source code
can be found on the GNU website,
The preferred platform for the library is a GNU system, which allows it
to take advantage of additional features in the GNU C compiler and GNU C
library. However, the library is fully portable and should compile on
most systems with a C compiler.
Announcements of new releases, updates and other relevant events are
made on the 'info-gsl@gnu.org' mailing list. To subscribe to this
low-volume list, send an email of the following form:
To: info-gsl-request@gnu.org
Subject: subscribe
You will receive a response asking you to reply in order to confirm your
subscription.
1.4 No Warranty
===============
The software described in this manual has no warranty, it is provided
"as is". It is your responsibility to validate the behavior of the
routines and their accuracy using the source code provided, or to
purchase support and warranties from commercial redistributors. Consult
the GNU General Public license for further details (*note GNU General
Public License::).
1.5 Reporting Bugs
==================
A list of known bugs can be found in the 'BUGS' file included in the GSL
distribution or online in the GSL bug tracker.(1) Details of
compilation problems can be found in the 'INSTALL' file.
If you find a bug which is not listed in these files, please report
it to .
All bug reports should include:
* The version number of GSL
* The hardware and operating system
* The compiler used, including version number and compilation options
* A description of the bug behavior
* A short program which exercises the bug
It is useful if you can check whether the same problem occurs when the
library is compiled without optimization. Thank you.
Any errors or omissions in this manual can also be reported to the
same address.
---------- Footnotes ----------
(1)
1.6 Further Information
=======================
Additional information, including online copies of this manual, links to
related projects, and mailing list archives are available from the
website mentioned above.
Any questions about the use and installation of the library can be
asked on the mailing list 'help-gsl@gnu.org'. To subscribe to this
list, send an email of the following form:
To: help-gsl-request@gnu.org
Subject: subscribe
This mailing list can be used to ask questions not covered by this
manual, and to contact the developers of the library.
If you would like to refer to the GNU Scientific Library in a journal
article, the recommended way is to cite this reference manual, e.g. 'M.
Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN
0954612078'.
If you want to give a url, use "".
1.7 Conventions used in this manual
===================================
This manual contains many examples which can be typed at the keyboard.
A command entered at the terminal is shown like this,
$ command
The first character on the line is the terminal prompt, and should not
be typed. The dollar sign '$' is used as the standard prompt in this
manual, although some systems may use a different character.
The examples assume the use of the GNU operating system. There may
be minor differences in the output on other systems. The commands for
setting environment variables use the Bourne shell syntax of the
standard GNU shell ('bash').
2 Using the library
*******************
This chapter describes how to compile programs that use GSL, and
introduces its conventions.
2.1 An Example Program
======================
The following short program demonstrates the use of the library by
computing the value of the Bessel function J_0(x) for x=5,
#include
#include
int
main (void)
{
double x = 5.0;
double y = gsl_sf_bessel_J0 (x);
printf ("J0(%g) = %.18e\n", x, y);
return 0;
}
The output is shown below, and should be correct to double-precision
accuracy,(1)
J0(5) = -1.775967713143382642e-01
The steps needed to compile this program are described in the following
sections.
---------- Footnotes ----------
(1) The last few digits may vary slightly depending on the compiler
and platform used--this is normal.
2.2 Compiling and Linking
=========================
The library header files are installed in their own 'gsl' directory.
You should write any preprocessor include statements with a 'gsl/'
directory prefix thus,
#include
If the directory is not installed on the standard search path of your
compiler you will also need to provide its location to the preprocessor
as a command line flag. The default location of the 'gsl' directory is
'/usr/local/include/gsl'. A typical compilation command for a source
file 'example.c' with the GNU C compiler 'gcc' is,
$ gcc -Wall -I/usr/local/include -c example.c
This results in an object file 'example.o'. The default include path
for 'gcc' searches '/usr/local/include' automatically so the '-I' option
can actually be omitted when GSL is installed in its default location.
2.2.1 Linking programs with the library
---------------------------------------
The library is installed as a single file, 'libgsl.a'. A shared version
of the library 'libgsl.so' is also installed on systems that support
shared libraries. The default location of these files is
'/usr/local/lib'. If this directory is not on the standard search path
of your linker you will also need to provide its location as a command
line flag.
To link against the library you need to specify both the main library
and a supporting CBLAS library, which provides standard basic linear
algebra subroutines. A suitable CBLAS implementation is provided in the
library 'libgslcblas.a' if your system does not provide one. The
following example shows how to link an application with the library,
$ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm
The default library path for 'gcc' searches '/usr/local/lib'
automatically so the '-L' option can be omitted when GSL is installed in
its default location.
The option '-lm' links with the system math library. On some systems
it is not needed.(1)
For a tutorial introduction to the GNU C Compiler and related
programs, see 'An Introduction to GCC' (ISBN 0954161793).(2)
---------- Footnotes ----------
(1) It is not needed on MacOS X.
(2)
2.2.2 Linking with an alternative BLAS library
----------------------------------------------
The following command line shows how you would link the same application
with an alternative CBLAS library 'libcblas.a',
$ gcc example.o -lgsl -lcblas -lm
For the best performance an optimized platform-specific CBLAS library
should be used for '-lcblas'. The library must conform to the CBLAS
standard. The ATLAS package provides a portable high-performance BLAS
library with a CBLAS interface. It is free software and should be
installed for any work requiring fast vector and matrix operations. The
following command line will link with the ATLAS library and its CBLAS
interface,
$ gcc example.o -lgsl -lcblas -latlas -lm
If the ATLAS library is installed in a non-standard directory use the
'-L' option to add it to the search path, as described above.
For more information about BLAS functions see *note BLAS Support::.
2.3 Shared Libraries
====================
To run a program linked with the shared version of the library the
operating system must be able to locate the corresponding '.so' file at
runtime. If the library cannot be found, the following error will
occur:
$ ./a.out
./a.out: error while loading shared libraries:
libgsl.so.0: cannot open shared object file: No such
file or directory
To avoid this error, either modify the system dynamic linker
configuration(1) or define the shell variable 'LD_LIBRARY_PATH' to
include the directory where the library is installed.
For example, in the Bourne shell ('/bin/sh' or '/bin/bash'), the
library search path can be set with the following commands:
$ LD_LIBRARY_PATH=/usr/local/lib
$ export LD_LIBRARY_PATH
$ ./example
In the C-shell ('/bin/csh' or '/bin/tcsh') the equivalent command is,
% setenv LD_LIBRARY_PATH /usr/local/lib
The standard prompt for the C-shell in the example above is the percent
character '%', and should not be typed as part of the command.
To save retyping these commands each session they can be placed in an
individual or system-wide login file.
To compile a statically linked version of the program, use the
'-static' flag in 'gcc',
$ gcc -static example.o -lgsl -lgslcblas -lm
---------- Footnotes ----------
(1) '/etc/ld.so.conf' on GNU/Linux systems.
2.4 ANSI C Compliance
=====================
The library is written in ANSI C and is intended to conform to the ANSI
C standard (C89). It should be portable to any system with a working
ANSI C compiler.
The library does not rely on any non-ANSI extensions in the interface
it exports to the user. Programs you write using GSL can be ANSI
compliant. Extensions which can be used in a way compatible with pure
ANSI C are supported, however, via conditional compilation. This allows
the library to take advantage of compiler extensions on those platforms
which support them.
When an ANSI C feature is known to be broken on a particular system
the library will exclude any related functions at compile-time. This
should make it impossible to link a program that would use these
functions and give incorrect results.
To avoid namespace conflicts all exported function names and
variables have the prefix 'gsl_', while exported macros have the prefix
'GSL_'.
2.5 Inline functions
====================
The 'inline' keyword is not part of the original ANSI C standard (C89)
so the library does not export any inline function definitions by
default. Inline functions were introduced officially in the newer C99
standard but most C89 compilers have also included 'inline' as an
extension for a long time.
To allow the use of inline functions, the library provides optional
inline versions of performance-critical routines by conditional
compilation in the exported header files. The inline versions of these
functions can be included by defining the macro 'HAVE_INLINE' when
compiling an application,
$ gcc -Wall -c -DHAVE_INLINE example.c
If you use 'autoconf' this macro can be defined automatically. If you
do not define the macro 'HAVE_INLINE' then the slower non-inlined
versions of the functions will be used instead.
By default, the actual form of the inline keyword is 'extern inline',
which is a 'gcc' extension that eliminates unnecessary function
definitions. If the form 'extern inline' causes problems with other
compilers a stricter autoconf test can be used, see *note Autoconf
Macros::.
When compiling with 'gcc' in C99 mode ('gcc -std=c99') the header
files automatically switch to C99-compatible inline function
declarations instead of 'extern inline'. With other C99 compilers,
define the macro 'GSL_C99_INLINE' to use these declarations.
2.6 Long double
===============
In general, the algorithms in the library are written for double
precision only. The 'long double' type is not supported for actual
computation.
One reason for this choice is that the precision of 'long double' is
platform dependent. The IEEE standard only specifies the minimum
precision of extended precision numbers, while the precision of 'double'
is the same on all platforms.
However, it is sometimes necessary to interact with external data in
long-double format, so the vector and matrix datatypes include
long-double versions.
It should be noted that in some system libraries the 'stdio.h'
formatted input/output functions 'printf' and 'scanf' are not
implemented correctly for 'long double'. Undefined or incorrect results
are avoided by testing these functions during the 'configure' stage of
library compilation and eliminating certain GSL functions which depend
on them if necessary. The corresponding line in the 'configure' output
looks like this,
checking whether printf works with long double... no
Consequently when 'long double' formatted input/output does not work on
a given system it should be impossible to link a program which uses GSL
functions dependent on this.
If it is necessary to work on a system which does not support
formatted 'long double' input/output then the options are to use binary
formats or to convert 'long double' results into 'double' for reading
and writing.
2.7 Portability functions
=========================
To help in writing portable applications GSL provides some
implementations of functions that are found in other libraries, such as
the BSD math library. You can write your application to use the native
versions of these functions, and substitute the GSL versions via a
preprocessor macro if they are unavailable on another platform.
For example, after determining whether the BSD function 'hypot' is
available you can include the following macro definitions in a file
'config.h' with your application,
/* Substitute gsl_hypot for missing system hypot */
#ifndef HAVE_HYPOT
#define hypot gsl_hypot
#endif
The application source files can then use the include command '#include
' to replace each occurrence of 'hypot' by 'gsl_hypot' when
'hypot' is not available. This substitution can be made automatically
if you use 'autoconf', see *note Autoconf Macros::.
In most circumstances the best strategy is to use the native versions
of these functions when available, and fall back to GSL versions
otherwise, since this allows your application to take advantage of any
platform-specific optimizations in the system library. This is the
strategy used within GSL itself.
2.8 Alternative optimized functions
===================================
The main implementation of some functions in the library will not be
optimal on all architectures. For example, there are several ways to
compute a Gaussian random variate and their relative speeds are
platform-dependent. In cases like this the library provides alternative
implementations of these functions with the same interface. If you
write your application using calls to the standard implementation you
can select an alternative version later via a preprocessor definition.
It is also possible to introduce your own optimized functions this way
while retaining portability. The following lines demonstrate the use of
a platform-dependent choice of methods for sampling from the Gaussian
distribution,
#ifdef SPARC
#define gsl_ran_gaussian gsl_ran_gaussian_ratio_method
#endif
#ifdef INTEL
#define gsl_ran_gaussian my_gaussian
#endif
These lines would be placed in the configuration header file 'config.h'
of the application, which should then be included by all the source
files. Note that the alternative implementations will not produce
bit-for-bit identical results, and in the case of random number
distributions will produce an entirely different stream of random
variates.
2.9 Support for different numeric types
=======================================
Many functions in the library are defined for different numeric types.
This feature is implemented by varying the name of the function with a
type-related modifier--a primitive form of C++ templates. The modifier
is inserted into the function name after the initial module prefix. The
following table shows the function names defined for all the numeric
types of an imaginary module 'gsl_foo' with function 'fn',
gsl_foo_fn double
gsl_foo_long_double_fn long double
gsl_foo_float_fn float
gsl_foo_long_fn long
gsl_foo_ulong_fn unsigned long
gsl_foo_int_fn int
gsl_foo_uint_fn unsigned int
gsl_foo_short_fn short
gsl_foo_ushort_fn unsigned short
gsl_foo_char_fn char
gsl_foo_uchar_fn unsigned char
The normal numeric precision 'double' is considered the default and does
not require a suffix. For example, the function 'gsl_stats_mean'
computes the mean of double precision numbers, while the function
'gsl_stats_int_mean' computes the mean of integers.
A corresponding scheme is used for library defined types, such as
'gsl_vector' and 'gsl_matrix'. In this case the modifier is appended to
the type name. For example, if a module defines a new type-dependent
struct or typedef 'gsl_foo' it is modified for other types in the
following way,
gsl_foo double
gsl_foo_long_double long double
gsl_foo_float float
gsl_foo_long long
gsl_foo_ulong unsigned long
gsl_foo_int int
gsl_foo_uint unsigned int
gsl_foo_short short
gsl_foo_ushort unsigned short
gsl_foo_char char
gsl_foo_uchar unsigned char
When a module contains type-dependent definitions the library provides
individual header files for each type. The filenames are modified as
shown in the below. For convenience the default header includes the
definitions for all the types. To include only the double precision
header file, or any other specific type, use its individual filename.
#include All types
#include double
#include long double
#include float
#include long
#include unsigned long
#include int
#include unsigned int
#include short
#include unsigned short
#include char
#include unsigned char
2.10 Compatibility with C++
===========================
The library header files automatically define functions to have 'extern
"C"' linkage when included in C++ programs. This allows the functions
to be called directly from C++.
To use C++ exception handling within user-defined functions passed to
the library as parameters, the library must be built with the additional
'CFLAGS' compilation option '-fexceptions'.
2.11 Aliasing of arrays
=======================
The library assumes that arrays, vectors and matrices passed as
modifiable arguments are not aliased and do not overlap with each other.
This removes the need for the library to handle overlapping memory
regions as a special case, and allows additional optimizations to be
used. If overlapping memory regions are passed as modifiable arguments
then the results of such functions will be undefined. If the arguments
will not be modified (for example, if a function prototype declares them
as 'const' arguments) then overlapping or aliased memory regions can be
safely used.
2.12 Thread-safety
==================
The library can be used in multi-threaded programs. All the functions
are thread-safe, in the sense that they do not use static variables.
Memory is always associated with objects and not with functions. For
functions which use "workspace" objects as temporary storage the
workspaces should be allocated on a per-thread basis. For functions
which use "table" objects as read-only memory the tables can be used by
multiple threads simultaneously. Table arguments are always declared
'const' in function prototypes, to indicate that they may be safely
accessed by different threads.
There are a small number of static global variables which are used to
control the overall behavior of the library (e.g. whether to use
range-checking, the function to call on fatal error, etc). These
variables are set directly by the user, so they should be initialized
once at program startup and not modified by different threads.
2.13 Deprecated Functions
=========================
From time to time, it may be necessary for the definitions of some
functions to be altered or removed from the library. In these
circumstances the functions will first be declared "deprecated" and then
removed from subsequent versions of the library. Functions that are
deprecated can be disabled in the current release by setting the
preprocessor definition 'GSL_DISABLE_DEPRECATED'. This allows existing
code to be tested for forwards compatibility.
2.14 Code Reuse
===============
Where possible the routines in the library have been written to avoid
dependencies between modules and files. This should make it possible to
extract individual functions for use in your own applications, without
needing to have the whole library installed. You may need to define
certain macros such as 'GSL_ERROR' and remove some '#include' statements
in order to compile the files as standalone units. Reuse of the library
code in this way is encouraged, subject to the terms of the GNU General
Public License.
3 Error Handling
****************
This chapter describes the way that GSL functions report and handle
errors. By examining the status information returned by every function
you can determine whether it succeeded or failed, and if it failed you
can find out what the precise cause of failure was. You can also define
your own error handling functions to modify the default behavior of the
library.
The functions described in this section are declared in the header
file 'gsl_errno.h'.
3.1 Error Reporting
===================
The library follows the thread-safe error reporting conventions of the
POSIX Threads library. Functions return a non-zero error code to
indicate an error and '0' to indicate success.
int status = gsl_function (...)
if (status) { /* an error occurred */
.....
/* status value specifies the type of error */
}
The routines report an error whenever they cannot perform the task
requested of them. For example, a root-finding function would return a
non-zero error code if could not converge to the requested accuracy, or
exceeded a limit on the number of iterations. Situations like this are
a normal occurrence when using any mathematical library and you should
check the return status of the functions that you call.
Whenever a routine reports an error the return value specifies the
type of error. The return value is analogous to the value of the
variable 'errno' in the C library. The caller can examine the return
code and decide what action to take, including ignoring the error if it
is not considered serious.
In addition to reporting errors by return codes the library also has
an error handler function 'gsl_error'. This function is called by other
library functions when they report an error, just before they return to
the caller. The default behavior of the error handler is to print a
message and abort the program,
gsl: file.c:67: ERROR: invalid argument supplied by user
Default GSL error handler invoked.
Aborted
The purpose of the 'gsl_error' handler is to provide a function where
a breakpoint can be set that will catch library errors when running
under the debugger. It is not intended for use in production programs,
which should handle any errors using the return codes.
3.2 Error Codes
===============
The error code numbers returned by library functions are defined in the
file 'gsl_errno.h'. They all have the prefix 'GSL_' and expand to
non-zero constant integer values. Error codes above 1024 are reserved
for applications, and are not used by the library. Many of the error
codes use the same base name as the corresponding error code in the C
library. Here are some of the most common error codes,
-- Macro: int GSL_EDOM
Domain error; used by mathematical functions when an argument value
does not fall into the domain over which the function is defined
(like EDOM in the C library)
-- Macro: int GSL_ERANGE
Range error; used by mathematical functions when the result value
is not representable because of overflow or underflow (like ERANGE
in the C library)
-- Macro: int GSL_ENOMEM
No memory available. The system cannot allocate more virtual
memory because its capacity is full (like ENOMEM in the C library).
This error is reported when a GSL routine encounters problems when
trying to allocate memory with 'malloc'.
-- Macro: int GSL_EINVAL
Invalid argument. This is used to indicate various kinds of
problems with passing the wrong argument to a library function
(like EINVAL in the C library).
The error codes can be converted into an error message using the
function 'gsl_strerror'.
-- Function: const char * gsl_strerror (const int GSL_ERRNO)
This function returns a pointer to a string describing the error
code GSL_ERRNO. For example,
printf ("error: %s\n", gsl_strerror (status));
would print an error message like 'error: output range error' for a
status value of 'GSL_ERANGE'.
3.3 Error Handlers
==================
The default behavior of the GSL error handler is to print a short
message and call 'abort'. When this default is in use programs will
stop with a core-dump whenever a library routine reports an error. This
is intended as a fail-safe default for programs which do not check the
return status of library routines (we don't encourage you to write
programs this way).
If you turn off the default error handler it is your responsibility
to check the return values of routines and handle them yourself. You
can also customize the error behavior by providing a new error handler.
For example, an alternative error handler could log all errors to a
file, ignore certain error conditions (such as underflows), or start the
debugger and attach it to the current process when an error occurs.
All GSL error handlers have the type 'gsl_error_handler_t', which is
defined in 'gsl_errno.h',
-- Data Type: gsl_error_handler_t
This is the type of GSL error handler functions. An error handler
will be passed four arguments which specify the reason for the
error (a string), the name of the source file in which it occurred
(also a string), the line number in that file (an integer) and the
error number (an integer). The source file and line number are set
at compile time using the '__FILE__' and '__LINE__' directives in
the preprocessor. An error handler function returns type 'void'.
Error handler functions should be defined like this,
void handler (const char * reason,
const char * file,
int line,
int gsl_errno)
To request the use of your own error handler you need to call the
function 'gsl_set_error_handler' which is also declared in
'gsl_errno.h',
-- Function: gsl_error_handler_t * gsl_set_error_handler
(gsl_error_handler_t * NEW_HANDLER)
This function sets a new error handler, NEW_HANDLER, for the GSL
library routines. The previous handler is returned (so that you
can restore it later). Note that the pointer to a user defined
error handler function is stored in a static variable, so there can
be only one error handler per program. This function should be not
be used in multi-threaded programs except to set up a program-wide
error handler from a master thread. The following example shows
how to set and restore a new error handler,
/* save original handler, install new handler */
old_handler = gsl_set_error_handler (&my_handler);
/* code uses new handler */
.....
/* restore original handler */
gsl_set_error_handler (old_handler);
To use the default behavior ('abort' on error) set the error
handler to 'NULL',
old_handler = gsl_set_error_handler (NULL);
-- Function: gsl_error_handler_t * gsl_set_error_handler_off ()
This function turns off the error handler by defining an error
handler which does nothing. This will cause the program to
continue after any error, so the return values from any library
routines must be checked. This is the recommended behavior for
production programs. The previous handler is returned (so that you
can restore it later).
The error behavior can be changed for specific applications by
recompiling the library with a customized definition of the 'GSL_ERROR'
macro in the file 'gsl_errno.h'.
3.4 Using GSL error reporting in your own functions
===================================================
If you are writing numerical functions in a program which also uses GSL
code you may find it convenient to adopt the same error reporting
conventions as in the library.
To report an error you need to call the function 'gsl_error' with a
string describing the error and then return an appropriate error code
from 'gsl_errno.h', or a special value, such as 'NaN'. For convenience
the file 'gsl_errno.h' defines two macros which carry out these steps:
-- Macro: GSL_ERROR (REASON, GSL_ERRNO)
This macro reports an error using the GSL conventions and returns a
status value of 'gsl_errno'. It expands to the following code
fragment,
gsl_error (reason, __FILE__, __LINE__, gsl_errno);
return gsl_errno;
The macro definition in 'gsl_errno.h' actually wraps the code in a
'do { ... } while (0)' block to prevent possible parsing problems.
Here is an example of how the macro could be used to report that a
routine did not achieve a requested tolerance. To report the error the
routine needs to return the error code 'GSL_ETOL'.
if (residual > tolerance)
{
GSL_ERROR("residual exceeds tolerance", GSL_ETOL);
}
-- Macro: GSL_ERROR_VAL (REASON, GSL_ERRNO, VALUE)
This macro is the same as 'GSL_ERROR' but returns a user-defined
value of VALUE instead of an error code. It can be used for
mathematical functions that return a floating point value.
The following example shows how to return a 'NaN' at a mathematical
singularity using the 'GSL_ERROR_VAL' macro,
if (x == 0)
{
GSL_ERROR_VAL("argument lies on singularity",
GSL_ERANGE, GSL_NAN);
}
3.5 Examples
============
Here is an example of some code which checks the return value of a
function where an error might be reported,
#include
#include
#include
...
int status;
size_t n = 37;
gsl_set_error_handler_off();
status = gsl_fft_complex_radix2_forward (data, stride, n);
if (status) {
if (status == GSL_EINVAL) {
fprintf (stderr, "invalid argument, n=%d\n", n);
} else {
fprintf (stderr, "failed, gsl_errno=%d\n",
status);
}
exit (-1);
}
...
The function 'gsl_fft_complex_radix2' only accepts integer lengths which
are a power of two. If the variable 'n' is not a power of two then the
call to the library function will return 'GSL_EINVAL', indicating that
the length argument is invalid. The function call to
'gsl_set_error_handler_off' stops the default error handler from
aborting the program. The 'else' clause catches any other possible
errors.
4 Mathematical Functions
************************
This chapter describes basic mathematical functions. Some of these
functions are present in system libraries, but the alternative versions
given here can be used as a substitute when the system functions are not
available.
The functions and macros described in this chapter are defined in the
header file 'gsl_math.h'.
4.1 Mathematical Constants
==========================
The library ensures that the standard BSD mathematical constants are
defined. For reference, here is a list of the constants:
'M_E'
The base of exponentials, e
'M_LOG2E'
The base-2 logarithm of e, \log_2 (e)
'M_LOG10E'
The base-10 logarithm of e, \log_10 (e)
'M_SQRT2'
The square root of two, \sqrt 2
'M_SQRT1_2'
The square root of one-half, \sqrt{1/2}
'M_SQRT3'
The square root of three, \sqrt 3
'M_PI'
The constant pi, \pi
'M_PI_2'
Pi divided by two, \pi/2
'M_PI_4'
Pi divided by four, \pi/4
'M_SQRTPI'
The square root of pi, \sqrt\pi
'M_2_SQRTPI'
Two divided by the square root of pi, 2/\sqrt\pi
'M_1_PI'
The reciprocal of pi, 1/\pi
'M_2_PI'
Twice the reciprocal of pi, 2/\pi
'M_LN10'
The natural logarithm of ten, \ln(10)
'M_LN2'
The natural logarithm of two, \ln(2)
'M_LNPI'
The natural logarithm of pi, \ln(\pi)
'M_EULER'
Euler's constant, \gamma
4.2 Infinities and Not-a-number
===============================
-- Macro: GSL_POSINF
This macro contains the IEEE representation of positive infinity,
+\infty. It is computed from the expression '+1.0/0.0'.
-- Macro: GSL_NEGINF
This macro contains the IEEE representation of negative infinity,
-\infty. It is computed from the expression '-1.0/0.0'.
-- Macro: GSL_NAN
This macro contains the IEEE representation of the Not-a-Number
symbol, 'NaN'. It is computed from the ratio '0.0/0.0'.
-- Function: int gsl_isnan (const double X)
This function returns 1 if X is not-a-number.
-- Function: int gsl_isinf (const double X)
This function returns +1 if X is positive infinity, -1 if X is
negative infinity and 0 otherwise.(1)
-- Function: int gsl_finite (const double X)
This function returns 1 if X is a real number, and 0 if it is
infinite or not-a-number.
---------- Footnotes ----------
(1) Note that the C99 standard only requires the system 'isinf'
function to return a non-zero value, without the sign of the infinity.
The implementation in some earlier versions of GSL used the system
'isinf' function and may have this behavior on some platforms.
Therefore, it is advisable to test the sign of X separately, if needed,
rather than relying the sign of the return value from 'gsl_isinf()'.
4.3 Elementary Functions
========================
The following routines provide portable implementations of functions
found in the BSD math library. When native versions are not available
the functions described here can be used instead. The substitution can
be made automatically if you use 'autoconf' to compile your application
(*note Portability functions::).
-- Function: double gsl_log1p (const double X)
This function computes the value of \log(1+x) in a way that is
accurate for small X. It provides an alternative to the BSD math
function 'log1p(x)'.
-- Function: double gsl_expm1 (const double X)
This function computes the value of \exp(x)-1 in a way that is
accurate for small X. It provides an alternative to the BSD math
function 'expm1(x)'.
-- Function: double gsl_hypot (const double X, const double Y)
This function computes the value of \sqrt{x^2 + y^2} in a way that
avoids overflow. It provides an alternative to the BSD math
function 'hypot(x,y)'.
-- Function: double gsl_hypot3 (const double X, const double Y, const
double Z)
This function computes the value of \sqrt{x^2 + y^2 + z^2} in a way
that avoids overflow.
-- Function: double gsl_acosh (const double X)
This function computes the value of \arccosh(x). It provides an
alternative to the standard math function 'acosh(x)'.
-- Function: double gsl_asinh (const double X)
This function computes the value of \arcsinh(x). It provides an
alternative to the standard math function 'asinh(x)'.
-- Function: double gsl_atanh (const double X)
This function computes the value of \arctanh(x). It provides an
alternative to the standard math function 'atanh(x)'.
-- Function: double gsl_ldexp (double X, int E)
This function computes the value of x * 2^e. It provides an
alternative to the standard math function 'ldexp(x,e)'.
-- Function: double gsl_frexp (double X, int * E)
This function splits the number x into its normalized fraction f
and exponent e, such that x = f * 2^e and 0.5 <= f < 1. The
function returns f and stores the exponent in e. If x is zero,
both f and e are set to zero. This function provides an
alternative to the standard math function 'frexp(x, e)'.
4.4 Small integer powers
========================
A common complaint about the standard C library is its lack of a
function for calculating (small) integer powers. GSL provides some
simple functions to fill this gap. For reasons of efficiency, these
functions do not check for overflow or underflow conditions.
-- Function: double gsl_pow_int (double X, int N)
-- Function: double gsl_pow_uint (double X, unsigned int N)
These routines computes the power x^n for integer N. The power is
computed efficiently--for example, x^8 is computed as ((x^2)^2)^2,
requiring only 3 multiplications. A version of this function which
also computes the numerical error in the result is available as
'gsl_sf_pow_int_e'.
-- Function: double gsl_pow_2 (const double X)
-- Function: double gsl_pow_3 (const double X)
-- Function: double gsl_pow_4 (const double X)
-- Function: double gsl_pow_5 (const double X)
-- Function: double gsl_pow_6 (const double X)
-- Function: double gsl_pow_7 (const double X)
-- Function: double gsl_pow_8 (const double X)
-- Function: double gsl_pow_9 (const double X)
These functions can be used to compute small integer powers x^2,
x^3, etc. efficiently. The functions will be inlined when
'HAVE_INLINE' is defined, so that use of these functions should be
as efficient as explicitly writing the corresponding product
expression.
#include
double y = gsl_pow_4 (3.141) /* compute 3.141**4 */
4.5 Testing the Sign of Numbers
===============================
-- Macro: GSL_SIGN (x)
This macro returns the sign of X. It is defined as '((x) >= 0 ? 1
: -1)'. Note that with this definition the sign of zero is
positive (regardless of its IEEE sign bit).
4.6 Testing for Odd and Even Numbers
====================================
-- Macro: GSL_IS_ODD (n)
This macro evaluates to 1 if N is odd and 0 if N is even. The
argument N must be of integer type.
-- Macro: GSL_IS_EVEN (n)
This macro is the opposite of 'GSL_IS_ODD(n)'. It evaluates to 1
if N is even and 0 if N is odd. The argument N must be of integer
type.
4.7 Maximum and Minimum functions
=================================
Note that the following macros perform multiple evaluations of their
arguments, so they should not be used with arguments that have side
effects (such as a call to a random number generator).
-- Macro: GSL_MAX (a, b)
This macro returns the maximum of A and B. It is defined as '((a)
> (b) ? (a):(b))'.
-- Macro: GSL_MIN (a, b)
This macro returns the minimum of A and B. It is defined as '((a)
< (b) ? (a):(b))'.
-- Function: extern inline double GSL_MAX_DBL (double A, double B)
This function returns the maximum of the double precision numbers A
and B using an inline function. The use of a function allows for
type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
'GSL_MAX' will be automatically substituted.
-- Function: extern inline double GSL_MIN_DBL (double A, double B)
This function returns the minimum of the double precision numbers A
and B using an inline function. The use of a function allows for
type checking of the arguments as an extra safety feature. On
platforms where inline functions are not available the macro
'GSL_MIN' will be automatically substituted.
-- Function: extern inline int GSL_MAX_INT (int A, int B)
-- Function: extern inline int GSL_MIN_INT (int A, int B)
These functions return the maximum or minimum of the integers A and
B using an inline function. On platforms where inline functions
are not available the macros 'GSL_MAX' or 'GSL_MIN' will be
automatically substituted.
-- Function: extern inline long double GSL_MAX_LDBL (long double A,
long double B)
-- Function: extern inline long double GSL_MIN_LDBL (long double A,
long double B)
These functions return the maximum or minimum of the long doubles A
and B using an inline function. On platforms where inline
functions are not available the macros 'GSL_MAX' or 'GSL_MIN' will
be automatically substituted.
4.8 Approximate Comparison of Floating Point Numbers
====================================================
It is sometimes useful to be able to compare two floating point numbers
approximately, to allow for rounding and truncation errors. The
following function implements the approximate floating-point comparison
algorithm proposed by D.E. Knuth in Section 4.2.2 of 'Seminumerical
Algorithms' (3rd edition).
-- Function: int gsl_fcmp (double X, double Y, double EPSILON)
This function determines whether x and y are approximately equal to
a relative accuracy EPSILON.
The relative accuracy is measured using an interval of size 2
\delta, where \delta = 2^k \epsilon and k is the maximum base-2
exponent of x and y as computed by the function 'frexp'.
If x and y lie within this interval, they are considered
approximately equal and the function returns 0. Otherwise if x <
y, the function returns -1, or if x > y, the function returns +1.
Note that x and y are compared to relative accuracy, so this
function is not suitable for testing whether a value is
approximately zero.
The implementation is based on the package 'fcmp' by T.C. Belding.
5 Complex Numbers
*****************
The functions described in this chapter provide support for complex
numbers. The algorithms take care to avoid unnecessary intermediate
underflows and overflows, allowing the functions to be evaluated over as
much of the complex plane as possible.
For multiple-valued functions the branch cuts have been chosen to
follow the conventions of Abramowitz and Stegun in the 'Handbook of
Mathematical Functions'. The functions return principal values which
are the same as those in GNU Calc, which in turn are the same as those
in 'Common Lisp, The Language (Second Edition)'(1) and the HP-28/48
series of calculators.
The complex types are defined in the header file 'gsl_complex.h',
while the corresponding complex functions and arithmetic operations are
defined in 'gsl_complex_math.h'.
---------- Footnotes ----------
(1) Note that the first edition uses different definitions.
5.1 Representation of complex numbers
=====================================
Complex numbers are represented using the type 'gsl_complex'. The
internal representation of this type may vary across platforms and
should not be accessed directly. The functions and macros described
below allow complex numbers to be manipulated in a portable way.
For reference, the default form of the 'gsl_complex' type is given by
the following struct,
typedef struct
{
double dat[2];
} gsl_complex;
The real and imaginary part are stored in contiguous elements of a two
element array. This eliminates any padding between the real and
imaginary parts, 'dat[0]' and 'dat[1]', allowing the struct to be mapped
correctly onto packed complex arrays.
-- Function: gsl_complex gsl_complex_rect (double X, double Y)
This function uses the rectangular Cartesian components (X,Y) to
return the complex number z = x + i y. An inline version of this
function is used when 'HAVE_INLINE' is defined.
-- Function: gsl_complex gsl_complex_polar (double R, double THETA)
This function returns the complex number z = r \exp(i \theta) = r
(\cos(\theta) + i \sin(\theta)) from the polar representation
(R,THETA).
-- Macro: GSL_REAL (Z)
-- Macro: GSL_IMAG (Z)
These macros return the real and imaginary parts of the complex
number Z.
-- Macro: GSL_SET_COMPLEX (ZP, X, Y)
This macro uses the Cartesian components (X,Y) to set the real and
imaginary parts of the complex number pointed to by ZP. For
example,
GSL_SET_COMPLEX(&z, 3, 4)
sets Z to be 3 + 4i.
-- Macro: GSL_SET_REAL (ZP,X)
-- Macro: GSL_SET_IMAG (ZP,Y)
These macros allow the real and imaginary parts of the complex
number pointed to by ZP to be set independently.
5.2 Properties of complex numbers
=================================
-- Function: double gsl_complex_arg (gsl_complex Z)
This function returns the argument of the complex number Z,
\arg(z), where -\pi < \arg(z) <= \pi.
-- Function: double gsl_complex_abs (gsl_complex Z)
This function returns the magnitude of the complex number Z, |z|.
-- Function: double gsl_complex_abs2 (gsl_complex Z)
This function returns the squared magnitude of the complex number
Z, |z|^2.
-- Function: double gsl_complex_logabs (gsl_complex Z)
This function returns the natural logarithm of the magnitude of the
complex number Z, \log|z|. It allows an accurate evaluation of
\log|z| when |z| is close to one. The direct evaluation of
'log(gsl_complex_abs(z))' would lead to a loss of precision in this
case.
5.3 Complex arithmetic operators
================================
-- Function: gsl_complex gsl_complex_add (gsl_complex A, gsl_complex B)
This function returns the sum of the complex numbers A and B,
z=a+b.
-- Function: gsl_complex gsl_complex_sub (gsl_complex A, gsl_complex B)
This function returns the difference of the complex numbers A and
B, z=a-b.
-- Function: gsl_complex gsl_complex_mul (gsl_complex A, gsl_complex B)
This function returns the product of the complex numbers A and B,
z=ab.
-- Function: gsl_complex gsl_complex_div (gsl_complex A, gsl_complex B)
This function returns the quotient of the complex numbers A and B,
z=a/b.
-- Function: gsl_complex gsl_complex_add_real (gsl_complex A, double X)
This function returns the sum of the complex number A and the real
number X, z=a+x.
-- Function: gsl_complex gsl_complex_sub_real (gsl_complex A, double X)
This function returns the difference of the complex number A and
the real number X, z=a-x.
-- Function: gsl_complex gsl_complex_mul_real (gsl_complex A, double X)
This function returns the product of the complex number A and the
real number X, z=ax.
-- Function: gsl_complex gsl_complex_div_real (gsl_complex A, double X)
This function returns the quotient of the complex number A and the
real number X, z=a/x.
-- Function: gsl_complex gsl_complex_add_imag (gsl_complex A, double Y)
This function returns the sum of the complex number A and the
imaginary number iY, z=a+iy.
-- Function: gsl_complex gsl_complex_sub_imag (gsl_complex A, double Y)
This function returns the difference of the complex number A and
the imaginary number iY, z=a-iy.
-- Function: gsl_complex gsl_complex_mul_imag (gsl_complex A, double Y)
This function returns the product of the complex number A and the
imaginary number iY, z=a*(iy).
-- Function: gsl_complex gsl_complex_div_imag (gsl_complex A, double Y)
This function returns the quotient of the complex number A and the
imaginary number iY, z=a/(iy).
-- Function: gsl_complex gsl_complex_conjugate (gsl_complex Z)
This function returns the complex conjugate of the complex number
Z, z^* = x - i y.
-- Function: gsl_complex gsl_complex_inverse (gsl_complex Z)
This function returns the inverse, or reciprocal, of the complex
number Z, 1/z = (x - i y)/(x^2 + y^2).
-- Function: gsl_complex gsl_complex_negative (gsl_complex Z)
This function returns the negative of the complex number Z, -z =
(-x) + i(-y).
5.4 Elementary Complex Functions
================================
-- Function: gsl_complex gsl_complex_sqrt (gsl_complex Z)
This function returns the square root of the complex number Z,
\sqrt z. The branch cut is the negative real axis. The result
always lies in the right half of the complex plane.
-- Function: gsl_complex gsl_complex_sqrt_real (double X)
This function returns the complex square root of the real number X,
where X may be negative.
-- Function: gsl_complex gsl_complex_pow (gsl_complex Z, gsl_complex A)
The function returns the complex number Z raised to the complex
power A, z^a. This is computed as \exp(\log(z)*a) using complex
logarithms and complex exponentials.
-- Function: gsl_complex gsl_complex_pow_real (gsl_complex Z, double X)
This function returns the complex number Z raised to the real power
X, z^x.
-- Function: gsl_complex gsl_complex_exp (gsl_complex Z)
This function returns the complex exponential of the complex number
Z, \exp(z).
-- Function: gsl_complex gsl_complex_log (gsl_complex Z)
This function returns the complex natural logarithm (base e) of the
complex number Z, \log(z). The branch cut is the negative real
axis.
-- Function: gsl_complex gsl_complex_log10 (gsl_complex Z)
This function returns the complex base-10 logarithm of the complex
number Z, \log_10 (z).
-- Function: gsl_complex gsl_complex_log_b (gsl_complex Z, gsl_complex
B)
This function returns the complex base-B logarithm of the complex
number Z, \log_b(z). This quantity is computed as the ratio
\log(z)/\log(b).
5.5 Complex Trigonometric Functions
===================================
-- Function: gsl_complex gsl_complex_sin (gsl_complex Z)
This function returns the complex sine of the complex number Z,
\sin(z) = (\exp(iz) - \exp(-iz))/(2i).
-- Function: gsl_complex gsl_complex_cos (gsl_complex Z)
This function returns the complex cosine of the complex number Z,
\cos(z) = (\exp(iz) + \exp(-iz))/2.
-- Function: gsl_complex gsl_complex_tan (gsl_complex Z)
This function returns the complex tangent of the complex number Z,
\tan(z) = \sin(z)/\cos(z).
-- Function: gsl_complex gsl_complex_sec (gsl_complex Z)
This function returns the complex secant of the complex number Z,
\sec(z) = 1/\cos(z).
-- Function: gsl_complex gsl_complex_csc (gsl_complex Z)
This function returns the complex cosecant of the complex number Z,
\csc(z) = 1/\sin(z).
-- Function: gsl_complex gsl_complex_cot (gsl_complex Z)
This function returns the complex cotangent of the complex number
Z, \cot(z) = 1/\tan(z).
5.6 Inverse Complex Trigonometric Functions
===========================================
-- Function: gsl_complex gsl_complex_arcsin (gsl_complex Z)
This function returns the complex arcsine of the complex number Z,
\arcsin(z). The branch cuts are on the real axis, less than -1 and
greater than 1.
-- Function: gsl_complex gsl_complex_arcsin_real (double Z)
This function returns the complex arcsine of the real number Z,
\arcsin(z). For z between -1 and 1, the function returns a real
value in the range [-\pi/2,\pi/2]. For z less than -1 the result
has a real part of -\pi/2 and a positive imaginary part. For z
greater than 1 the result has a real part of \pi/2 and a negative
imaginary part.
-- Function: gsl_complex gsl_complex_arccos (gsl_complex Z)
This function returns the complex arccosine of the complex number
Z, \arccos(z). The branch cuts are on the real axis, less than -1
and greater than 1.
-- Function: gsl_complex gsl_complex_arccos_real (double Z)
This function returns the complex arccosine of the real number Z,
\arccos(z). For z between -1 and 1, the function returns a real
value in the range [0,\pi]. For z less than -1 the result has a
real part of \pi and a negative imaginary part. For z greater than
1 the result is purely imaginary and positive.
-- Function: gsl_complex gsl_complex_arctan (gsl_complex Z)
This function returns the complex arctangent of the complex number
Z, \arctan(z). The branch cuts are on the imaginary axis, below -i
and above i.
-- Function: gsl_complex gsl_complex_arcsec (gsl_complex Z)
This function returns the complex arcsecant of the complex number
Z, \arcsec(z) = \arccos(1/z).
-- Function: gsl_complex gsl_complex_arcsec_real (double Z)
This function returns the complex arcsecant of the real number Z,
\arcsec(z) = \arccos(1/z).
-- Function: gsl_complex gsl_complex_arccsc (gsl_complex Z)
This function returns the complex arccosecant of the complex number
Z, \arccsc(z) = \arcsin(1/z).
-- Function: gsl_complex gsl_complex_arccsc_real (double Z)
This function returns the complex arccosecant of the real number Z,
\arccsc(z) = \arcsin(1/z).
-- Function: gsl_complex gsl_complex_arccot (gsl_complex Z)
This function returns the complex arccotangent of the complex
number Z, \arccot(z) = \arctan(1/z).
5.7 Complex Hyperbolic Functions
================================
-- Function: gsl_complex gsl_complex_sinh (gsl_complex Z)
This function returns the complex hyperbolic sine of the complex
number Z, \sinh(z) = (\exp(z) - \exp(-z))/2.
-- Function: gsl_complex gsl_complex_cosh (gsl_complex Z)
This function returns the complex hyperbolic cosine of the complex
number Z, \cosh(z) = (\exp(z) + \exp(-z))/2.
-- Function: gsl_complex gsl_complex_tanh (gsl_complex Z)
This function returns the complex hyperbolic tangent of the complex
number Z, \tanh(z) = \sinh(z)/\cosh(z).
-- Function: gsl_complex gsl_complex_sech (gsl_complex Z)
This function returns the complex hyperbolic secant of the complex
number Z, \sech(z) = 1/\cosh(z).
-- Function: gsl_complex gsl_complex_csch (gsl_complex Z)
This function returns the complex hyperbolic cosecant of the
complex number Z, \csch(z) = 1/\sinh(z).
-- Function: gsl_complex gsl_complex_coth (gsl_complex Z)
This function returns the complex hyperbolic cotangent of the
complex number Z, \coth(z) = 1/\tanh(z).
5.8 Inverse Complex Hyperbolic Functions
========================================
-- Function: gsl_complex gsl_complex_arcsinh (gsl_complex Z)
This function returns the complex hyperbolic arcsine of the complex
number Z, \arcsinh(z). The branch cuts are on the imaginary axis,
below -i and above i.
-- Function: gsl_complex gsl_complex_arccosh (gsl_complex Z)
This function returns the complex hyperbolic arccosine of the
complex number Z, \arccosh(z). The branch cut is on the real axis,
less than 1. Note that in this case we use the negative square
root in formula 4.6.21 of Abramowitz & Stegun giving
\arccosh(z)=\log(z-\sqrt{z^2-1}).
-- Function: gsl_complex gsl_complex_arccosh_real (double Z)
This function returns the complex hyperbolic arccosine of the real
number Z, \arccosh(z).
-- Function: gsl_complex gsl_complex_arctanh (gsl_complex Z)
This function returns the complex hyperbolic arctangent of the
complex number Z, \arctanh(z). The branch cuts are on the real
axis, less than -1 and greater than 1.
-- Function: gsl_complex gsl_complex_arctanh_real (double Z)
This function returns the complex hyperbolic arctangent of the real
number Z, \arctanh(z).
-- Function: gsl_complex gsl_complex_arcsech (gsl_complex Z)
This function returns the complex hyperbolic arcsecant of the
complex number Z, \arcsech(z) = \arccosh(1/z).
-- Function: gsl_complex gsl_complex_arccsch (gsl_complex Z)
This function returns the complex hyperbolic arccosecant of the
complex number Z, \arccsch(z) = \arcsin(1/z).
-- Function: gsl_complex gsl_complex_arccoth (gsl_complex Z)
This function returns the complex hyperbolic arccotangent of the
complex number Z, \arccoth(z) = \arctanh(1/z).
5.9 References and Further Reading
==================================
The implementations of the elementary and trigonometric functions are
based on the following papers,
T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
"Implementing Complex Elementary Functions Using Exception
Handling", 'ACM Transactions on Mathematical Software', Volume 20
(1994), pp 215-244, Corrigenda, p553
T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
"Implementing the complex arcsin and arccosine functions using
exception handling", 'ACM Transactions on Mathematical Software',
Volume 23 (1997) pp 299-335
The general formulas and details of branch cuts can be found in the
following books,
Abramowitz and Stegun, 'Handbook of Mathematical Functions',
"Circular Functions in Terms of Real and Imaginary Parts", Formulas
4.3.55-58, "Inverse Circular Functions in Terms of Real and
Imaginary Parts", Formulas 4.4.37-39, "Hyperbolic Functions in
Terms of Real and Imaginary Parts", Formulas 4.5.49-52, "Inverse
Hyperbolic Functions--relation to Inverse Circular Functions",
Formulas 4.6.14-19.
Dave Gillespie, 'Calc Manual', Free Software Foundation, ISBN
1-882114-18-3
6 Polynomials
*************
This chapter describes functions for evaluating and solving polynomials.
There are routines for finding real and complex roots of quadratic and
cubic equations using analytic methods. An iterative polynomial solver
is also available for finding the roots of general polynomials with real
coefficients (of any order). The functions are declared in the header
file 'gsl_poly.h'.
6.1 Polynomial Evaluation
=========================
The functions described here evaluate the polynomial P(x) = c[0] + c[1]
x + c[2] x^2 + \dots + c[len-1] x^{len-1} using Horner's method for
stability. Inline versions of these functions are used when
'HAVE_INLINE' is defined.
-- Function: double gsl_poly_eval (const double C[], const int LEN,
const double X)
This function evaluates a polynomial with real coefficients for the
real variable X.
-- Function: gsl_complex gsl_poly_complex_eval (const double C[], const
int LEN, const gsl_complex Z)
This function evaluates a polynomial with real coefficients for the
complex variable Z.
-- Function: gsl_complex gsl_complex_poly_complex_eval (const
gsl_complex C[], const int LEN, const gsl_complex Z)
This function evaluates a polynomial with complex coefficients for
the complex variable Z.
-- Function: int gsl_poly_eval_derivs (const double C[], const size_t
LENC, const double X, double RES[], const size_t LENRES)
This function evaluates a polynomial and its derivatives storing
the results in the array RES of size LENRES. The output array
contains the values of d^k P/d x^k for the specified value of X
starting with k = 0.
6.2 Divided Difference Representation of Polynomials
====================================================
The functions described here manipulate polynomials stored in Newton's
divided-difference representation. The use of divided-differences is
described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden
and Faires, chapter 3, and discussed briefly below.
Given a function f(x), an nth degree interpolating polynomial P_{n}(x)
can be constructed which agrees with f at n+1 distinct points
x_0,x_1,...,x_{n}. This polynomial can be written in a form known as
Newton's divided-difference representation:
P_n(x) = f(x_0) + \sum_(k=1)^n [x_0,x_1,...,x_k] (x-x_0)(x-x_1)...(x-x_(k-1))
where the divided differences [x_0,x_1,...,x_k] are defined in
section 25.1.4 of Abramowitz and Stegun. Additionally, it is possible
to construct an interpolating polynomial of degree 2n+1 which also
matches the first derivatives of f at the points x_0,x_1,...,x_n. This
is called the Hermite interpolating polynomial and is defined as
H_(2n+1)(x) = f(z_0) + \sum_(k=1)^(2n+1) [z_0,z_1,...,z_k] (x-z_0)(x-z_1)...(x-z_(k-1))
where the elements of z = \{x_0,x_0,x_1,x_1,...,x_n,x_n\} are defined
by z_{2k} = z_{2k+1} = x_k. The divided-differences [z_0,z_1,...,z_k]
are discussed in Burden and Faires, section 3.4.
-- Function: int gsl_poly_dd_init (double DD[], const double XA[],
const double YA[], size_t SIZE)
This function computes a divided-difference representation of the
interpolating polynomial for the points (X, Y) stored in the arrays
XA and YA of length SIZE. On output the divided-differences of
(XA,YA) are stored in the array DD, also of length SIZE. Using the
notation above, dd[k] = [x_0,x_1,...,x_k].
-- Function: double gsl_poly_dd_eval (const double DD[], const double
XA[], const size_t SIZE, const double X)
This function evaluates the polynomial stored in divided-difference
form in the arrays DD and XA of length SIZE at the point X. An
inline version of this function is used when 'HAVE_INLINE' is
defined.
-- Function: int gsl_poly_dd_taylor (double C[], double XP, const
double DD[], const double XA[], size_t SIZE, double W[])
This function converts the divided-difference representation of a
polynomial to a Taylor expansion. The divided-difference
representation is supplied in the arrays DD and XA of length SIZE.
On output the Taylor coefficients of the polynomial expanded about
the point XP are stored in the array C also of length SIZE. A
workspace of length SIZE must be provided in the array W.
-- Function: int gsl_poly_dd_hermite_init (double DD[], double ZA[],
const double XA[], const double YA[], const double DYA[],
const size_t SIZE)
This function computes a divided-difference representation of the
interpolating Hermite polynomial for the points (X, Y) stored in
the arrays XA and YA of length SIZE. Hermite interpolation
constructs polynomials which also match first derivatives dy/dx
which are provided in the array DYA also of length SIZE. The first
derivatives can be incorported into the usual divided-difference
algorithm by forming a new dataset z = \{x_0,x_0,x_1,x_1,...\},
which is stored in the array ZA of length 2*SIZE on output. On
output the divided-differences of the Hermite representation are
stored in the array DD, also of length 2*SIZE. Using the notation
above, dd[k] = [z_0,z_1,...,z_k]. The resulting Hermite polynomial
can be evaluated by calling 'gsl_poly_dd_eval' and using ZA for the
input argument XA.
6.3 Quadratic Equations
=======================
-- Function: int gsl_poly_solve_quadratic (double A, double B, double
C, double * X0, double * X1)
This function finds the real roots of the quadratic equation,
a x^2 + b x + c = 0
The number of real roots (either zero, one or two) is returned, and
their locations are stored in X0 and X1. If no real roots are
found then X0 and X1 are not modified. If one real root is found
(i.e. if a=0) then it is stored in X0. When two real roots are
found they are stored in X0 and X1 in ascending order. The case of
coincident roots is not considered special. For example (x-1)^2=0
will have two roots, which happen to have exactly equal values.
The number of roots found depends on the sign of the discriminant
b^2 - 4 a c. This will be subject to rounding and cancellation
errors when computed in double precision, and will also be subject
to errors if the coefficients of the polynomial are inexact. These
errors may cause a discrete change in the number of roots.
However, for polynomials with small integer coefficients the
discriminant can always be computed exactly.
-- Function: int gsl_poly_complex_solve_quadratic (double A, double B,
double C, gsl_complex * Z0, gsl_complex * Z1)
This function finds the complex roots of the quadratic equation,
a z^2 + b z + c = 0
The number of complex roots is returned (either one or two) and the
locations of the roots are stored in Z0 and Z1. The roots are
returned in ascending order, sorted first by their real components
and then by their imaginary components. If only one real root is
found (i.e. if a=0) then it is stored in Z0.
6.4 Cubic Equations
===================
-- Function: int gsl_poly_solve_cubic (double A, double B, double C,
double * X0, double * X1, double * X2)
This function finds the real roots of the cubic equation,
x^3 + a x^2 + b x + c = 0
with a leading coefficient of unity. The number of real roots
(either one or three) is returned, and their locations are stored
in X0, X1 and X2. If one real root is found then only X0 is
modified. When three real roots are found they are stored in X0,
X1 and X2 in ascending order. The case of coincident roots is not
considered special. For example, the equation (x-1)^3=0 will have
three roots with exactly equal values. As in the quadratic case,
finite precision may cause equal or closely-spaced real roots to
move off the real axis into the complex plane, leading to a
discrete change in the number of real roots.
-- Function: int gsl_poly_complex_solve_cubic (double A, double B,
double C, gsl_complex * Z0, gsl_complex * Z1, gsl_complex *
Z2)
This function finds the complex roots of the cubic equation,
z^3 + a z^2 + b z + c = 0
The number of complex roots is returned (always three) and the
locations of the roots are stored in Z0, Z1 and Z2. The roots are
returned in ascending order, sorted first by their real components
and then by their imaginary components.
6.5 General Polynomial Equations
================================
The roots of polynomial equations cannot be found analytically beyond
the special cases of the quadratic, cubic and quartic equation. The
algorithm described in this section uses an iterative method to find the
approximate locations of roots of higher order polynomials.
-- Function: gsl_poly_complex_workspace *
gsl_poly_complex_workspace_alloc (size_t N)
This function allocates space for a 'gsl_poly_complex_workspace'
struct and a workspace suitable for solving a polynomial with N
coefficients using the routine 'gsl_poly_complex_solve'.
The function returns a pointer to the newly allocated
'gsl_poly_complex_workspace' if no errors were detected, and a null
pointer in the case of error.
-- Function: void gsl_poly_complex_workspace_free
(gsl_poly_complex_workspace * W)
This function frees all the memory associated with the workspace W.
-- Function: int gsl_poly_complex_solve (const double * A, size_t N,
gsl_poly_complex_workspace * W, gsl_complex_packed_ptr Z)
This function computes the roots of the general polynomial P(x) =
a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using balanced-QR
reduction of the companion matrix. The parameter N specifies the
length of the coefficient array. The coefficient of the highest
order term must be non-zero. The function requires a workspace W
of the appropriate size. The n-1 roots are returned in the packed
complex array Z of length 2(n-1), alternating real and imaginary
parts.
The function returns 'GSL_SUCCESS' if all the roots are found. If
the QR reduction does not converge, the error handler is invoked
with an error code of 'GSL_EFAILED'. Note that due to finite
precision, roots of higher multiplicity are returned as a cluster
of simple roots with reduced accuracy. The solution of polynomials
with higher-order roots requires specialized algorithms that take
the multiplicity structure into account (see e.g. Z. Zeng,
Algorithm 835, ACM Transactions on Mathematical Software, Volume
30, Issue 2 (2004), pp 218-236).
6.6 Examples
============
To demonstrate the use of the general polynomial solver we will take the
polynomial P(x) = x^5 - 1 which has the following roots,
1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}
The following program will find these roots.
#include
#include
int
main (void)
{
int i;
/* coefficients of P(x) = -1 + x^5 */
double a[6] = { -1, 0, 0, 0, 0, 1 };
double z[10];
gsl_poly_complex_workspace * w
= gsl_poly_complex_workspace_alloc (6);
gsl_poly_complex_solve (a, 6, w, z);
gsl_poly_complex_workspace_free (w);
for (i = 0; i < 5; i++)
{
printf ("z%d = %+.18f %+.18f\n",
i, z[2*i], z[2*i+1]);
}
return 0;
}
The output of the program is,
$ ./a.out
z0 = -0.809016994374947673 +0.587785252292473359
z1 = -0.809016994374947673 -0.587785252292473359
z2 = +0.309016994374947507 +0.951056516295152976
z3 = +0.309016994374947507 -0.951056516295152976
z4 = +0.999999999999999889 +0.000000000000000000
which agrees with the analytic result, z_n = \exp(2 \pi n i/5).
6.7 References and Further Reading
==================================
The balanced-QR method and its error analysis are described in the
following papers,
R.S. Martin, G. Peters and J.H. Wilkinson, "The QR Algorithm for
Real Hessenberg Matrices", 'Numerische Mathematik', 14 (1970),
219-231.
B.N. Parlett and C. Reinsch, "Balancing a Matrix for Calculation of
Eigenvalues and Eigenvectors", 'Numerische Mathematik', 13 (1969),
293-304.
A. Edelman and H. Murakami, "Polynomial roots from companion matrix
eigenvalues", 'Mathematics of Computation', Vol. 64, No. 210
(1995), 763-776.
The formulas for divided differences are given in the following texts,
Abramowitz and Stegun, 'Handbook of Mathematical Functions',
Sections 25.1.4 and 25.2.26.
R. L. Burden and J. D. Faires, 'Numerical Analysis', 9th edition,
ISBN 0-538-73351-9, 2011.
7 Special Functions
*******************
This chapter describes the GSL special function library. The library
includes routines for calculating the values of Airy functions, Bessel
functions, Clausen functions, Coulomb wave functions, Coupling
coefficients, the Dawson function, Debye functions, Dilogarithms,
Elliptic integrals, Jacobi elliptic functions, Error functions,
Exponential integrals, Fermi-Dirac functions, Gamma functions,
Gegenbauer functions, Hypergeometric functions, Laguerre functions,
Legendre functions and Spherical Harmonics, the Psi (Digamma) Function,
Synchrotron functions, Transport functions, Trigonometric functions and
Zeta functions. Each routine also computes an estimate of the numerical
error in the calculated value of the function.
The functions in this chapter are declared in individual header
files, such as 'gsl_sf_airy.h', 'gsl_sf_bessel.h', etc. The complete
set of header files can be included using the file 'gsl_sf.h'.
7.1 Usage
=========
The special functions are available in two calling conventions, a
"natural form" which returns the numerical value of the function and an
"error-handling form" which returns an error code. The two types of
function provide alternative ways of accessing the same underlying code.
The "natural form" returns only the value of the function and can be
used directly in mathematical expressions. For example, the following
function call will compute the value of the Bessel function J_0(x),
double y = gsl_sf_bessel_J0 (x);
There is no way to access an error code or to estimate the error using
this method. To allow access to this information the alternative
error-handling form stores the value and error in a modifiable argument,
gsl_sf_result result;
int status = gsl_sf_bessel_J0_e (x, &result);
The error-handling functions have the suffix '_e'. The returned status
value indicates error conditions such as overflow, underflow or loss of
precision. If there are no errors the error-handling functions return
'GSL_SUCCESS'.
7.2 The gsl_sf_result struct
============================
The error handling form of the special functions always calculate an
error estimate along with the value of the result. Therefore,
structures are provided for amalgamating a value and error estimate.
These structures are declared in the header file 'gsl_sf_result.h'.
The 'gsl_sf_result' struct contains value and error fields.
typedef struct
{
double val;
double err;
} gsl_sf_result;
The field VAL contains the value and the field ERR contains an estimate
of the absolute error in the value.
In some cases, an overflow or underflow can be detected and handled
by a function. In this case, it may be possible to return a scaling
exponent as well as an error/value pair in order to save the result from
exceeding the dynamic range of the built-in types. The
'gsl_sf_result_e10' struct contains value and error fields as well as an
exponent field such that the actual result is obtained as 'result *
10^(e10)'.
typedef struct
{
double val;
double err;
int e10;
} gsl_sf_result_e10;
7.3 Modes
=========
The goal of the library is to achieve double precision accuracy wherever
possible. However the cost of evaluating some special functions to
double precision can be significant, particularly where very high order
terms are required. In these cases a 'mode' argument allows the
accuracy of the function to be reduced in order to improve performance.
The following precision levels are available for the mode argument,
'GSL_PREC_DOUBLE'
Double-precision, a relative accuracy of approximately 2 * 10^-16.
'GSL_PREC_SINGLE'
Single-precision, a relative accuracy of approximately 10^-7.
'GSL_PREC_APPROX'
Approximate values, a relative accuracy of approximately 5 * 10^-4.
The approximate mode provides the fastest evaluation at the lowest
accuracy.
7.4 Airy Functions and Derivatives
==================================
The Airy functions Ai(x) and Bi(x) are defined by the integral
representations,
Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt
Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3 + xt) + \sin((1/3) t^3 + xt)) dt
For further information see Abramowitz & Stegun, Section 10.4. The Airy
functions are defined in the header file 'gsl_sf_airy.h'.
7.4.1 Airy Functions
--------------------
-- Function: double gsl_sf_airy_Ai (double X, gsl_mode_t MODE)
-- Function: int gsl_sf_airy_Ai_e (double X, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute the Airy function Ai(x) with an accuracy
specified by MODE.
-- Function: double gsl_sf_airy_Bi (double X, gsl_mode_t MODE)
-- Function: int gsl_sf_airy_Bi_e (double X, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute the Airy function Bi(x) with an accuracy
specified by MODE.
-- Function: double gsl_sf_airy_Ai_scaled (double X, gsl_mode_t MODE)
-- Function: int gsl_sf_airy_Ai_scaled_e (double X, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute a scaled version of the Airy function S_A(x)
Ai(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)),
and is 1 for x<0.
-- Function: double gsl_sf_airy_Bi_scaled (double X, gsl_mode_t MODE)
-- Function: int gsl_sf_airy_Bi_scaled_e (double X, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute a scaled version of the Airy function S_B(x)
Bi(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)),
and is 1 for x<0.
7.4.2 Derivatives of Airy Functions
-----------------------------------
-- Function: double gsl_sf_airy_Ai_deriv (double X, gsl_mode_t MODE)
-- Function: int gsl_sf_airy_Ai_deriv_e (double X, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute the Airy function derivative Ai'(x) with an
accuracy specified by MODE.
-- Function: double gsl_sf_airy_Bi_deriv (double X, gsl_mode_t MODE)
-- Function: int gsl_sf_airy_Bi_deriv_e (double X, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute the Airy function derivative Bi'(x) with an
accuracy specified by MODE.
-- Function: double gsl_sf_airy_Ai_deriv_scaled (double X, gsl_mode_t
MODE)
-- Function: int gsl_sf_airy_Ai_deriv_scaled_e (double X, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These routines compute the scaled Airy function derivative S_A(x)
Ai'(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)),
and is 1 for x<0.
-- Function: double gsl_sf_airy_Bi_deriv_scaled (double X, gsl_mode_t
MODE)
-- Function: int gsl_sf_airy_Bi_deriv_scaled_e (double X, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These routines compute the scaled Airy function derivative S_B(x)
Bi'(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)),
and is 1 for x<0.
7.4.3 Zeros of Airy Functions
-----------------------------
-- Function: double gsl_sf_airy_zero_Ai (unsigned int S)
-- Function: int gsl_sf_airy_zero_Ai_e (unsigned int S, gsl_sf_result *
RESULT)
These routines compute the location of the S-th zero of the Airy
function Ai(x).
-- Function: double gsl_sf_airy_zero_Bi (unsigned int S)
-- Function: int gsl_sf_airy_zero_Bi_e (unsigned int S, gsl_sf_result *
RESULT)
These routines compute the location of the S-th zero of the Airy
function Bi(x).
7.4.4 Zeros of Derivatives of Airy Functions
--------------------------------------------
-- Function: double gsl_sf_airy_zero_Ai_deriv (unsigned int S)
-- Function: int gsl_sf_airy_zero_Ai_deriv_e (unsigned int S,
gsl_sf_result * RESULT)
These routines compute the location of the S-th zero of the Airy
function derivative Ai'(x).
-- Function: double gsl_sf_airy_zero_Bi_deriv (unsigned int S)
-- Function: int gsl_sf_airy_zero_Bi_deriv_e (unsigned int S,
gsl_sf_result * RESULT)
These routines compute the location of the S-th zero of the Airy
function derivative Bi'(x).
7.5 Bessel Functions
====================
The routines described in this section compute the Cylindrical Bessel
functions J_n(x), Y_n(x), Modified cylindrical Bessel functions I_n(x),
K_n(x), Spherical Bessel functions j_l(x), y_l(x), and Modified
Spherical Bessel functions i_l(x), k_l(x). For more information see
Abramowitz & Stegun, Chapters 9 and 10. The Bessel functions are
defined in the header file 'gsl_sf_bessel.h'.
7.5.1 Regular Cylindrical Bessel Functions
------------------------------------------
-- Function: double gsl_sf_bessel_J0 (double X)
-- Function: int gsl_sf_bessel_J0_e (double X, gsl_sf_result * RESULT)
These routines compute the regular cylindrical Bessel function of
zeroth order, J_0(x).
-- Function: double gsl_sf_bessel_J1 (double X)
-- Function: int gsl_sf_bessel_J1_e (double X, gsl_sf_result * RESULT)
These routines compute the regular cylindrical Bessel function of
first order, J_1(x).
-- Function: double gsl_sf_bessel_Jn (int N, double X)
-- Function: int gsl_sf_bessel_Jn_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the regular cylindrical Bessel function of
order N, J_n(x).
-- Function: int gsl_sf_bessel_Jn_array (int NMIN, int NMAX, double X,
double RESULT_ARRAY[])
This routine computes the values of the regular cylindrical Bessel
functions J_n(x) for n from NMIN to NMAX inclusive, storing the
results in the array RESULT_ARRAY. The values are computed using
recurrence relations for efficiency, and therefore may differ
slightly from the exact values.
7.5.2 Irregular Cylindrical Bessel Functions
--------------------------------------------
-- Function: double gsl_sf_bessel_Y0 (double X)
-- Function: int gsl_sf_bessel_Y0_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular cylindrical Bessel function of
zeroth order, Y_0(x), for x>0.
-- Function: double gsl_sf_bessel_Y1 (double X)
-- Function: int gsl_sf_bessel_Y1_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular cylindrical Bessel function of
first order, Y_1(x), for x>0.
-- Function: double gsl_sf_bessel_Yn (int N, double X)
-- Function: int gsl_sf_bessel_Yn_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the irregular cylindrical Bessel function of
order N, Y_n(x), for x>0.
-- Function: int gsl_sf_bessel_Yn_array (int NMIN, int NMAX, double X,
double RESULT_ARRAY[])
This routine computes the values of the irregular cylindrical
Bessel functions Y_n(x) for n from NMIN to NMAX inclusive, storing
the results in the array RESULT_ARRAY. The domain of the function
is x>0. The values are computed using recurrence relations for
efficiency, and therefore may differ slightly from the exact
values.
7.5.3 Regular Modified Cylindrical Bessel Functions
---------------------------------------------------
-- Function: double gsl_sf_bessel_I0 (double X)
-- Function: int gsl_sf_bessel_I0_e (double X, gsl_sf_result * RESULT)
These routines compute the regular modified cylindrical Bessel
function of zeroth order, I_0(x).
-- Function: double gsl_sf_bessel_I1 (double X)
-- Function: int gsl_sf_bessel_I1_e (double X, gsl_sf_result * RESULT)
These routines compute the regular modified cylindrical Bessel
function of first order, I_1(x).
-- Function: double gsl_sf_bessel_In (int N, double X)
-- Function: int gsl_sf_bessel_In_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the regular modified cylindrical Bessel
function of order N, I_n(x).
-- Function: int gsl_sf_bessel_In_array (int NMIN, int NMAX, double X,
double RESULT_ARRAY[])
This routine computes the values of the regular modified
cylindrical Bessel functions I_n(x) for n from NMIN to NMAX
inclusive, storing the results in the array RESULT_ARRAY. The
start of the range NMIN must be positive or zero. The values are
computed using recurrence relations for efficiency, and therefore
may differ slightly from the exact values.
-- Function: double gsl_sf_bessel_I0_scaled (double X)
-- Function: int gsl_sf_bessel_I0_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled regular modified cylindrical
Bessel function of zeroth order \exp(-|x|) I_0(x).
-- Function: double gsl_sf_bessel_I1_scaled (double X)
-- Function: int gsl_sf_bessel_I1_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled regular modified cylindrical
Bessel function of first order \exp(-|x|) I_1(x).
-- Function: double gsl_sf_bessel_In_scaled (int N, double X)
-- Function: int gsl_sf_bessel_In_scaled_e (int N, double X,
gsl_sf_result * RESULT)
These routines compute the scaled regular modified cylindrical
Bessel function of order N, \exp(-|x|) I_n(x)
-- Function: int gsl_sf_bessel_In_scaled_array (int NMIN, int NMAX,
double X, double RESULT_ARRAY[])
This routine computes the values of the scaled regular cylindrical
Bessel functions \exp(-|x|) I_n(x) for n from NMIN to NMAX
inclusive, storing the results in the array RESULT_ARRAY. The
start of the range NMIN must be positive or zero. The values are
computed using recurrence relations for efficiency, and therefore
may differ slightly from the exact values.
7.5.4 Irregular Modified Cylindrical Bessel Functions
-----------------------------------------------------
-- Function: double gsl_sf_bessel_K0 (double X)
-- Function: int gsl_sf_bessel_K0_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular modified cylindrical Bessel
function of zeroth order, K_0(x), for x > 0.
-- Function: double gsl_sf_bessel_K1 (double X)
-- Function: int gsl_sf_bessel_K1_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular modified cylindrical Bessel
function of first order, K_1(x), for x > 0.
-- Function: double gsl_sf_bessel_Kn (int N, double X)
-- Function: int gsl_sf_bessel_Kn_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the irregular modified cylindrical Bessel
function of order N, K_n(x), for x > 0.
-- Function: int gsl_sf_bessel_Kn_array (int NMIN, int NMAX, double X,
double RESULT_ARRAY[])
This routine computes the values of the irregular modified
cylindrical Bessel functions K_n(x) for n from NMIN to NMAX
inclusive, storing the results in the array RESULT_ARRAY. The
start of the range NMIN must be positive or zero. The domain of
the function is x>0. The values are computed using recurrence
relations for efficiency, and therefore may differ slightly from
the exact values.
-- Function: double gsl_sf_bessel_K0_scaled (double X)
-- Function: int gsl_sf_bessel_K0_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled irregular modified cylindrical
Bessel function of zeroth order \exp(x) K_0(x) for x>0.
-- Function: double gsl_sf_bessel_K1_scaled (double X)
-- Function: int gsl_sf_bessel_K1_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled irregular modified cylindrical
Bessel function of first order \exp(x) K_1(x) for x>0.
-- Function: double gsl_sf_bessel_Kn_scaled (int N, double X)
-- Function: int gsl_sf_bessel_Kn_scaled_e (int N, double X,
gsl_sf_result * RESULT)
These routines compute the scaled irregular modified cylindrical
Bessel function of order N, \exp(x) K_n(x), for x>0.
-- Function: int gsl_sf_bessel_Kn_scaled_array (int NMIN, int NMAX,
double X, double RESULT_ARRAY[])
This routine computes the values of the scaled irregular
cylindrical Bessel functions \exp(x) K_n(x) for n from NMIN to NMAX
inclusive, storing the results in the array RESULT_ARRAY. The
start of the range NMIN must be positive or zero. The domain of
the function is x>0. The values are computed using recurrence
relations for efficiency, and therefore may differ slightly from
the exact values.
7.5.5 Regular Spherical Bessel Functions
----------------------------------------
-- Function: double gsl_sf_bessel_j0 (double X)
-- Function: int gsl_sf_bessel_j0_e (double X, gsl_sf_result * RESULT)
These routines compute the regular spherical Bessel function of
zeroth order, j_0(x) = \sin(x)/x.
-- Function: double gsl_sf_bessel_j1 (double X)
-- Function: int gsl_sf_bessel_j1_e (double X, gsl_sf_result * RESULT)
These routines compute the regular spherical Bessel function of
first order, j_1(x) = (\sin(x)/x - \cos(x))/x.
-- Function: double gsl_sf_bessel_j2 (double X)
-- Function: int gsl_sf_bessel_j2_e (double X, gsl_sf_result * RESULT)
These routines compute the regular spherical Bessel function of
second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.
-- Function: double gsl_sf_bessel_jl (int L, double X)
-- Function: int gsl_sf_bessel_jl_e (int L, double X, gsl_sf_result *
RESULT)
These routines compute the regular spherical Bessel function of
order L, j_l(x), for l >= 0 and x >= 0.
-- Function: int gsl_sf_bessel_jl_array (int LMAX, double X, double
RESULT_ARRAY[])
This routine computes the values of the regular spherical Bessel
functions j_l(x) for l from 0 to LMAX inclusive for lmax >= 0 and x
>= 0, storing the results in the array RESULT_ARRAY. The values
are computed using recurrence relations for efficiency, and
therefore may differ slightly from the exact values.
-- Function: int gsl_sf_bessel_jl_steed_array (int LMAX, double X,
double * RESULT_ARRAY)
This routine uses Steed's method to compute the values of the
regular spherical Bessel functions j_l(x) for l from 0 to LMAX
inclusive for lmax >= 0 and x >= 0, storing the results in the
array RESULT_ARRAY. The Steed/Barnett algorithm is described in
'Comp. Phys. Comm.' 21, 297 (1981). Steed's method is more stable
than the recurrence used in the other functions but is also slower.
7.5.6 Irregular Spherical Bessel Functions
------------------------------------------
-- Function: double gsl_sf_bessel_y0 (double X)
-- Function: int gsl_sf_bessel_y0_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular spherical Bessel function of
zeroth order, y_0(x) = -\cos(x)/x.
-- Function: double gsl_sf_bessel_y1 (double X)
-- Function: int gsl_sf_bessel_y1_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular spherical Bessel function of
first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.
-- Function: double gsl_sf_bessel_y2 (double X)
-- Function: int gsl_sf_bessel_y2_e (double X, gsl_sf_result * RESULT)
These routines compute the irregular spherical Bessel function of
second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x).
-- Function: double gsl_sf_bessel_yl (int L, double X)
-- Function: int gsl_sf_bessel_yl_e (int L, double X, gsl_sf_result *
RESULT)
These routines compute the irregular spherical Bessel function of
order L, y_l(x), for l >= 0.
-- Function: int gsl_sf_bessel_yl_array (int LMAX, double X, double
RESULT_ARRAY[])
This routine computes the values of the irregular spherical Bessel
functions y_l(x) for l from 0 to LMAX inclusive for lmax >= 0,
storing the results in the array RESULT_ARRAY. The values are
computed using recurrence relations for efficiency, and therefore
may differ slightly from the exact values.
7.5.7 Regular Modified Spherical Bessel Functions
-------------------------------------------------
The regular modified spherical Bessel functions i_l(x) are related to
the modified Bessel functions of fractional order, i_l(x) =
\sqrt{\pi/(2x)} I_{l+1/2}(x)
-- Function: double gsl_sf_bessel_i0_scaled (double X)
-- Function: int gsl_sf_bessel_i0_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled regular modified spherical Bessel
function of zeroth order, \exp(-|x|) i_0(x).
-- Function: double gsl_sf_bessel_i1_scaled (double X)
-- Function: int gsl_sf_bessel_i1_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled regular modified spherical Bessel
function of first order, \exp(-|x|) i_1(x).
-- Function: double gsl_sf_bessel_i2_scaled (double X)
-- Function: int gsl_sf_bessel_i2_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled regular modified spherical Bessel
function of second order, \exp(-|x|) i_2(x)
-- Function: double gsl_sf_bessel_il_scaled (int L, double X)
-- Function: int gsl_sf_bessel_il_scaled_e (int L, double X,
gsl_sf_result * RESULT)
These routines compute the scaled regular modified spherical Bessel
function of order L, \exp(-|x|) i_l(x)
-- Function: int gsl_sf_bessel_il_scaled_array (int LMAX, double X,
double RESULT_ARRAY[])
This routine computes the values of the scaled regular modified
spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to LMAX
inclusive for lmax >= 0, storing the results in the array
RESULT_ARRAY. The values are computed using recurrence relations
for efficiency, and therefore may differ slightly from the exact
values.
7.5.8 Irregular Modified Spherical Bessel Functions
---------------------------------------------------
The irregular modified spherical Bessel functions k_l(x) are related to
the irregular modified Bessel functions of fractional order, k_l(x) =
\sqrt{\pi/(2x)} K_{l+1/2}(x).
-- Function: double gsl_sf_bessel_k0_scaled (double X)
-- Function: int gsl_sf_bessel_k0_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled irregular modified spherical
Bessel function of zeroth order, \exp(x) k_0(x), for x>0.
-- Function: double gsl_sf_bessel_k1_scaled (double X)
-- Function: int gsl_sf_bessel_k1_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled irregular modified spherical
Bessel function of first order, \exp(x) k_1(x), for x>0.
-- Function: double gsl_sf_bessel_k2_scaled (double X)
-- Function: int gsl_sf_bessel_k2_scaled_e (double X, gsl_sf_result *
RESULT)
These routines compute the scaled irregular modified spherical
Bessel function of second order, \exp(x) k_2(x), for x>0.
-- Function: double gsl_sf_bessel_kl_scaled (int L, double X)
-- Function: int gsl_sf_bessel_kl_scaled_e (int L, double X,
gsl_sf_result * RESULT)
These routines compute the scaled irregular modified spherical
Bessel function of order L, \exp(x) k_l(x), for x>0.
-- Function: int gsl_sf_bessel_kl_scaled_array (int LMAX, double X,
double RESULT_ARRAY[])
This routine computes the values of the scaled irregular modified
spherical Bessel functions \exp(x) k_l(x) for l from 0 to LMAX
inclusive for lmax >= 0 and x>0, storing the results in the array
RESULT_ARRAY. The values are computed using recurrence relations
for efficiency, and therefore may differ slightly from the exact
values.
7.5.9 Regular Bessel Function--Fractional Order
-----------------------------------------------
-- Function: double gsl_sf_bessel_Jnu (double NU, double X)
-- Function: int gsl_sf_bessel_Jnu_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the regular cylindrical Bessel function of
fractional order \nu, J_\nu(x).
-- Function: int gsl_sf_bessel_sequence_Jnu_e (double NU, gsl_mode_t
MODE, size_t SIZE, double V[])
This function computes the regular cylindrical Bessel function of
fractional order \nu, J_\nu(x), evaluated at a series of x values.
The array V of length SIZE contains the x values. They are assumed
to be strictly ordered and positive. The array is over-written
with the values of J_\nu(x_i).
7.5.10 Irregular Bessel Functions--Fractional Order
---------------------------------------------------
-- Function: double gsl_sf_bessel_Ynu (double NU, double X)
-- Function: int gsl_sf_bessel_Ynu_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the irregular cylindrical Bessel function of
fractional order \nu, Y_\nu(x).
7.5.11 Regular Modified Bessel Functions--Fractional Order
----------------------------------------------------------
-- Function: double gsl_sf_bessel_Inu (double NU, double X)
-- Function: int gsl_sf_bessel_Inu_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the regular modified Bessel function of
fractional order \nu, I_\nu(x) for x>0, \nu>0.
-- Function: double gsl_sf_bessel_Inu_scaled (double NU, double X)
-- Function: int gsl_sf_bessel_Inu_scaled_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the scaled regular modified Bessel function
of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.
7.5.12 Irregular Modified Bessel Functions--Fractional Order
------------------------------------------------------------
-- Function: double gsl_sf_bessel_Knu (double NU, double X)
-- Function: int gsl_sf_bessel_Knu_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the irregular modified Bessel function of
fractional order \nu, K_\nu(x) for x>0, \nu>0.
-- Function: double gsl_sf_bessel_lnKnu (double NU, double X)
-- Function: int gsl_sf_bessel_lnKnu_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the logarithm of the irregular modified
Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0,
\nu>0.
-- Function: double gsl_sf_bessel_Knu_scaled (double NU, double X)
-- Function: int gsl_sf_bessel_Knu_scaled_e (double NU, double X,
gsl_sf_result * RESULT)
These routines compute the scaled irregular modified Bessel
function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0,
\nu>0.
7.5.13 Zeros of Regular Bessel Functions
----------------------------------------
-- Function: double gsl_sf_bessel_zero_J0 (unsigned int S)
-- Function: int gsl_sf_bessel_zero_J0_e (unsigned int S, gsl_sf_result
* RESULT)
These routines compute the location of the S-th positive zero of
the Bessel function J_0(x).
-- Function: double gsl_sf_bessel_zero_J1 (unsigned int S)
-- Function: int gsl_sf_bessel_zero_J1_e (unsigned int S, gsl_sf_result
* RESULT)
These routines compute the location of the S-th positive zero of
the Bessel function J_1(x).
-- Function: double gsl_sf_bessel_zero_Jnu (double NU, unsigned int S)
-- Function: int gsl_sf_bessel_zero_Jnu_e (double NU, unsigned int S,
gsl_sf_result * RESULT)
These routines compute the location of the S-th positive zero of
the Bessel function J_\nu(x). The current implementation does not
support negative values of NU.
7.6 Clausen Functions
=====================
The Clausen function is defined by the following integral,
Cl_2(x) = - \int_0^x dt \log(2 \sin(t/2))
It is related to the dilogarithm by Cl_2(\theta) = \Im
Li_2(\exp(i\theta)). The Clausen functions are declared in the header
file 'gsl_sf_clausen.h'.
-- Function: double gsl_sf_clausen (double X)
-- Function: int gsl_sf_clausen_e (double X, gsl_sf_result * RESULT)
These routines compute the Clausen integral Cl_2(x).
7.7 Coulomb Functions
=====================
The prototypes of the Coulomb functions are declared in the header file
'gsl_sf_coulomb.h'. Both bound state and scattering solutions are
available.
7.7.1 Normalized Hydrogenic Bound States
----------------------------------------
-- Function: double gsl_sf_hydrogenicR_1 (double Z, double R)
-- Function: int gsl_sf_hydrogenicR_1_e (double Z, double R,
gsl_sf_result * RESULT)
These routines compute the lowest-order normalized hydrogenic bound
state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).
-- Function: double gsl_sf_hydrogenicR (int N, int L, double Z, double
R)
-- Function: int gsl_sf_hydrogenicR_e (int N, int L, double Z, double
R, gsl_sf_result * RESULT)
These routines compute the N-th normalized hydrogenic bound state
radial wavefunction,
R_n := 2 (Z^{3/2}/n^2) \sqrt{(n-l-1)!/(n+l)!} \exp(-Z r/n) (2Zr/n)^l
L^{2l+1}_{n-l-1}(2Zr/n).
where L^a_b(x) is the generalized Laguerre polynomial (*note
Laguerre Functions::). The normalization is chosen such that the
wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.
7.7.2 Coulomb Wave Functions
----------------------------
The Coulomb wave functions F_L(\eta,x), G_L(\eta,x) are described in
Abramowitz & Stegun, Chapter 14. Because there can be a large dynamic
range of values for these functions, overflows are handled gracefully.
If an overflow occurs, 'GSL_EOVRFLW' is signalled and exponent(s) are
returned through the modifiable parameters EXP_F, EXP_G. The full
solution can be reconstructed from the following relations,
F_L(eta,x) = fc[k_L] * exp(exp_F)
G_L(eta,x) = gc[k_L] * exp(exp_G)
F_L'(eta,x) = fcp[k_L] * exp(exp_F)
G_L'(eta,x) = gcp[k_L] * exp(exp_G)
-- Function: int gsl_sf_coulomb_wave_FG_e (double ETA, double X, double
L_F, int K, gsl_sf_result * F, gsl_sf_result * FP,
gsl_sf_result * G, gsl_sf_result * GP, double * EXP_F, double
* EXP_G)
This function computes the Coulomb wave functions F_L(\eta,x),
G_{L-k}(\eta,x) and their derivatives F'_L(\eta,x),
G'_{L-k}(\eta,x) with respect to x. The parameters are restricted
to L, L-k > -1/2, x > 0 and integer k. Note that L itself is not
restricted to being an integer. The results are stored in the
parameters F, G for the function values and FP, GP for the
derivative values. If an overflow occurs, 'GSL_EOVRFLW' is
returned and scaling exponents are stored in the modifiable
parameters EXP_F, EXP_G.
-- Function: int gsl_sf_coulomb_wave_F_array (double L_MIN, int KMAX,
double ETA, double X, double FC_ARRAY[], double * F_EXPONENT)
This function computes the Coulomb wave function F_L(\eta,x) for L
= Lmin \dots Lmin + kmax, storing the results in FC_ARRAY. In the
case of overflow the exponent is stored in F_EXPONENT.
-- Function: int gsl_sf_coulomb_wave_FG_array (double L_MIN, int KMAX,
double ETA, double X, double FC_ARRAY[], double GC_ARRAY[],
double * F_EXPONENT, double * G_EXPONENT)
This function computes the functions F_L(\eta,x), G_L(\eta,x) for L
= Lmin \dots Lmin + kmax storing the results in FC_ARRAY and
GC_ARRAY. In the case of overflow the exponents are stored in
F_EXPONENT and G_EXPONENT.
-- Function: int gsl_sf_coulomb_wave_FGp_array (double L_MIN, int KMAX,
double ETA, double X, double FC_ARRAY[], double FCP_ARRAY[],
double GC_ARRAY[], double GCP_ARRAY[], double * F_EXPONENT,
double * G_EXPONENT)
This function computes the functions F_L(\eta,x), G_L(\eta,x) and
their derivatives F'_L(\eta,x), G'_L(\eta,x) for L = Lmin \dots
Lmin + kmax storing the results in FC_ARRAY, GC_ARRAY, FCP_ARRAY
and GCP_ARRAY. In the case of overflow the exponents are stored in
F_EXPONENT and G_EXPONENT.
-- Function: int gsl_sf_coulomb_wave_sphF_array (double L_MIN, int
KMAX, double ETA, double X, double FC_ARRAY[], double
F_EXPONENT[])
This function computes the Coulomb wave function divided by the
argument F_L(\eta, x)/x for L = Lmin \dots Lmin + kmax, storing the
results in FC_ARRAY. In the case of overflow the exponent is
stored in F_EXPONENT. This function reduces to spherical Bessel
functions in the limit \eta \to 0.
7.7.3 Coulomb Wave Function Normalization Constant
--------------------------------------------------
The Coulomb wave function normalization constant is defined in
Abramowitz 14.1.7.
-- Function: int gsl_sf_coulomb_CL_e (double L, double ETA,
gsl_sf_result * RESULT)
This function computes the Coulomb wave function normalization
constant C_L(\eta) for L > -1.
-- Function: int gsl_sf_coulomb_CL_array (double LMIN, int KMAX, double
ETA, double CL[])
This function computes the Coulomb wave function normalization
constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.
7.8 Coupling Coefficients
=========================
The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for
combined angular momentum vectors. Since the arguments of the standard
coupling coefficient functions are integer or half-integer, the
arguments of the following functions are, by convention, integers equal
to twice the actual spin value. For information on the 3-j coefficients
see Abramowitz & Stegun, Section 27.9. The functions described in this
section are declared in the header file 'gsl_sf_coupling.h'.
7.8.1 3-j Symbols
-----------------
-- Function: double gsl_sf_coupling_3j (int TWO_JA, int TWO_JB, int
TWO_JC, int TWO_MA, int TWO_MB, int TWO_MC)
-- Function: int gsl_sf_coupling_3j_e (int TWO_JA, int TWO_JB, int
TWO_JC, int TWO_MA, int TWO_MB, int TWO_MC, gsl_sf_result *
RESULT)
These routines compute the Wigner 3-j coefficient,
(ja jb jc
ma mb mc)
where the arguments are given in half-integer units, ja = TWO_JA/2,
ma = TWO_MA/2, etc.
7.8.2 6-j Symbols
-----------------
-- Function: double gsl_sf_coupling_6j (int TWO_JA, int TWO_JB, int
TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF)
-- Function: int gsl_sf_coupling_6j_e (int TWO_JA, int TWO_JB, int
TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, gsl_sf_result *
RESULT)
These routines compute the Wigner 6-j coefficient,
{ja jb jc
jd je jf}
where the arguments are given in half-integer units, ja = TWO_JA/2,
ma = TWO_MA/2, etc.
7.8.3 9-j Symbols
-----------------
-- Function: double gsl_sf_coupling_9j (int TWO_JA, int TWO_JB, int
TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, int TWO_JG, int
TWO_JH, int TWO_JI)
-- Function: int gsl_sf_coupling_9j_e (int TWO_JA, int TWO_JB, int
TWO_JC, int TWO_JD, int TWO_JE, int TWO_JF, int TWO_JG, int
TWO_JH, int TWO_JI, gsl_sf_result * RESULT)
These routines compute the Wigner 9-j coefficient,
{ja jb jc
jd je jf
jg jh ji}
where the arguments are given in half-integer units, ja = TWO_JA/2,
ma = TWO_MA/2, etc.
7.9 Dawson Function
===================
The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2). A
table of Dawson's integral can be found in Abramowitz & Stegun, Table
7.5. The Dawson functions are declared in the header file
'gsl_sf_dawson.h'.
-- Function: double gsl_sf_dawson (double X)
-- Function: int gsl_sf_dawson_e (double X, gsl_sf_result * RESULT)
These routines compute the value of Dawson's integral for X.
7.10 Debye Functions
====================
The Debye functions D_n(x) are defined by the following integral,
D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1))
For further information see Abramowitz & Stegun, Section 27.1. The
Debye functions are declared in the header file 'gsl_sf_debye.h'.
-- Function: double gsl_sf_debye_1 (double X)
-- Function: int gsl_sf_debye_1_e (double X, gsl_sf_result * RESULT)
These routines compute the first-order Debye function D_1(x) =
(1/x) \int_0^x dt (t/(e^t - 1)).
-- Function: double gsl_sf_debye_2 (double X)
-- Function: int gsl_sf_debye_2_e (double X, gsl_sf_result * RESULT)
These routines compute the second-order Debye function D_2(x) =
(2/x^2) \int_0^x dt (t^2/(e^t - 1)).
-- Function: double gsl_sf_debye_3 (double X)
-- Function: int gsl_sf_debye_3_e (double X, gsl_sf_result * RESULT)
These routines compute the third-order Debye function D_3(x) =
(3/x^3) \int_0^x dt (t^3/(e^t - 1)).
-- Function: double gsl_sf_debye_4 (double X)
-- Function: int gsl_sf_debye_4_e (double X, gsl_sf_result * RESULT)
These routines compute the fourth-order Debye function D_4(x) =
(4/x^4) \int_0^x dt (t^4/(e^t - 1)).
-- Function: double gsl_sf_debye_5 (double X)
-- Function: int gsl_sf_debye_5_e (double X, gsl_sf_result * RESULT)
These routines compute the fifth-order Debye function D_5(x) =
(5/x^5) \int_0^x dt (t^5/(e^t - 1)).
-- Function: double gsl_sf_debye_6 (double X)
-- Function: int gsl_sf_debye_6_e (double X, gsl_sf_result * RESULT)
These routines compute the sixth-order Debye function D_6(x) =
(6/x^6) \int_0^x dt (t^6/(e^t - 1)).
7.11 Dilogarithm
================
The functions described in this section are declared in the header file
'gsl_sf_dilog.h'.
7.11.1 Real Argument
--------------------
-- Function: double gsl_sf_dilog (double X)
-- Function: int gsl_sf_dilog_e (double X, gsl_sf_result * RESULT)
These routines compute the dilogarithm for a real argument. In
Lewin's notation this is Li_2(x), the real part of the dilogarithm
of a real x. It is defined by the integral representation Li_2(x)
= - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for
x <= 1, and -\pi\log(x) for x > 1.
Note that Abramowitz & Stegun refer to the Spence integral
S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x).
7.11.2 Complex Argument
-----------------------
-- Function: int gsl_sf_complex_dilog_e (double R, double THETA,
gsl_sf_result * RESULT_RE, gsl_sf_result * RESULT_IM)
This function computes the full complex-valued dilogarithm for the
complex argument z = r \exp(i \theta). The real and imaginary
parts of the result are returned in RESULT_RE, RESULT_IM.
7.12 Elementary Operations
==========================
The following functions allow for the propagation of errors when
combining quantities by multiplication. The functions are declared in
the header file 'gsl_sf_elementary.h'.
-- Function: int gsl_sf_multiply_e (double X, double Y, gsl_sf_result *
RESULT)
This function multiplies X and Y storing the product and its
associated error in RESULT.
-- Function: int gsl_sf_multiply_err_e (double X, double DX, double Y,
double DY, gsl_sf_result * RESULT)
This function multiplies X and Y with associated absolute errors DX
and DY. The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is stored
in RESULT.
7.13 Elliptic Integrals
=======================
The functions described in this section are declared in the header file
'gsl_sf_ellint.h'. Further information about the elliptic integrals can
be found in Abramowitz & Stegun, Chapter 17.
7.13.1 Definition of Legendre Forms
-----------------------------------
The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and
\Pi(\phi,k,n) are defined by,
F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t)))
E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t)))
Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) =
E(\pi/2, k).
The notation used here is based on Carlson, 'Numerische Mathematik'
33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun,
where the functions are given in terms of the parameter m = k^2 and n is
replaced by -n.
7.13.2 Definition of Carlson Forms
----------------------------------
The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z),
RF(x,y,z) and RJ(x,y,z,p) are defined by,
RC(x,y) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1)
RD(x,y,z) = 3/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-3/2)
RF(x,y,z) = 1/2 \int_0^\infty dt (t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2)
RJ(x,y,z,p) = 3/2 \int_0^\infty dt
(t+x)^(-1/2) (t+y)^(-1/2) (t+z)^(-1/2) (t+p)^(-1)
7.13.3 Legendre Form of Complete Elliptic Integrals
---------------------------------------------------
-- Function: double gsl_sf_ellint_Kcomp (double K, gsl_mode_t MODE)
-- Function: int gsl_sf_ellint_Kcomp_e (double K, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute the complete elliptic integral K(k) to the
accuracy specified by the mode variable MODE. Note that Abramowitz
& Stegun define this function in terms of the parameter m = k^2.
-- Function: double gsl_sf_ellint_Ecomp (double K, gsl_mode_t MODE)
-- Function: int gsl_sf_ellint_Ecomp_e (double K, gsl_mode_t MODE,
gsl_sf_result * RESULT)
These routines compute the complete elliptic integral E(k) to the
accuracy specified by the mode variable MODE. Note that Abramowitz
& Stegun define this function in terms of the parameter m = k^2.
-- Function: double gsl_sf_ellint_Pcomp (double K, double N, gsl_mode_t
MODE)
-- Function: int gsl_sf_ellint_Pcomp_e (double K, double N, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These routines compute the complete elliptic integral \Pi(k,n) to
the accuracy specified by the mode variable MODE. Note that
Abramowitz & Stegun define this function in terms of the parameters
m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.
7.13.4 Legendre Form of Incomplete Elliptic Integrals
-----------------------------------------------------
-- Function: double gsl_sf_ellint_F (double PHI, double K, gsl_mode_t
MODE)
-- Function: int gsl_sf_ellint_F_e (double PHI, double K, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral F(\phi,k)
to the accuracy specified by the mode variable MODE. Note that
Abramowitz & Stegun define this function in terms of the parameter
m = k^2.
-- Function: double gsl_sf_ellint_E (double PHI, double K, gsl_mode_t
MODE)
-- Function: int gsl_sf_ellint_E_e (double PHI, double K, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral E(\phi,k)
to the accuracy specified by the mode variable MODE. Note that
Abramowitz & Stegun define this function in terms of the parameter
m = k^2.
-- Function: double gsl_sf_ellint_P (double PHI, double K, double N,
gsl_mode_t MODE)
-- Function: int gsl_sf_ellint_P_e (double PHI, double K, double N,
gsl_mode_t MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral
\Pi(\phi,k,n) to the accuracy specified by the mode variable MODE.
Note that Abramowitz & Stegun define this function in terms of the
parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of
sign n \to -n.
-- Function: double gsl_sf_ellint_D (double PHI, double K, gsl_mode_t
MODE)
-- Function: int gsl_sf_ellint_D_e (double PHI, double K, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These functions compute the incomplete elliptic integral D(\phi,k)
which is defined through the Carlson form RD(x,y,z) by the
following relation,
D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
7.13.5 Carlson Forms
--------------------
-- Function: double gsl_sf_ellint_RC (double X, double Y, gsl_mode_t
MODE)
-- Function: int gsl_sf_ellint_RC_e (double X, double Y, gsl_mode_t
MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral RC(x,y) to
the accuracy specified by the mode variable MODE.
-- Function: double gsl_sf_ellint_RD (double X, double Y, double Z,
gsl_mode_t MODE)
-- Function: int gsl_sf_ellint_RD_e (double X, double Y, double Z,
gsl_mode_t MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral RD(x,y,z)
to the accuracy specified by the mode variable MODE.
-- Function: double gsl_sf_ellint_RF (double X, double Y, double Z,
gsl_mode_t MODE)
-- Function: int gsl_sf_ellint_RF_e (double X, double Y, double Z,
gsl_mode_t MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral RF(x,y,z)
to the accuracy specified by the mode variable MODE.
-- Function: double gsl_sf_ellint_RJ (double X, double Y, double Z,
double P, gsl_mode_t MODE)
-- Function: int gsl_sf_ellint_RJ_e (double X, double Y, double Z,
double P, gsl_mode_t MODE, gsl_sf_result * RESULT)
These routines compute the incomplete elliptic integral RJ(x,y,z,p)
to the accuracy specified by the mode variable MODE.
7.14 Elliptic Functions (Jacobi)
================================
The Jacobian Elliptic functions are defined in Abramowitz & Stegun,
Chapter 16. The functions are declared in the header file
'gsl_sf_elljac.h'.
-- Function: int gsl_sf_elljac_e (double U, double M, double * SN,
double * CN, double * DN)
This function computes the Jacobian elliptic functions sn(u|m),
cn(u|m), dn(u|m) by descending Landen transformations.
7.15 Error Functions
====================
The error function is described in Abramowitz & Stegun, Chapter 7. The
functions in this section are declared in the header file
'gsl_sf_erf.h'.
7.15.1 Error Function
---------------------
-- Function: double gsl_sf_erf (double X)
-- Function: int gsl_sf_erf_e (double X, gsl_sf_result * RESULT)
These routines compute the error function erf(x), where erf(x) =
(2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).
7.15.2 Complementary Error Function
-----------------------------------
-- Function: double gsl_sf_erfc (double X)
-- Function: int gsl_sf_erfc_e (double X, gsl_sf_result * RESULT)
These routines compute the complementary error function erfc(x) = 1
- erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).
7.15.3 Log Complementary Error Function
---------------------------------------
-- Function: double gsl_sf_log_erfc (double X)
-- Function: int gsl_sf_log_erfc_e (double X, gsl_sf_result * RESULT)
These routines compute the logarithm of the complementary error
function \log(\erfc(x)).
7.15.4 Probability functions
----------------------------
The probability functions for the Normal or Gaussian distribution are
described in Abramowitz & Stegun, Section 26.2.
-- Function: double gsl_sf_erf_Z (double X)
-- Function: int gsl_sf_erf_Z_e (double X, gsl_sf_result * RESULT)
These routines compute the Gaussian probability density function
Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).
-- Function: double gsl_sf_erf_Q (double X)
-- Function: int gsl_sf_erf_Q_e (double X, gsl_sf_result * RESULT)
These routines compute the upper tail of the Gaussian probability
function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2).
The "hazard function" for the normal distribution, also known as the
inverse Mills' ratio, is defined as,
h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim
x as x approaches +\infty.
-- Function: double gsl_sf_hazard (double X)
-- Function: int gsl_sf_hazard_e (double X, gsl_sf_result * RESULT)
These routines compute the hazard function for the normal
distribution.
7.16 Exponential Functions
==========================
The functions described in this section are declared in the header file
'gsl_sf_exp.h'.
7.16.1 Exponential Function
---------------------------
-- Function: double gsl_sf_exp (double X)
-- Function: int gsl_sf_exp_e (double X, gsl_sf_result * RESULT)
These routines provide an exponential function \exp(x) using GSL
semantics and error checking.
-- Function: int gsl_sf_exp_e10_e (double X, gsl_sf_result_e10 *
RESULT)
This function computes the exponential \exp(x) using the
'gsl_sf_result_e10' type to return a result with extended range.
This function may be useful if the value of \exp(x) would overflow
the numeric range of 'double'.
-- Function: double gsl_sf_exp_mult (double X, double Y)
-- Function: int gsl_sf_exp_mult_e (double X, double Y, gsl_sf_result *
RESULT)
These routines exponentiate X and multiply by the factor Y to
return the product y \exp(x).
-- Function: int gsl_sf_exp_mult_e10_e (const double X, const double Y,
gsl_sf_result_e10 * RESULT)
This function computes the product y \exp(x) using the
'gsl_sf_result_e10' type to return a result with extended numeric
range.
7.16.2 Relative Exponential Functions
-------------------------------------
-- Function: double gsl_sf_expm1 (double X)
-- Function: int gsl_sf_expm1_e (double X, gsl_sf_result * RESULT)
These routines compute the quantity \exp(x)-1 using an algorithm
that is accurate for small x.
-- Function: double gsl_sf_exprel (double X)
-- Function: int gsl_sf_exprel_e (double X, gsl_sf_result * RESULT)
These routines compute the quantity (\exp(x)-1)/x using an
algorithm that is accurate for small x. For small x the algorithm
is based on the expansion (\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) +
x^3/(2*3*4) + \dots.
-- Function: double gsl_sf_exprel_2 (double X)
-- Function: int gsl_sf_exprel_2_e (double X, gsl_sf_result * RESULT)
These routines compute the quantity 2(\exp(x)-1-x)/x^2 using an
algorithm that is accurate for small x. For small x the algorithm
is based on the expansion 2(\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4)
+ x^3/(3*4*5) + \dots.
-- Function: double gsl_sf_exprel_n (int N, double X)
-- Function: int gsl_sf_exprel_n_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the N-relative exponential, which is the
N-th generalization of the functions 'gsl_sf_exprel' and
'gsl_sf_exprel_2'. The N-relative exponential is given by,
exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!)
= 1 + x/(N+1) + x^2/((N+1)(N+2)) + ...
= 1F1 (1,1+N,x)
7.16.3 Exponentiation With Error Estimate
-----------------------------------------
-- Function: int gsl_sf_exp_err_e (double X, double DX, gsl_sf_result *
RESULT)
This function exponentiates X with an associated absolute error DX.
-- Function: int gsl_sf_exp_err_e10_e (double X, double DX,
gsl_sf_result_e10 * RESULT)
This function exponentiates a quantity X with an associated
absolute error DX using the 'gsl_sf_result_e10' type to return a
result with extended range.
-- Function: int gsl_sf_exp_mult_err_e (double X, double DX, double Y,
double DY, gsl_sf_result * RESULT)
This routine computes the product y \exp(x) for the quantities X, Y
with associated absolute errors DX, DY.
-- Function: int gsl_sf_exp_mult_err_e10_e (double X, double DX, double
Y, double DY, gsl_sf_result_e10 * RESULT)
This routine computes the product y \exp(x) for the quantities X, Y
with associated absolute errors DX, DY using the
'gsl_sf_result_e10' type to return a result with extended range.
7.17 Exponential Integrals
==========================
Information on the exponential integrals can be found in Abramowitz &
Stegun, Chapter 5. These functions are declared in the header file
'gsl_sf_expint.h'.
7.17.1 Exponential Integral
---------------------------
-- Function: double gsl_sf_expint_E1 (double X)
-- Function: int gsl_sf_expint_E1_e (double X, gsl_sf_result * RESULT)
These routines compute the exponential integral E_1(x),
E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
-- Function: double gsl_sf_expint_E2 (double X)
-- Function: int gsl_sf_expint_E2_e (double X, gsl_sf_result * RESULT)
These routines compute the second-order exponential integral
E_2(x),
E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
-- Function: double gsl_sf_expint_En (int N, double X)
-- Function: int gsl_sf_expint_En_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the exponential integral E_n(x) of order n,
E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.
7.17.2 Ei(x)
------------
-- Function: double gsl_sf_expint_Ei (double X)
-- Function: int gsl_sf_expint_Ei_e (double X, gsl_sf_result * RESULT)
These routines compute the exponential integral Ei(x),
Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t)
where PV denotes the principal value of the integral.
7.17.3 Hyperbolic Integrals
---------------------------
-- Function: double gsl_sf_Shi (double X)
-- Function: int gsl_sf_Shi_e (double X, gsl_sf_result * RESULT)
These routines compute the integral Shi(x) = \int_0^x dt
\sinh(t)/t.
-- Function: double gsl_sf_Chi (double X)
-- Function: int gsl_sf_Chi_e (double X, gsl_sf_result * RESULT)
These routines compute the integral Chi(x) := \Re[ \gamma_E +
\log(x) + \int_0^x dt (\cosh(t)-1)/t] , where \gamma_E is the Euler
constant (available as the macro 'M_EULER').
7.17.4 Ei_3(x)
--------------
-- Function: double gsl_sf_expint_3 (double X)
-- Function: int gsl_sf_expint_3_e (double X, gsl_sf_result * RESULT)
These routines compute the third-order exponential integral Ei_3(x)
= \int_0^xdt \exp(-t^3) for x >= 0.
7.17.5 Trigonometric Integrals
------------------------------
-- Function: double gsl_sf_Si (const double X)
-- Function: int gsl_sf_Si_e (double X, gsl_sf_result * RESULT)
These routines compute the Sine integral Si(x) = \int_0^x dt
\sin(t)/t.
-- Function: double gsl_sf_Ci (const double X)
-- Function: int gsl_sf_Ci_e (double X, gsl_sf_result * RESULT)
These routines compute the Cosine integral Ci(x) = -\int_x^\infty
dt \cos(t)/t for x > 0.
7.17.6 Arctangent Integral
--------------------------
-- Function: double gsl_sf_atanint (double X)
-- Function: int gsl_sf_atanint_e (double X, gsl_sf_result * RESULT)
These routines compute the Arctangent integral, which is defined as
AtanInt(x) = \int_0^x dt \arctan(t)/t.
7.18 Fermi-Dirac Function
=========================
The functions described in this section are declared in the header file
'gsl_sf_fermi_dirac.h'.
7.18.1 Complete Fermi-Dirac Integrals
-------------------------------------
The complete Fermi-Dirac integral F_j(x) is given by,
F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
Note that the Fermi-Dirac integral is sometimes defined without the
normalisation factor in other texts.
-- Function: double gsl_sf_fermi_dirac_m1 (double X)
-- Function: int gsl_sf_fermi_dirac_m1_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral with an
index of -1. This integral is given by F_{-1}(x) = e^x / (1 +
e^x).
-- Function: double gsl_sf_fermi_dirac_0 (double X)
-- Function: int gsl_sf_fermi_dirac_0_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral with an
index of 0. This integral is given by F_0(x) = \ln(1 + e^x).
-- Function: double gsl_sf_fermi_dirac_1 (double X)
-- Function: int gsl_sf_fermi_dirac_1_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral with an
index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)).
-- Function: double gsl_sf_fermi_dirac_2 (double X)
-- Function: int gsl_sf_fermi_dirac_2_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral with an
index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)).
-- Function: double gsl_sf_fermi_dirac_int (int J, double X)
-- Function: int gsl_sf_fermi_dirac_int_e (int J, double X,
gsl_sf_result * RESULT)
These routines compute the complete Fermi-Dirac integral with an
integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j
/(\exp(t-x)+1)).
-- Function: double gsl_sf_fermi_dirac_mhalf (double X)
-- Function: int gsl_sf_fermi_dirac_mhalf_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral
F_{-1/2}(x).
-- Function: double gsl_sf_fermi_dirac_half (double X)
-- Function: int gsl_sf_fermi_dirac_half_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral
F_{1/2}(x).
-- Function: double gsl_sf_fermi_dirac_3half (double X)
-- Function: int gsl_sf_fermi_dirac_3half_e (double X, gsl_sf_result *
RESULT)
These routines compute the complete Fermi-Dirac integral
F_{3/2}(x).
7.18.2 Incomplete Fermi-Dirac Integrals
---------------------------------------
The incomplete Fermi-Dirac integral F_j(x,b) is given by,
F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))
-- Function: double gsl_sf_fermi_dirac_inc_0 (double X, double B)
-- Function: int gsl_sf_fermi_dirac_inc_0_e (double X, double B,
gsl_sf_result * RESULT)
These routines compute the incomplete Fermi-Dirac integral with an
index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).
7.19 Gamma and Beta Functions
=============================
This following routines compute the gamma and beta functions in their
full and incomplete forms, as well as various kinds of factorials. The
functions described in this section are declared in the header file
'gsl_sf_gamma.h'.
7.19.1 Gamma Functions
----------------------
The Gamma function is defined by the following integral,
\Gamma(x) = \int_0^\infty dt t^{x-1} \exp(-t)
It is related to the factorial function by \Gamma(n)=(n-1)! for positive
integer n. Further information on the Gamma function can be found in
Abramowitz & Stegun, Chapter 6.
-- Function: double gsl_sf_gamma (double X)
-- Function: int gsl_sf_gamma_e (double X, gsl_sf_result * RESULT)
These routines compute the Gamma function \Gamma(x), subject to x
not being a negative integer or zero. The function is computed
using the real Lanczos method. The maximum value of x such that
\Gamma(x) is not considered an overflow is given by the macro
'GSL_SF_GAMMA_XMAX' and is 171.0.
-- Function: double gsl_sf_lngamma (double X)
-- Function: int gsl_sf_lngamma_e (double X, gsl_sf_result * RESULT)
These routines compute the logarithm of the Gamma function,
\log(\Gamma(x)), subject to x not being a negative integer or zero.
For x<0 the real part of \log(\Gamma(x)) is returned, which is
equivalent to \log(|\Gamma(x)|). The function is computed using
the real Lanczos method.
-- Function: int gsl_sf_lngamma_sgn_e (double X, gsl_sf_result *
RESULT_LG, double * SGN)
This routine computes the sign of the gamma function and the
logarithm of its magnitude, subject to x not being a negative
integer or zero. The function is computed using the real Lanczos
method. The value of the gamma function and its error can be
reconstructed using the relation \Gamma(x) = sgn *
\exp(result\_lg), taking into account the two components of
RESULT_LG.
-- Function: double gsl_sf_gammastar (double X)
-- Function: int gsl_sf_gammastar_e (double X, gsl_sf_result * RESULT)
These routines compute the regulated Gamma Function \Gamma^*(x) for
x > 0. The regulated gamma function is given by,
\Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x))
= (1 + (1/12x) + ...) for x \to \infty
and is a useful suggestion of Temme.
-- Function: double gsl_sf_gammainv (double X)
-- Function: int gsl_sf_gammainv_e (double X, gsl_sf_result * RESULT)
These routines compute the reciprocal of the gamma function,
1/\Gamma(x) using the real Lanczos method.
-- Function: int gsl_sf_lngamma_complex_e (double ZR, double ZI,
gsl_sf_result * LNR, gsl_sf_result * ARG)
This routine computes \log(\Gamma(z)) for complex z=z_r+i z_i and z
not a negative integer or zero, using the complex Lanczos method.
The returned parameters are lnr = \log|\Gamma(z)| and arg =
\arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase part (ARG) is
not well-determined when |z| is very large, due to inevitable
roundoff in restricting to (-\pi,\pi]. This will result in a
'GSL_ELOSS' error when it occurs. The absolute value part (LNR),
however, never suffers from loss of precision.
7.19.2 Factorials
-----------------
Although factorials can be computed from the Gamma function, using the
relation n! = \Gamma(n+1) for non-negative integer n, it is usually more
efficient to call the functions in this section, particularly for small
values of n, whose factorial values are maintained in hardcoded tables.
-- Function: double gsl_sf_fact (unsigned int N)
-- Function: int gsl_sf_fact_e (unsigned int N, gsl_sf_result * RESULT)
These routines compute the factorial n!. The factorial is related
to the Gamma function by n! = \Gamma(n+1). The maximum value of n
such that n! is not considered an overflow is given by the macro
'GSL_SF_FACT_NMAX' and is 170.
-- Function: double gsl_sf_doublefact (unsigned int N)
-- Function: int gsl_sf_doublefact_e (unsigned int N, gsl_sf_result *
RESULT)
These routines compute the double factorial n!! = n(n-2)(n-4)
\dots. The maximum value of n such that n!! is not considered an
overflow is given by the macro 'GSL_SF_DOUBLEFACT_NMAX' and is 297.
-- Function: double gsl_sf_lnfact (unsigned int N)
-- Function: int gsl_sf_lnfact_e (unsigned int N, gsl_sf_result *
RESULT)
These routines compute the logarithm of the factorial of N,
\log(n!). The algorithm is faster than computing \ln(\Gamma(n+1))
via 'gsl_sf_lngamma' for n < 170, but defers for larger N.
-- Function: double gsl_sf_lndoublefact (unsigned int N)
-- Function: int gsl_sf_lndoublefact_e (unsigned int N, gsl_sf_result *
RESULT)
These routines compute the logarithm of the double factorial of N,
\log(n!!).
-- Function: double gsl_sf_choose (unsigned int N, unsigned int M)
-- Function: int gsl_sf_choose_e (unsigned int N, unsigned int M,
gsl_sf_result * RESULT)
These routines compute the combinatorial factor 'n choose m' =
n!/(m!(n-m)!)
-- Function: double gsl_sf_lnchoose (unsigned int N, unsigned int M)
-- Function: int gsl_sf_lnchoose_e (unsigned int N, unsigned int M,
gsl_sf_result * RESULT)
These routines compute the logarithm of 'n choose m'. This is
equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).
-- Function: double gsl_sf_taylorcoeff (int N, double X)
-- Function: int gsl_sf_taylorcoeff_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the Taylor coefficient x^n / n! for x >= 0,
n >= 0.
7.19.3 Pochhammer Symbol
------------------------
-- Function: double gsl_sf_poch (double A, double X)
-- Function: int gsl_sf_poch_e (double A, double X, gsl_sf_result *
RESULT)
These routines compute the Pochhammer symbol (a)_x = \Gamma(a +
x)/\Gamma(a). The Pochhammer symbol is also known as the Apell
symbol and sometimes written as (a,x). When a and a+x are negative
integers or zero, the limiting value of the ratio is returned.
-- Function: double gsl_sf_lnpoch (double A, double X)
-- Function: int gsl_sf_lnpoch_e (double A, double X, gsl_sf_result *
RESULT)
These routines compute the logarithm of the Pochhammer symbol,
\log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)).
-- Function: int gsl_sf_lnpoch_sgn_e (double A, double X, gsl_sf_result
* RESULT, double * SGN)
These routines compute the sign of the Pochhammer symbol and the
logarithm of its magnitude. The computed parameters are result =
\log(|(a)_x|) with a corresponding error term, and sgn =
\sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).
-- Function: double gsl_sf_pochrel (double A, double X)
-- Function: int gsl_sf_pochrel_e (double A, double X, gsl_sf_result *
RESULT)
These routines compute the relative Pochhammer symbol ((a)_x - 1)/x
where (a)_x = \Gamma(a + x)/\Gamma(a).
7.19.4 Incomplete Gamma Functions
---------------------------------
-- Function: double gsl_sf_gamma_inc (double A, double X)
-- Function: int gsl_sf_gamma_inc_e (double A, double X, gsl_sf_result
* RESULT)
These functions compute the unnormalized incomplete Gamma Function
\Gamma(a,x) = \int_x^\infty dt t^{a-1} \exp(-t) for a real and x >=
0.
-- Function: double gsl_sf_gamma_inc_Q (double A, double X)
-- Function: int gsl_sf_gamma_inc_Q_e (double A, double X,
gsl_sf_result * RESULT)
These routines compute the normalized incomplete Gamma Function
Q(a,x) = 1/\Gamma(a) \int_x^\infty dt t^{a-1} \exp(-t) for a > 0, x
>= 0.
-- Function: double gsl_sf_gamma_inc_P (double A, double X)
-- Function: int gsl_sf_gamma_inc_P_e (double A, double X,
gsl_sf_result * RESULT)
These routines compute the complementary normalized incomplete
Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt
t^{a-1} \exp(-t) for a > 0, x >= 0.
Note that Abramowitz & Stegun call P(a,x) the incomplete gamma
function (section 6.5).
7.19.5 Beta Functions
---------------------
-- Function: double gsl_sf_beta (double A, double B)
-- Function: int gsl_sf_beta_e (double A, double B, gsl_sf_result *
RESULT)
These routines compute the Beta Function, B(a,b) =
\Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being
negative integers.
-- Function: double gsl_sf_lnbeta (double A, double B)
-- Function: int gsl_sf_lnbeta_e (double A, double B, gsl_sf_result *
RESULT)
These routines compute the logarithm of the Beta Function,
\log(B(a,b)) subject to a and b not being negative integers.
7.19.6 Incomplete Beta Function
-------------------------------
-- Function: double gsl_sf_beta_inc (double A, double B, double X)
-- Function: int gsl_sf_beta_inc_e (double A, double B, double X,
gsl_sf_result * RESULT)
These routines compute the normalized incomplete Beta function
I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1}
(1-t)^{b-1} dt for 0 <= x <= 1. For a > 0, b > 0 the value is
computed using a continued fraction expansion. For all other
values it is computed using the relation I_x(a,b,x) = (1/a) x^a
2F1(a,1-b,a+1,x)/B(a,b).
7.20 Gegenbauer Functions
=========================
The Gegenbauer polynomials are defined in Abramowitz & Stegun, Chapter
22, where they are known as Ultraspherical polynomials. The functions
described in this section are declared in the header file
'gsl_sf_gegenbauer.h'.
-- Function: double gsl_sf_gegenpoly_1 (double LAMBDA, double X)
-- Function: double gsl_sf_gegenpoly_2 (double LAMBDA, double X)
-- Function: double gsl_sf_gegenpoly_3 (double LAMBDA, double X)
-- Function: int gsl_sf_gegenpoly_1_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
-- Function: int gsl_sf_gegenpoly_2_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
-- Function: int gsl_sf_gegenpoly_3_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
These functions evaluate the Gegenbauer polynomials
C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.
-- Function: double gsl_sf_gegenpoly_n (int N, double LAMBDA, double X)
-- Function: int gsl_sf_gegenpoly_n_e (int N, double LAMBDA, double X,
gsl_sf_result * RESULT)
These functions evaluate the Gegenbauer polynomial
C^{(\lambda)}_n(x) for a specific value of N, LAMBDA, X subject to
\lambda > -1/2, n >= 0.
-- Function: int gsl_sf_gegenpoly_array (int NMAX, double LAMBDA,
double X, double RESULT_ARRAY[])
This function computes an array of Gegenbauer polynomials
C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda
> -1/2, nmax >= 0.
7.21 Hypergeometric Functions
=============================
Hypergeometric functions are described in Abramowitz & Stegun, Chapters
13 and 15. These functions are declared in the header file
'gsl_sf_hyperg.h'.
-- Function: double gsl_sf_hyperg_0F1 (double C, double X)
-- Function: int gsl_sf_hyperg_0F1_e (double C, double X, gsl_sf_result
* RESULT)
These routines compute the hypergeometric function 0F1(c,x).
-- Function: double gsl_sf_hyperg_1F1_int (int M, int N, double X)
-- Function: int gsl_sf_hyperg_1F1_int_e (int M, int N, double X,
gsl_sf_result * RESULT)
These routines compute the confluent hypergeometric function
1F1(m,n,x) = M(m,n,x) for integer parameters M, N.
-- Function: double gsl_sf_hyperg_1F1 (double A, double B, double X)
-- Function: int gsl_sf_hyperg_1F1_e (double A, double B, double X,
gsl_sf_result * RESULT)
These routines compute the confluent hypergeometric function
1F1(a,b,x) = M(a,b,x) for general parameters A, B.
-- Function: double gsl_sf_hyperg_U_int (int M, int N, double X)
-- Function: int gsl_sf_hyperg_U_int_e (int M, int N, double X,
gsl_sf_result * RESULT)
These routines compute the confluent hypergeometric function
U(m,n,x) for integer parameters M, N.
-- Function: int gsl_sf_hyperg_U_int_e10_e (int M, int N, double X,
gsl_sf_result_e10 * RESULT)
This routine computes the confluent hypergeometric function
U(m,n,x) for integer parameters M, N using the 'gsl_sf_result_e10'
type to return a result with extended range.
-- Function: double gsl_sf_hyperg_U (double A, double B, double X)
-- Function: int gsl_sf_hyperg_U_e (double A, double B, double X,
gsl_sf_result * RESULT)
These routines compute the confluent hypergeometric function
U(a,b,x).
-- Function: int gsl_sf_hyperg_U_e10_e (double A, double B, double X,
gsl_sf_result_e10 * RESULT)
This routine computes the confluent hypergeometric function
U(a,b,x) using the 'gsl_sf_result_e10' type to return a result with
extended range.
-- Function: double gsl_sf_hyperg_2F1 (double A, double B, double C,
double X)
-- Function: int gsl_sf_hyperg_2F1_e (double A, double B, double C,
double X, gsl_sf_result * RESULT)
These routines compute the Gauss hypergeometric function
2F1(a,b,c,x) = F(a,b,c,x) for |x| < 1.
If the arguments (a,b,c,x) are too close to a singularity then the
function can return the error code 'GSL_EMAXITER' when the series
approximation converges too slowly. This occurs in the region of
x=1, c - a - b = m for integer m.
-- Function: double gsl_sf_hyperg_2F1_conj (double AR, double AI,
double C, double X)
-- Function: int gsl_sf_hyperg_2F1_conj_e (double AR, double AI, double
C, double X, gsl_sf_result * RESULT)
These routines compute the Gauss hypergeometric function 2F1(a_R +
i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1.
-- Function: double gsl_sf_hyperg_2F1_renorm (double A, double B,
double C, double X)
-- Function: int gsl_sf_hyperg_2F1_renorm_e (double A, double B, double
C, double X, gsl_sf_result * RESULT)
These routines compute the renormalized Gauss hypergeometric
function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.
-- Function: double gsl_sf_hyperg_2F1_conj_renorm (double AR, double
AI, double C, double X)
-- Function: int gsl_sf_hyperg_2F1_conj_renorm_e (double AR, double AI,
double C, double X, gsl_sf_result * RESULT)
These routines compute the renormalized Gauss hypergeometric
function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| <
1.
-- Function: double gsl_sf_hyperg_2F0 (double A, double B, double X)
-- Function: int gsl_sf_hyperg_2F0_e (double A, double B, double X,
gsl_sf_result * RESULT)
These routines compute the hypergeometric function 2F0(a,b,x). The
series representation is a divergent hypergeometric series.
However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)
7.22 Laguerre Functions
=======================
The generalized Laguerre polynomials are defined in terms of confluent
hypergeometric functions as L^a_n(x) = ((a+1)_n / n!) 1F1(-n,a+1,x), and
are sometimes referred to as the associated Laguerre polynomials. They
are related to the plain Laguerre polynomials L_n(x) by L^0_n(x) =
L_n(x) and L^k_n(x) = (-1)^k (d^k/dx^k) L_(n+k)(x). For more
information see Abramowitz & Stegun, Chapter 22.
The functions described in this section are declared in the header
file 'gsl_sf_laguerre.h'.
-- Function: double gsl_sf_laguerre_1 (double A, double X)
-- Function: double gsl_sf_laguerre_2 (double A, double X)
-- Function: double gsl_sf_laguerre_3 (double A, double X)
-- Function: int gsl_sf_laguerre_1_e (double A, double X, gsl_sf_result
* RESULT)
-- Function: int gsl_sf_laguerre_2_e (double A, double X, gsl_sf_result
* RESULT)
-- Function: int gsl_sf_laguerre_3_e (double A, double X, gsl_sf_result
* RESULT)
These routines evaluate the generalized Laguerre polynomials
L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.
-- Function: double gsl_sf_laguerre_n (const int N, const double A,
const double X)
-- Function: int gsl_sf_laguerre_n_e (int N, double A, double X,
gsl_sf_result * RESULT)
These routines evaluate the generalized Laguerre polynomials
L^a_n(x) for a > -1, n >= 0.
7.23 Lambert W Functions
========================
Lambert's W functions, W(x), are defined to be solutions of the equation
W(x) \exp(W(x)) = x. This function has multiple branches for x < 0;
however, it has only two real-valued branches. We define W_0(x) to be
the principal branch, where W > -1 for x < 0, and W_{-1}(x) to be the
other real branch, where W < -1 for x < 0. The Lambert functions are
declared in the header file 'gsl_sf_lambert.h'.
-- Function: double gsl_sf_lambert_W0 (double X)
-- Function: int gsl_sf_lambert_W0_e (double X, gsl_sf_result * RESULT)
These compute the principal branch of the Lambert W function,
W_0(x).
-- Function: double gsl_sf_lambert_Wm1 (double X)
-- Function: int gsl_sf_lambert_Wm1_e (double X, gsl_sf_result *
RESULT)
These compute the secondary real-valued branch of the Lambert W
function, W_{-1}(x).
7.24 Legendre Functions and Spherical Harmonics
===============================================
The Legendre Functions and Legendre Polynomials are described in
Abramowitz & Stegun, Chapter 8. These functions are declared in the
header file 'gsl_sf_legendre.h'.
7.24.1 Legendre Polynomials
---------------------------
-- Function: double gsl_sf_legendre_P1 (double X)
-- Function: double gsl_sf_legendre_P2 (double X)
-- Function: double gsl_sf_legendre_P3 (double X)
-- Function: int gsl_sf_legendre_P1_e (double X, gsl_sf_result *
RESULT)
-- Function: int gsl_sf_legendre_P2_e (double X, gsl_sf_result *
RESULT)
-- Function: int gsl_sf_legendre_P3_e (double X, gsl_sf_result *
RESULT)
These functions evaluate the Legendre polynomials P_l(x) using
explicit representations for l=1, 2, 3.
-- Function: double gsl_sf_legendre_Pl (int L, double X)
-- Function: int gsl_sf_legendre_Pl_e (int L, double X, gsl_sf_result *
RESULT)
These functions evaluate the Legendre polynomial P_l(x) for a
specific value of L, X subject to l >= 0, |x| <= 1
-- Function: int gsl_sf_legendre_Pl_array (int LMAX, double X, double
RESULT_ARRAY[])
-- Function: int gsl_sf_legendre_Pl_deriv_array (int LMAX, double X,
double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
These functions compute arrays of Legendre polynomials P_l(x) and
derivatives dP_l(x)/dx, for l = 0, \dots, lmax, |x| <= 1
-- Function: double gsl_sf_legendre_Q0 (double X)
-- Function: int gsl_sf_legendre_Q0_e (double X, gsl_sf_result *
RESULT)
These routines compute the Legendre function Q_0(x) for x > -1, x
!= 1.
-- Function: double gsl_sf_legendre_Q1 (double X)
-- Function: int gsl_sf_legendre_Q1_e (double X, gsl_sf_result *
RESULT)
These routines compute the Legendre function Q_1(x) for x > -1, x
!= 1.
-- Function: double gsl_sf_legendre_Ql (int L, double X)
-- Function: int gsl_sf_legendre_Ql_e (int L, double X, gsl_sf_result *
RESULT)
These routines compute the Legendre function Q_l(x) for x > -1, x
!= 1 and l >= 0.
7.24.2 Associated Legendre Polynomials and Spherical Harmonics
--------------------------------------------------------------
The following functions compute the associated Legendre polynomials
P_l^m(x) which are solutions of the differential equation
(1 - x^2) d^2 P_l^m(x) / dx^2 P_l^m(x) - 2x d/dx P_l^m(x) +
( l(l+1) - m^2 / (1 - x^2) ) P_l^m(x) = 0
where the degree l and order m satisfy 0 \le l and 0 \le m \le l.
The functions P_l^m(x) grow combinatorially with l and can overflow for
l larger than about 150. Alternatively, one may calculate normalized
associated Legendre polynomials. There are a number of different
normalization conventions, and these functions can be stably computed up
to degree and order 2700. The following normalizations are provided:
'Schmidt semi-normalization'
Schmidt semi-normalized associated Legendre polynomials are often
used in the magnetics community and are defined as
S_l^0(x) = P_l^0(x)
S_l^m(x) = (-1)^m \sqrt((2(l-m)! / (l+m)!)) P_l^m(x), m > 0
The factor of (-1)^m is called the Condon-Shortley phase factor and
can be excluded if desired by setting the parameter 'csphase = 1'
in the functions below.
'Spherical Harmonic Normalization'
The associated Legendre polynomials suitable for calculating
spherical harmonics are defined as
Y_l^m(x) = (-1)^m \sqrt((2l + 1) * (l-m)! / (4 \pi) / (l+m)!) P_l^m(x)
where again the phase factor (-1)^m can be included or excluded if
desired.
'Full Normalization'
The fully normalized associated Legendre polynomials are defined as
N_l^m(x) = (-1)^m \sqrt((l + 1/2) * (l-m)! / (l+m)!) P_l^m(x)
and have the property
\int_(-1)^1 ( N_l^m(x) )^2 dx = 1
The normalized associated Legendre routines below use a recurrence
relation which is stable up to a degree and order of about 2700. Beyond
this, the computed functions could suffer from underflow leading to
incorrect results. Routines are provided to compute first and second
derivatives dP_l^m(x)/dx and d^2 P_l^m(x)/dx^2 as well as their
alternate versions d P_l^m(\cos{\theta})/d\theta and d^2
P_l^m(\cos{\theta})/d\theta^2. While there is a simple scaling
relationship between the two forms, the derivatives involving \theta are
heavily used in spherical harmonic expansions and so these routines are
also provided.
In the functions below, a parameter of type 'gsl_sf_legendre_t'
specifies the type of normalization to use. The possible values are
'GSL_SF_LEGENDRE_NONE'
This specifies the computation of the unnormalized associated
Legendre polynomials P_l^m(x).
'GSL_SF_LEGENDRE_SCHMIDT'
This specifies the computation of the Schmidt semi-normalized
associated Legendre polynomials S_l^m(x).
'GSL_SF_LEGENDRE_SPHARM'
This specifies the computation of the spherical harmonic associated
Legendre polynomials Y_l^m(x).
'GSL_SF_LEGENDRE_FULL'
This specifies the computation of the fully normalized associated
Legendre polynomials N_l^m(x).
-- Function: int gsl_sf_legendre_array (const gsl_sf_legendre_t NORM,
const size_t LMAX, const double X, double RESULT_ARRAY[])
-- Function: int gsl_sf_legendre_array_e (const gsl_sf_legendre_t NORM,
const size_t LMAX, const double X, const double CSPHASE,
double RESULT_ARRAY[])
These functions calculate all normalized associated Legendre
polynomials for 0 \le l \le lmax and 0 \le m \le l for |x| <= 1.
The NORM parameter specifies which normalization is used. The
normalized P_l^m(x) values are stored in RESULT_ARRAY, whose
minimum size can be obtained from calling
'gsl_sf_legendre_array_n'. The array index of P_l^m(x) is obtained
from calling 'gsl_sf_legendre_array_index(l, m)'. To include or
exclude the Condon-Shortley phase factor of (-1)^m, set the
parameter CSPHASE to either -1 or 1 respectively in the '_e'
function. This factor is included by default.
-- Function: int gsl_sf_legendre_deriv_array (const gsl_sf_legendre_t
NORM, const size_t LMAX, const double X, double
RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
-- Function: int gsl_sf_legendre_deriv_array_e (const gsl_sf_legendre_t
NORM, const size_t LMAX, const double X, const double CSPHASE,
double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
These functions calculate all normalized associated Legendre
functions and their first derivatives up to degree LMAX for |x| <
1. The parameter NORM specifies the normalization used. The
normalized P_l^m(x) values and their derivatives dP_l^m(x)/dx are
stored in RESULT_ARRAY and RESULT_DERIV_ARRAY respectively. To
include or exclude the Condon-Shortley phase factor of (-1)^m, set
the parameter CSPHASE to either -1 or 1 respectively in the '_e'
function. This factor is included by default.
-- Function: int gsl_sf_legendre_deriv_alt_array (const
gsl_sf_legendre_t NORM, const size_t LMAX, const double X,
double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
-- Function: int gsl_sf_legendre_deriv_alt_array_e (const
gsl_sf_legendre_t NORM, const size_t LMAX, const double X,
const double CSPHASE, double RESULT_ARRAY[], double
RESULT_DERIV_ARRAY[])
These functions calculate all normalized associated Legendre
functions and their (alternate) first derivatives up to degree LMAX
for |x| < 1. The normalized P_l^m(x) values and their derivatives
dP_l^m(\cos{\theta})/d\theta are stored in RESULT_ARRAY and
RESULT_DERIV_ARRAY respectively. To include or exclude the
Condon-Shortley phase factor of (-1)^m, set the parameter CSPHASE
to either -1 or 1 respectively in the '_e' function. This factor
is included by default.
-- Function: int gsl_sf_legendre_deriv2_array (const gsl_sf_legendre_t
NORM, const size_t LMAX, const double X, double
RESULT_ARRAY[], double RESULT_DERIV_ARRAY[], double
RESULT_DERIV2_ARRAY[])
-- Function: int gsl_sf_legendre_deriv2_array_e (const
gsl_sf_legendre_t NORM, const size_t LMAX, const double X,
const double CSPHASE, double RESULT_ARRAY[], double
RESULT_DERIV_ARRAY[], double RESULT_DERIV2_ARRAY[])
These functions calculate all normalized associated Legendre
functions and their first and second derivatives up to degree LMAX
for |x| < 1. The parameter NORM specifies the normalization used.
The normalized P_l^m(x), their first derivatives dP_l^m(x)/dx, and
their second derivatives d^2 P_l^m(x)/dx^2 are stored in
RESULT_ARRAY, RESULT_DERIV_ARRAY, and RESULT_DERIV2_ARRAY
respectively. To include or exclude the Condon-Shortley phase
factor of (-1)^m, set the parameter CSPHASE to either -1 or 1
respectively in the '_e' function. This factor is included by
default.
-- Function: int gsl_sf_legendre_deriv2_alt_array (const
gsl_sf_legendre_t NORM, const size_t LMAX, const double X,
double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[], double
RESULT_DERIV2_ARRAY[])
-- Function: int gsl_sf_legendre_deriv2_alt_array_e (const
gsl_sf_legendre_t NORM, const size_t LMAX, const double X,
const double CSPHASE, double RESULT_ARRAY[], double
RESULT_DERIV_ARRAY[], double RESULT_DERIV2_ARRAY[])
These functions calculate all normalized associated Legendre
functions and their (alternate) first and second derivatives up to
degree LMAX for |x| < 1. The parameter NORM specifies the
normalization used. The normalized P_l^m(x), their first
derivatives dP_l^m(\cos{\theta})/d\theta, and their second
derivatives d^2 P_l^m(\cos{\theta})/d\theta^2 are stored in
RESULT_ARRAY, RESULT_DERIV_ARRAY, and RESULT_DERIV2_ARRAY
respectively. To include or exclude the Condon-Shortley phase
factor of (-1)^m, set the parameter CSPHASE to either -1 or 1
respectively in the '_e' function. This factor is included by
default.
-- Function: size_t gsl_sf_legendre_array_n (const size_t LMAX)
This function returns the minimum array size for maximum degree
LMAX needed for the array versions of the associated Legendre
functions. Size is calculated as the total number of P_l^m(x)
functions, plus extra space for precomputing multiplicative factors
used in the recurrence relations.
-- Function: size_t gsl_sf_legendre_array_index (const size_t L, const
size_t M)
This function returns the index into RESULT_ARRAY,
RESULT_DERIV_ARRAY, or RESULT_DERIV2_ARRAY corresponding to
P_l^m(x), P_l^{'m}(x), or P_l^{''m}(x). The index is given by
l(l+1)/2 + m.
-- Function: double gsl_sf_legendre_Plm (int L, int M, double X)
-- Function: int gsl_sf_legendre_Plm_e (int L, int M, double X,
gsl_sf_result * RESULT)
These routines compute the associated Legendre polynomial P_l^m(x)
for m >= 0, l >= m, |x| <= 1.
-- Function: double gsl_sf_legendre_sphPlm (int L, int M, double X)
-- Function: int gsl_sf_legendre_sphPlm_e (int L, int M, double X,
gsl_sf_result * RESULT)
These routines compute the normalized associated Legendre
polynomial \sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)
suitable for use in spherical harmonics. The parameters must
satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the
overflows that occur for the standard normalization of P_l^m(x).
-- Function: int gsl_sf_legendre_Plm_array (int LMAX, int M, double X,
double RESULT_ARRAY[])
-- Function: int gsl_sf_legendre_Plm_deriv_array (int LMAX, int M,
double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
These functions are now deprecated and will be removed in a future
release; see 'gsl_sf_legendre_array' and
'gsl_sf_legendre_deriv_array'.
-- Function: int gsl_sf_legendre_sphPlm_array (int LMAX, int M, double
X, double RESULT_ARRAY[])
-- Function: int gsl_sf_legendre_sphPlm_deriv_array (int LMAX, int M,
double X, double RESULT_ARRAY[], double RESULT_DERIV_ARRAY[])
These functions are now deprecated and will be removed in a future
release; see 'gsl_sf_legendre_array' and
'gsl_sf_legendre_deriv_array'.
-- Function: int gsl_sf_legendre_array_size (const int LMAX, const int
M)
This function is now deprecated and will be removed in a future
release.
7.24.3 Conical Functions
------------------------
The Conical Functions P^\mu_{-(1/2)+i\lambda}(x) and
Q^\mu_{-(1/2)+i\lambda} are described in Abramowitz & Stegun, Section
8.12.
-- Function: double gsl_sf_conicalP_half (double LAMBDA, double X)
-- Function: int gsl_sf_conicalP_half_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
These routines compute the irregular Spherical Conical Function
P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.
-- Function: double gsl_sf_conicalP_mhalf (double LAMBDA, double X)
-- Function: int gsl_sf_conicalP_mhalf_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
These routines compute the regular Spherical Conical Function
P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.
-- Function: double gsl_sf_conicalP_0 (double LAMBDA, double X)
-- Function: int gsl_sf_conicalP_0_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
These routines compute the conical function P^0_{-1/2 + i
\lambda}(x) for x > -1.
-- Function: double gsl_sf_conicalP_1 (double LAMBDA, double X)
-- Function: int gsl_sf_conicalP_1_e (double LAMBDA, double X,
gsl_sf_result * RESULT)
These routines compute the conical function P^1_{-1/2 + i
\lambda}(x) for x > -1.
-- Function: double gsl_sf_conicalP_sph_reg (int L, double LAMBDA,
double X)
-- Function: int gsl_sf_conicalP_sph_reg_e (int L, double LAMBDA,
double X, gsl_sf_result * RESULT)
These routines compute the Regular Spherical Conical Function
P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1.
-- Function: double gsl_sf_conicalP_cyl_reg (int M, double LAMBDA,
double X)
-- Function: int gsl_sf_conicalP_cyl_reg_e (int M, double LAMBDA,
double X, gsl_sf_result * RESULT)
These routines compute the Regular Cylindrical Conical Function
P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1.
7.24.4 Radial Functions for Hyperbolic Space
--------------------------------------------
The following spherical functions are specializations of Legendre
functions which give the regular eigenfunctions of the Laplacian on a
3-dimensional hyperbolic space H3d. Of particular interest is the flat
limit, \lambda \to \infty, \eta \to 0, \lambda\eta fixed.
-- Function: double gsl_sf_legendre_H3d_0 (double LAMBDA, double ETA)
-- Function: int gsl_sf_legendre_H3d_0_e (double LAMBDA, double ETA,
gsl_sf_result * RESULT)
These routines compute the zeroth radial eigenfunction of the
Laplacian on the 3-dimensional hyperbolic space,
L^{H3d}_0(\lambda,\eta) := \sin(\lambda\eta)/(\lambda\sinh(\eta))
for \eta >= 0. In the flat limit this takes the form
L^{H3d}_0(\lambda,\eta) = j_0(\lambda\eta).
-- Function: double gsl_sf_legendre_H3d_1 (double LAMBDA, double ETA)
-- Function: int gsl_sf_legendre_H3d_1_e (double LAMBDA, double ETA,
gsl_sf_result * RESULT)
These routines compute the first radial eigenfunction of the
Laplacian on the 3-dimensional hyperbolic space,
L^{H3d}_1(\lambda,\eta) := 1/\sqrt{\lambda^2 + 1} \sin(\lambda
\eta)/(\lambda \sinh(\eta)) (\coth(\eta) - \lambda
\cot(\lambda\eta)) for \eta >= 0. In the flat limit this takes the
form L^{H3d}_1(\lambda,\eta) = j_1(\lambda\eta).
-- Function: double gsl_sf_legendre_H3d (int L, double LAMBDA, double
ETA)
-- Function: int gsl_sf_legendre_H3d_e (int L, double LAMBDA, double
ETA, gsl_sf_result * RESULT)
These routines compute the L-th radial eigenfunction of the
Laplacian on the 3-dimensional hyperbolic space \eta >= 0, l >= 0.
In the flat limit this takes the form L^{H3d}_l(\lambda,\eta) =
j_l(\lambda\eta).
-- Function: int gsl_sf_legendre_H3d_array (int LMAX, double LAMBDA,
double ETA, double RESULT_ARRAY[])
This function computes an array of radial eigenfunctions
L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.
7.25 Logarithm and Related Functions
====================================
Information on the properties of the Logarithm function can be found in
Abramowitz & Stegun, Chapter 4. The functions described in this section
are declared in the header file 'gsl_sf_log.h'.
-- Function: double gsl_sf_log (double X)
-- Function: int gsl_sf_log_e (double X, gsl_sf_result * RESULT)
These routines compute the logarithm of X, \log(x), for x > 0.
-- Function: double gsl_sf_log_abs (double X)
-- Function: int gsl_sf_log_abs_e (double X, gsl_sf_result * RESULT)
These routines compute the logarithm of the magnitude of X,
\log(|x|), for x \ne 0.
-- Function: int gsl_sf_complex_log_e (double ZR, double ZI,
gsl_sf_result * LNR, gsl_sf_result * THETA)
This routine computes the complex logarithm of z = z_r + i z_i.
The results are returned as LNR, THETA such that \exp(lnr + i
\theta) = z_r + i z_i, where \theta lies in the range [-\pi,\pi].
-- Function: double gsl_sf_log_1plusx (double X)
-- Function: int gsl_sf_log_1plusx_e (double X, gsl_sf_result * RESULT)
These routines compute \log(1 + x) for x > -1 using an algorithm
that is accurate for small x.
-- Function: double gsl_sf_log_1plusx_mx (double X)
-- Function: int gsl_sf_log_1plusx_mx_e (double X, gsl_sf_result *
RESULT)
These routines compute \log(1 + x) - x for x > -1 using an
algorithm that is accurate for small x.
7.26 Mathieu Functions
======================
The routines described in this section compute the angular and radial
Mathieu functions, and their characteristic values. Mathieu functions
are the solutions of the following two differential equations:
d^2y/dv^2 + (a - 2q\cos 2v)y = 0
d^2f/du^2 - (a - 2q\cosh 2u)f = 0
The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd
periodic solutions of the first equation, which is known as Mathieu's
equation. These exist only for the discrete sequence of characteristic
values a=a_r(q) (even-periodic) and a=b_r(q) (odd-periodic).
The radial Mathieu functions Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q) are
the solutions of the second equation, which is referred to as Mathieu's
modified equation. The radial Mathieu functions of the first, second,
third and fourth kind are denoted by the parameter j, which takes the
value 1, 2, 3 or 4.
For more information on the Mathieu functions, see Abramowitz and
Stegun, Chapter 20. These functions are defined in the header file
'gsl_sf_mathieu.h'.
7.26.1 Mathieu Function Workspace
---------------------------------
The Mathieu functions can be computed for a single order or for multiple
orders, using array-based routines. The array-based routines require a
preallocated workspace.
-- Function: gsl_sf_mathieu_workspace * gsl_sf_mathieu_alloc (size_t N,
double QMAX)
This function returns a workspace for the array versions of the
Mathieu routines. The arguments N and QMAX specify the maximum
order and q-value of Mathieu functions which can be computed with
this workspace.
-- Function: void gsl_sf_mathieu_free (gsl_sf_mathieu_workspace * WORK)
This function frees the workspace WORK.
7.26.2 Mathieu Function Characteristic Values
---------------------------------------------
-- Function: int gsl_sf_mathieu_a (int N, double Q)
-- Function: int gsl_sf_mathieu_a_e (int N, double Q, gsl_sf_result *
RESULT)
-- Function: int gsl_sf_mathieu_b (int N, double Q)
-- Function: int gsl_sf_mathieu_b_e (int N, double Q, gsl_sf_result *
RESULT)
These routines compute the characteristic values a_n(q), b_n(q) of
the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.
-- Function: int gsl_sf_mathieu_a_array (int ORDER_MIN, int ORDER_MAX,
double Q, gsl_sf_mathieu_workspace * WORK, double
RESULT_ARRAY[])
-- Function: int gsl_sf_mathieu_b_array (int ORDER_MIN, int ORDER_MAX,
double Q, gsl_sf_mathieu_workspace * WORK, double
RESULT_ARRAY[])
These routines compute a series of Mathieu characteristic values
a_n(q), b_n(q) for n from ORDER_MIN to ORDER_MAX inclusive, storing
the results in the array RESULT_ARRAY.
7.26.3 Angular Mathieu Functions
--------------------------------
-- Function: int gsl_sf_mathieu_ce (int N, double Q, double X)
-- Function: int gsl_sf_mathieu_ce_e (int N, double Q, double X,
gsl_sf_result * RESULT)
-- Function: int gsl_sf_mathieu_se (int N, double Q, double X)
-- Function: int gsl_sf_mathieu_se_e (int N, double Q, double X,
gsl_sf_result * RESULT)
These routines compute the angular Mathieu functions ce_n(q,x) and
se_n(q,x), respectively.
-- Function: int gsl_sf_mathieu_ce_array (int NMIN, int NMAX, double Q,
double X, gsl_sf_mathieu_workspace * WORK, double
RESULT_ARRAY[])
-- Function: int gsl_sf_mathieu_se_array (int NMIN, int NMAX, double Q,
double X, gsl_sf_mathieu_workspace * WORK, double
RESULT_ARRAY[])
These routines compute a series of the angular Mathieu functions
ce_n(q,x) and se_n(q,x) of order n from NMIN to NMAX inclusive,
storing the results in the array RESULT_ARRAY.
7.26.4 Radial Mathieu Functions
-------------------------------
-- Function: int gsl_sf_mathieu_Mc (int J, int N, double Q, double X)
-- Function: int gsl_sf_mathieu_Mc_e (int J, int N, double Q, double X,
gsl_sf_result * RESULT)
-- Function: int gsl_sf_mathieu_Ms (int J, int N, double Q, double X)
-- Function: int gsl_sf_mathieu_Ms_e (int J, int N, double Q, double X,
gsl_sf_result * RESULT)
These routines compute the radial J-th kind Mathieu functions
Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order N.
The allowed values of J are 1 and 2. The functions for j = 3,4 can
be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and M_n^{(4)} =
M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or Ms_n^{(j)}.
-- Function: int gsl_sf_mathieu_Mc_array (int J, int NMIN, int NMAX,
double Q, double X, gsl_sf_mathieu_workspace * WORK, double
RESULT_ARRAY[])
-- Function: int gsl_sf_mathieu_Ms_array (int J, int NMIN, int NMAX,
double Q, double X, gsl_sf_mathieu_workspace * WORK, double
RESULT_ARRAY[])
These routines compute a series of the radial Mathieu functions of
kind J, with order from NMIN to NMAX inclusive, storing the results
in the array RESULT_ARRAY.
7.27 Power Function
===================
The following functions are equivalent to the function 'gsl_pow_int'
(*note Small integer powers::) with an error estimate. These functions
are declared in the header file 'gsl_sf_pow_int.h'.
-- Function: double gsl_sf_pow_int (double X, int N)
-- Function: int gsl_sf_pow_int_e (double X, int N, gsl_sf_result *
RESULT)
These routines compute the power x^n for integer N. The power is
computed using the minimum number of multiplications. For example,
x^8 is computed as ((x^2)^2)^2, requiring only 3 multiplications.
For reasons of efficiency, these functions do not check for
overflow or underflow conditions.
#include
/* compute 3.0**12 */
double y = gsl_sf_pow_int(3.0, 12);
7.28 Psi (Digamma) Function
===========================
The polygamma functions of order n are defined by
\psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))
where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function.
These functions are declared in the header file 'gsl_sf_psi.h'.
7.28.1 Digamma Function
-----------------------
-- Function: double gsl_sf_psi_int (int N)
-- Function: int gsl_sf_psi_int_e (int N, gsl_sf_result * RESULT)
These routines compute the digamma function \psi(n) for positive
integer N. The digamma function is also called the Psi function.
-- Function: double gsl_sf_psi (double X)
-- Function: int gsl_sf_psi_e (double X, gsl_sf_result * RESULT)
These routines compute the digamma function \psi(x) for general x,
x \ne 0.
-- Function: double gsl_sf_psi_1piy (double Y)
-- Function: int gsl_sf_psi_1piy_e (double Y, gsl_sf_result * RESULT)
These routines compute the real part of the digamma function on the
line 1+i y, \Re[\psi(1 + i y)].
7.28.2 Trigamma Function
------------------------
-- Function: double gsl_sf_psi_1_int (int N)
-- Function: int gsl_sf_psi_1_int_e (int N, gsl_sf_result * RESULT)
These routines compute the Trigamma function \psi'(n) for positive
integer n.
-- Function: double gsl_sf_psi_1 (double X)
-- Function: int gsl_sf_psi_1_e (double X, gsl_sf_result * RESULT)
These routines compute the Trigamma function \psi'(x) for general
x.
7.28.3 Polygamma Function
-------------------------
-- Function: double gsl_sf_psi_n (int N, double X)
-- Function: int gsl_sf_psi_n_e (int N, double X, gsl_sf_result *
RESULT)
These routines compute the polygamma function \psi^{(n)}(x) for n
>= 0, x > 0.
7.29 Synchrotron Functions
==========================
The functions described in this section are declared in the header file
'gsl_sf_synchrotron.h'.
-- Function: double gsl_sf_synchrotron_1 (double X)
-- Function: int gsl_sf_synchrotron_1_e (double X, gsl_sf_result *
RESULT)
These routines compute the first synchrotron function x
\int_x^\infty dt K_{5/3}(t) for x >= 0.
-- Function: double gsl_sf_synchrotron_2 (double X)
-- Function: int gsl_sf_synchrotron_2_e (double X, gsl_sf_result *
RESULT)
These routines compute the second synchrotron function x K_{2/3}(x)
for x >= 0.
7.30 Transport Functions
========================
The transport functions J(n,x) are defined by the integral
representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2. They are
declared in the header file 'gsl_sf_transport.h'.
-- Function: double gsl_sf_transport_2 (double X)
-- Function: int gsl_sf_transport_2_e (double X, gsl_sf_result *
RESULT)
These routines compute the transport function J(2,x).
-- Function: double gsl_sf_transport_3 (double X)
-- Function: int gsl_sf_transport_3_e (double X, gsl_sf_result *
RESULT)
These routines compute the transport function J(3,x).
-- Function: double gsl_sf_transport_4 (double X)
-- Function: int gsl_sf_transport_4_e (double X, gsl_sf_result *
RESULT)
These routines compute the transport function J(4,x).
-- Function: double gsl_sf_transport_5 (double X)
-- Function: int gsl_sf_transport_5_e (double X, gsl_sf_result *
RESULT)
These routines compute the transport function J(5,x).
7.31 Trigonometric Functions
============================
The library includes its own trigonometric functions in order to provide
consistency across platforms and reliable error estimates. These
functions are declared in the header file 'gsl_sf_trig.h'.
7.31.1 Circular Trigonometric Functions
---------------------------------------
-- Function: double gsl_sf_sin (double X)
-- Function: int gsl_sf_sin_e (double X, gsl_sf_result * RESULT)
These routines compute the sine function \sin(x).
-- Function: double gsl_sf_cos (double X)
-- Function: int gsl_sf_cos_e (double X, gsl_sf_result * RESULT)
These routines compute the cosine function \cos(x).
-- Function: double gsl_sf_hypot (double X, double Y)
-- Function: int gsl_sf_hypot_e (double X, double Y, gsl_sf_result *
RESULT)
These routines compute the hypotenuse function \sqrt{x^2 + y^2}
avoiding overflow and underflow.
-- Function: double gsl_sf_sinc (double X)
-- Function: int gsl_sf_sinc_e (double X, gsl_sf_result * RESULT)
These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any
value of X.
7.31.2 Trigonometric Functions for Complex Arguments
----------------------------------------------------
-- Function: int gsl_sf_complex_sin_e (double ZR, double ZI,
gsl_sf_result * SZR, gsl_sf_result * SZI)
This function computes the complex sine, \sin(z_r + i z_i) storing
the real and imaginary parts in SZR, SZI.
-- Function: int gsl_sf_complex_cos_e (double ZR, double ZI,
gsl_sf_result * CZR, gsl_sf_result * CZI)
This function computes the complex cosine, \cos(z_r + i z_i)
storing the real and imaginary parts in CZR, CZI.
-- Function: int gsl_sf_complex_logsin_e (double ZR, double ZI,
gsl_sf_result * LSZR, gsl_sf_result * LSZI)
This function computes the logarithm of the complex sine,
\log(\sin(z_r + i z_i)) storing the real and imaginary parts in
LSZR, LSZI.
7.31.3 Hyperbolic Trigonometric Functions
-----------------------------------------
-- Function: double gsl_sf_lnsinh (double X)
-- Function: int gsl_sf_lnsinh_e (double X, gsl_sf_result * RESULT)
These routines compute \log(\sinh(x)) for x > 0.
-- Function: double gsl_sf_lncosh (double X)
-- Function: int gsl_sf_lncosh_e (double X, gsl_sf_result * RESULT)
These routines compute \log(\cosh(x)) for any X.
7.31.4 Conversion Functions
---------------------------
-- Function: int gsl_sf_polar_to_rect (double R, double THETA,
gsl_sf_result * X, gsl_sf_result * Y);
This function converts the polar coordinates (R,THETA) to
rectilinear coordinates (X,Y), x = r\cos(\theta), y =
r\sin(\theta).
-- Function: int gsl_sf_rect_to_polar (double X, double Y,
gsl_sf_result * R, gsl_sf_result * THETA)
This function converts the rectilinear coordinates (X,Y) to polar
coordinates (R,THETA), such that x = r\cos(\theta), y =
r\sin(\theta). The argument THETA lies in the range [-\pi, \pi].
7.31.5 Restriction Functions
----------------------------
-- Function: double gsl_sf_angle_restrict_symm (double THETA)
-- Function: int gsl_sf_angle_restrict_symm_e (double * THETA)
These routines force the angle THETA to lie in the range
(-\pi,\pi].
Note that the mathematical value of \pi is slightly greater than
'M_PI', so the machine numbers 'M_PI' and '-M_PI' are included in
the range.
-- Function: double gsl_sf_angle_restrict_pos (double THETA)
-- Function: int gsl_sf_angle_restrict_pos_e (double * THETA)
These routines force the angle THETA to lie in the range [0, 2\pi).
Note that the mathematical value of 2\pi is slightly greater than
'2*M_PI', so the machine number '2*M_PI' is included in the range.
7.31.6 Trigonometric Functions With Error Estimates
---------------------------------------------------
-- Function: int gsl_sf_sin_err_e (double X, double DX, gsl_sf_result *
RESULT)
This routine computes the sine of an angle X with an associated
absolute error DX, \sin(x \pm dx). Note that this function is
provided in the error-handling form only since its purpose is to
compute the propagated error.
-- Function: int gsl_sf_cos_err_e (double X, double DX, gsl_sf_result *
RESULT)
This routine computes the cosine of an angle X with an associated
absolute error DX, \cos(x \pm dx). Note that this function is
provided in the error-handling form only since its purpose is to
compute the propagated error.
7.32 Zeta Functions
===================
The Riemann zeta function is defined in Abramowitz & Stegun, Section
23.2. The functions described in this section are declared in the
header file 'gsl_sf_zeta.h'.
7.32.1 Riemann Zeta Function
----------------------------
The Riemann zeta function is defined by the infinite sum \zeta(s) =
\sum_{k=1}^\infty k^{-s}.
-- Function: double gsl_sf_zeta_int (int N)
-- Function: int gsl_sf_zeta_int_e (int N, gsl_sf_result * RESULT)
These routines compute the Riemann zeta function \zeta(n) for
integer N, n \ne 1.
-- Function: double gsl_sf_zeta (double S)
-- Function: int gsl_sf_zeta_e (double S, gsl_sf_result * RESULT)
These routines compute the Riemann zeta function \zeta(s) for
arbitrary S, s \ne 1.
7.32.2 Riemann Zeta Function Minus One
--------------------------------------
For large positive argument, the Riemann zeta function approaches one.
In this region the fractional part is interesting, and therefore we need
a function to evaluate it explicitly.
-- Function: double gsl_sf_zetam1_int (int N)
-- Function: int gsl_sf_zetam1_int_e (int N, gsl_sf_result * RESULT)
These routines compute \zeta(n) - 1 for integer N, n \ne 1.
-- Function: double gsl_sf_zetam1 (double S)
-- Function: int gsl_sf_zetam1_e (double S, gsl_sf_result * RESULT)
These routines compute \zeta(s) - 1 for arbitrary S, s \ne 1.
7.32.3 Hurwitz Zeta Function
----------------------------
The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty
(k+q)^{-s}.
-- Function: double gsl_sf_hzeta (double S, double Q)
-- Function: int gsl_sf_hzeta_e (double S, double Q, gsl_sf_result *
RESULT)
These routines compute the Hurwitz zeta function \zeta(s,q) for s >
1, q > 0.
7.32.4 Eta Function
-------------------
The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s).
-- Function: double gsl_sf_eta_int (int N)
-- Function: int gsl_sf_eta_int_e (int N, gsl_sf_result * RESULT)
These routines compute the eta function \eta(n) for integer N.
-- Function: double gsl_sf_eta (double S)
-- Function: int gsl_sf_eta_e (double S, gsl_sf_result * RESULT)
These routines compute the eta function \eta(s) for arbitrary S.
7.33 Examples
=============
The following example demonstrates the use of the error handling form of
the special functions, in this case to compute the Bessel function
J_0(5.0),
#include
#include
#include
int
main (void)
{
double x = 5.0;
gsl_sf_result result;
double expected = -0.17759677131433830434739701;
int status = gsl_sf_bessel_J0_e (x, &result);
printf ("status = %s\n", gsl_strerror(status));
printf ("J0(5.0) = %.18f\n"
" +/- % .18f\n",
result.val, result.err);
printf ("exact = %.18f\n", expected);
return status;
}
Here are the results of running the program,
$ ./a.out
status = success
J0(5.0) = -0.177596771314338264
+/- 0.000000000000000193
exact = -0.177596771314338292
The next program computes the same quantity using the natural form of
the function. In this case the error term RESULT.ERR and return status
are not accessible.
#include
#include
int
main (void)
{
double x = 5.0;
double expected = -0.17759677131433830434739701;
double y = gsl_sf_bessel_J0 (x);
printf ("J0(5.0) = %.18f\n", y);
printf ("exact = %.18f\n", expected);
return 0;
}
The results of the function are the same,
$ ./a.out
J0(5.0) = -0.177596771314338264
exact = -0.177596771314338292
7.34 References and Further Reading
===================================
The library follows the conventions of 'Abramowitz & Stegun' where
possible,
Abramowitz & Stegun (eds.), 'Handbook of Mathematical Functions'
The following papers contain information on the algorithms used to
compute the special functions,
Allan J. MacLeod, MISCFUN: A software package to compute uncommon
special functions. 'ACM Trans. Math. Soft.', vol. 22, 1996,
288-301
G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd
Edition (Cambridge University Press, 1944).
G. Nemeth, Mathematical Approximations of Special Functions, Nova
Science Publishers, ISBN 1-56072-052-2
B.C. Carlson, Special Functions of Applied Mathematics (1977)
N. M. Temme, Special Functions: An Introduction to the Classical
Functions of Mathematical Physics (1996), ISBN 978-0471113133.
W.J. Thompson, Atlas for Computing Mathematical Functions, John
Wiley & Sons, New York (1997).
Y.Y. Luke, Algorithms for the Computation of Mathematical
Functions, Academic Press, New York (1977).
S. A. Holmes and W. E. Featherstone, A unified approach to the
Clenshaw summation and the recursive computation of very high
degree and order normalised associated Legendre functions, Journal
of Geodesy, 76, pg. 279-299, 2002.
8 Vectors and Matrices
**********************
The functions described in this chapter provide a simple vector and
matrix interface to ordinary C arrays. The memory management of these
arrays is implemented using a single underlying type, known as a block.
By writing your functions in terms of vectors and matrices you can pass
a single structure containing both data and dimensions as an argument
without needing additional function parameters. The structures are
compatible with the vector and matrix formats used by BLAS routines.
8.1 Data types
==============
All the functions are available for each of the standard data-types.
The versions for 'double' have the prefix 'gsl_block', 'gsl_vector' and
'gsl_matrix'. Similarly the versions for single-precision 'float'
arrays have the prefix 'gsl_block_float', 'gsl_vector_float' and
'gsl_matrix_float'. The full list of available types is given below,
gsl_block double
gsl_block_float float
gsl_block_long_double long double
gsl_block_int int
gsl_block_uint unsigned int
gsl_block_long long
gsl_block_ulong unsigned long
gsl_block_short short
gsl_block_ushort unsigned short
gsl_block_char char
gsl_block_uchar unsigned char
gsl_block_complex complex double
gsl_block_complex_float complex float
gsl_block_complex_long_double complex long double
Corresponding types exist for the 'gsl_vector' and 'gsl_matrix'
functions.
8.2 Blocks
==========
For consistency all memory is allocated through a 'gsl_block' structure.
The structure contains two components, the size of an area of memory and
a pointer to the memory. The 'gsl_block' structure looks like this,
typedef struct
{
size_t size;
double * data;
} gsl_block;
Vectors and matrices are made by "slicing" an underlying block. A slice
is a set of elements formed from an initial offset and a combination of
indices and step-sizes. In the case of a matrix the step-size for the
column index represents the row-length. The step-size for a vector is
known as the "stride".
The functions for allocating and deallocating blocks are defined in
'gsl_block.h'
8.2.1 Block allocation
----------------------
The functions for allocating memory to a block follow the style of
'malloc' and 'free'. In addition they also perform their own error
checking. If there is insufficient memory available to allocate a block
then the functions call the GSL error handler (with an error number of
'GSL_ENOMEM') in addition to returning a null pointer. Thus if you use
the library error handler to abort your program then it isn't necessary
to check every 'alloc'.
-- Function: gsl_block * gsl_block_alloc (size_t N)
This function allocates memory for a block of N double-precision
elements, returning a pointer to the block struct. The block is
not initialized and so the values of its elements are undefined.
Use the function 'gsl_block_calloc' if you want to ensure that all
the elements are initialized to zero.
A null pointer is returned if insufficient memory is available to
create the block.
-- Function: gsl_block * gsl_block_calloc (size_t N)
This function allocates memory for a block and initializes all the
elements of the block to zero.
-- Function: void gsl_block_free (gsl_block * B)
This function frees the memory used by a block B previously
allocated with 'gsl_block_alloc' or 'gsl_block_calloc'.
8.2.2 Reading and writing blocks
--------------------------------
The library provides functions for reading and writing blocks to a file
as binary data or formatted text.
-- Function: int gsl_block_fwrite (FILE * STREAM, const gsl_block * B)
This function writes the elements of the block B to the stream
STREAM in binary format. The return value is 0 for success and
'GSL_EFAILED' if there was a problem writing to the file. Since
the data is written in the native binary format it may not be
portable between different architectures.
-- Function: int gsl_block_fread (FILE * STREAM, gsl_block * B)
This function reads into the block B from the open stream STREAM in
binary format. The block B must be preallocated with the correct
length since the function uses the size of B to determine how many
bytes to read. The return value is 0 for success and 'GSL_EFAILED'
if there was a problem reading from the file. The data is assumed
to have been written in the native binary format on the same
architecture.
-- Function: int gsl_block_fprintf (FILE * STREAM, const gsl_block * B,
const char * FORMAT)
This function writes the elements of the block B line-by-line to
the stream STREAM using the format specifier FORMAT, which should
be one of the '%g', '%e' or '%f' formats for floating point numbers
and '%d' for integers. The function returns 0 for success and
'GSL_EFAILED' if there was a problem writing to the file.
-- Function: int gsl_block_fscanf (FILE * STREAM, gsl_block * B)
This function reads formatted data from the stream STREAM into the
block B. The block B must be preallocated with the correct length
since the function uses the size of B to determine how many numbers
to read. The function returns 0 for success and 'GSL_EFAILED' if
there was a problem reading from the file.
8.2.3 Example programs for blocks
---------------------------------
The following program shows how to allocate a block,
#include
#include
int
main (void)
{
gsl_block * b = gsl_block_alloc (100);
printf ("length of block = %zu\n", b->size);
printf ("block data address = %p\n", b->data);
gsl_block_free (b);
return 0;
}
Here is the output from the program,
length of block = 100
block data address = 0x804b0d8
8.3 Vectors
===========
Vectors are defined by a 'gsl_vector' structure which describes a slice
of a block. Different vectors can be created which point to the same
block. A vector slice is a set of equally-spaced elements of an area of
memory.
The 'gsl_vector' structure contains five components, the "size", the
"stride", a pointer to the memory where the elements are stored, DATA, a
pointer to the block owned by the vector, BLOCK, if any, and an
ownership flag, OWNER. The structure is very simple and looks like
this,
typedef struct
{
size_t size;
size_t stride;
double * data;
gsl_block * block;
int owner;
} gsl_vector;
The SIZE is simply the number of vector elements. The range of valid
indices runs from 0 to 'size-1'. The STRIDE is the step-size from one
element to the next in physical memory, measured in units of the
appropriate datatype. The pointer DATA gives the location of the first
element of the vector in memory. The pointer BLOCK stores the location
of the memory block in which the vector elements are located (if any).
If the vector owns this block then the OWNER field is set to one and the
block will be deallocated when the vector is freed. If the vector
points to a block owned by another object then the OWNER field is zero
and any underlying block will not be deallocated with the vector.
The functions for allocating and accessing vectors are defined in
'gsl_vector.h'
8.3.1 Vector allocation
-----------------------
The functions for allocating memory to a vector follow the style of
'malloc' and 'free'. In addition they also perform their own error
checking. If there is insufficient memory available to allocate a
vector then the functions call the GSL error handler (with an error
number of 'GSL_ENOMEM') in addition to returning a null pointer. Thus
if you use the library error handler to abort your program then it isn't
necessary to check every 'alloc'.
-- Function: gsl_vector * gsl_vector_alloc (size_t N)
This function creates a vector of length N, returning a pointer to
a newly initialized vector struct. A new block is allocated for
the elements of the vector, and stored in the BLOCK component of
the vector struct. The block is "owned" by the vector, and will be
deallocated when the vector is deallocated.
-- Function: gsl_vector * gsl_vector_calloc (size_t N)
This function allocates memory for a vector of length N and
initializes all the elements of the vector to zero.
-- Function: void gsl_vector_free (gsl_vector * V)
This function frees a previously allocated vector V. If the vector
was created using 'gsl_vector_alloc' then the block underlying the
vector will also be deallocated. If the vector has been created
from another object then the memory is still owned by that object
and will not be deallocated.
8.3.2 Accessing vector elements
-------------------------------
Unlike FORTRAN compilers, C compilers do not usually provide support for
range checking of vectors and matrices.(1) The functions
'gsl_vector_get' and 'gsl_vector_set' can perform portable range
checking for you and report an error if you attempt to access elements
outside the allowed range.
The functions for accessing the elements of a vector or matrix are
defined in 'gsl_vector.h' and declared 'extern inline' to eliminate
function-call overhead. You must compile your program with the
preprocessor macro 'HAVE_INLINE' defined to use these functions.
If necessary you can turn off range checking completely without
modifying any source files by recompiling your program with the
preprocessor definition 'GSL_RANGE_CHECK_OFF'. Provided your compiler
supports inline functions the effect of turning off range checking is to
replace calls to 'gsl_vector_get(v,i)' by 'v->data[i*v->stride]' and
calls to 'gsl_vector_set(v,i,x)' by 'v->data[i*v->stride]=x'. Thus
there should be no performance penalty for using the range checking
functions when range checking is turned off.
If you use a C99 compiler which requires inline functions in header
files to be declared 'inline' instead of 'extern inline', define the
macro 'GSL_C99_INLINE' (*note Inline functions::). With GCC this is
selected automatically when compiling in C99 mode ('-std=c99').
If inline functions are not used, calls to the functions
'gsl_vector_get' and 'gsl_vector_set' will link to the compiled versions
of these functions in the library itself. The range checking in these
functions is controlled by the global integer variable
'gsl_check_range'. It is enabled by default--to disable it, set
'gsl_check_range' to zero. Due to function-call overhead, there is less
benefit in disabling range checking here than for inline functions.
-- Function: double gsl_vector_get (const gsl_vector * V, const size_t
I)
This function returns the I-th element of a vector V. If I lies
outside the allowed range of 0 to N-1 then the error handler is
invoked and 0 is returned. An inline version of this function is
used when 'HAVE_INLINE' is defined.
-- Function: void gsl_vector_set (gsl_vector * V, const size_t I,
double X)
This function sets the value of the I-th element of a vector V to
X. If I lies outside the allowed range of 0 to N-1 then the error
handler is invoked. An inline version of this function is used
when 'HAVE_INLINE' is defined.
-- Function: double * gsl_vector_ptr (gsl_vector * V, size_t I)
-- Function: const double * gsl_vector_const_ptr (const gsl_vector * V,
size_t I)
These functions return a pointer to the I-th element of a vector V.
If I lies outside the allowed range of 0 to N-1 then the error
handler is invoked and a null pointer is returned. Inline versions
of these functions are used when 'HAVE_INLINE' is defined.
---------- Footnotes ----------
(1) Range checking is available in the GNU C Compiler bounds-checking
extension, but it is not part of the default installation of GCC. Memory
accesses can also be checked with Valgrind or the 'gcc -fmudflap' memory
protection option.
8.3.3 Initializing vector elements
----------------------------------
-- Function: void gsl_vector_set_all (gsl_vector * V, double X)
This function sets all the elements of the vector V to the value X.
-- Function: void gsl_vector_set_zero (gsl_vector * V)
This function sets all the elements of the vector V to zero.
-- Function: int gsl_vector_set_basis (gsl_vector * V, size_t I)
This function makes a basis vector by setting all the elements of
the vector V to zero except for the I-th element which is set to
one.
8.3.4 Reading and writing vectors
---------------------------------
The library provides functions for reading and writing vectors to a file
as binary data or formatted text.
-- Function: int gsl_vector_fwrite (FILE * STREAM, const gsl_vector *
V)
This function writes the elements of the vector V to the stream
STREAM in binary format. The return value is 0 for success and
'GSL_EFAILED' if there was a problem writing to the file. Since
the data is written in the native binary format it may not be
portable between different architectures.
-- Function: int gsl_vector_fread (FILE * STREAM, gsl_vector * V)
This function reads into the vector V from the open stream STREAM
in binary format. The vector V must be preallocated with the
correct length since the function uses the size of V to determine
how many bytes to read. The return value is 0 for success and
'GSL_EFAILED' if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on
the same architecture.
-- Function: int gsl_vector_fprintf (FILE * STREAM, const gsl_vector *
V, const char * FORMAT)
This function writes the elements of the vector V line-by-line to
the stream STREAM using the format specifier FORMAT, which should
be one of the '%g', '%e' or '%f' formats for floating point numbers
and '%d' for integers. The function returns 0 for success and
'GSL_EFAILED' if there was a problem writing to the file.
-- Function: int gsl_vector_fscanf (FILE * STREAM, gsl_vector * V)
This function reads formatted data from the stream STREAM into the
vector V. The vector V must be preallocated with the correct
length since the function uses the size of V to determine how many
numbers to read. The function returns 0 for success and
'GSL_EFAILED' if there was a problem reading from the file.
8.3.5 Vector views
------------------
In addition to creating vectors from slices of blocks it is also
possible to slice vectors and create vector views. For example, a
subvector of another vector can be described with a view, or two views
can be made which provide access to the even and odd elements of a
vector.
A vector view is a temporary object, stored on the stack, which can
be used to operate on a subset of vector elements. Vector views can be
defined for both constant and non-constant vectors, using separate types
that preserve constness. A vector view has the type 'gsl_vector_view'
and a constant vector view has the type 'gsl_vector_const_view'. In
both cases the elements of the view can be accessed as a 'gsl_vector'
using the 'vector' component of the view object. A pointer to a vector
of type 'gsl_vector *' or 'const gsl_vector *' can be obtained by taking
the address of this component with the '&' operator.
When using this pointer it is important to ensure that the view
itself remains in scope--the simplest way to do so is by always writing
the pointer as '&'VIEW'.vector', and never storing this value in another
variable.
-- Function: gsl_vector_view gsl_vector_subvector (gsl_vector * V,
size_t OFFSET, size_t N)
-- Function: gsl_vector_const_view gsl_vector_const_subvector (const
gsl_vector * V, size_t OFFSET, size_t N)
These functions return a vector view of a subvector of another
vector V. The start of the new vector is offset by OFFSET elements
from the start of the original vector. The new vector has N
elements. Mathematically, the I-th element of the new vector V' is
given by,
v'(i) = v->data[(offset + i)*v->stride]
where the index I runs from 0 to 'n-1'.
The 'data' pointer of the returned vector struct is set to null if
the combined parameters (OFFSET,N) overrun the end of the original
vector.
The new vector is only a view of the block underlying the original
vector, V. The block containing the elements of V is not owned by
the new vector. When the view goes out of scope the original
vector V and its block will continue to exist. The original memory
can only be deallocated by freeing the original vector. Of course,
the original vector should not be deallocated while the view is
still in use.
The function 'gsl_vector_const_subvector' is equivalent to
'gsl_vector_subvector' but can be used for vectors which are
declared 'const'.
-- Function: gsl_vector_view gsl_vector_subvector_with_stride
(gsl_vector * V, size_t OFFSET, size_t STRIDE, size_t N)
-- Function: gsl_vector_const_view
gsl_vector_const_subvector_with_stride (const gsl_vector * V,
size_t OFFSET, size_t STRIDE, size_t N)
These functions return a vector view of a subvector of another
vector V with an additional stride argument. The subvector is
formed in the same way as for 'gsl_vector_subvector' but the new
vector has N elements with a step-size of STRIDE from one element
to the next in the original vector. Mathematically, the I-th
element of the new vector V' is given by,
v'(i) = v->data[(offset + i*stride)*v->stride]
where the index I runs from 0 to 'n-1'.
Note that subvector views give direct access to the underlying
elements of the original vector. For example, the following code
will zero the even elements of the vector 'v' of length 'n', while
leaving the odd elements untouched,
gsl_vector_view v_even
= gsl_vector_subvector_with_stride (v, 0, 2, n/2);
gsl_vector_set_zero (&v_even.vector);
A vector view can be passed to any subroutine which takes a vector
argument just as a directly allocated vector would be, using
'&'VIEW'.vector'. For example, the following code computes the
norm of the odd elements of 'v' using the BLAS routine DNRM2,
gsl_vector_view v_odd
= gsl_vector_subvector_with_stride (v, 1, 2, n/2);
double r = gsl_blas_dnrm2 (&v_odd.vector);
The function 'gsl_vector_const_subvector_with_stride' is equivalent
to 'gsl_vector_subvector_with_stride' but can be used for vectors
which are declared 'const'.
-- Function: gsl_vector_view gsl_vector_complex_real
(gsl_vector_complex * V)
-- Function: gsl_vector_const_view gsl_vector_complex_const_real (const
gsl_vector_complex * V)
These functions return a vector view of the real parts of the
complex vector V.
The function 'gsl_vector_complex_const_real' is equivalent to
'gsl_vector_complex_real' but can be used for vectors which are
declared 'const'.
-- Function: gsl_vector_view gsl_vector_complex_imag
(gsl_vector_complex * V)
-- Function: gsl_vector_const_view gsl_vector_complex_const_imag (const
gsl_vector_complex * V)
These functions return a vector view of the imaginary parts of the
complex vector V.
The function 'gsl_vector_complex_const_imag' is equivalent to
'gsl_vector_complex_imag' but can be used for vectors which are
declared 'const'.
-- Function: gsl_vector_view gsl_vector_view_array (double * BASE,
size_t N)
-- Function: gsl_vector_const_view gsl_vector_const_view_array (const
double * BASE, size_t N)
These functions return a vector view of an array. The start of the
new vector is given by BASE and has N elements. Mathematically,
the I-th element of the new vector V' is given by,
v'(i) = base[i]
where the index I runs from 0 to 'n-1'.
The array containing the elements of V is not owned by the new
vector view. When the view goes out of scope the original array
will continue to exist. The original memory can only be
deallocated by freeing the original pointer BASE. Of course, the
original array should not be deallocated while the view is still in
use.
The function 'gsl_vector_const_view_array' is equivalent to
'gsl_vector_view_array' but can be used for arrays which are
declared 'const'.
-- Function: gsl_vector_view gsl_vector_view_array_with_stride (double
* BASE, size_t STRIDE, size_t N)
-- Function: gsl_vector_const_view
gsl_vector_const_view_array_with_stride (const double * BASE,
size_t STRIDE, size_t N)
These functions return a vector view of an array BASE with an
additional stride argument. The subvector is formed in the same
way as for 'gsl_vector_view_array' but the new vector has N
elements with a step-size of STRIDE from one element to the next in
the original array. Mathematically, the I-th element of the new
vector V' is given by,
v'(i) = base[i*stride]
where the index I runs from 0 to 'n-1'.
Note that the view gives direct access to the underlying elements
of the original array. A vector view can be passed to any
subroutine which takes a vector argument just as a directly
allocated vector would be, using '&'VIEW'.vector'.
The function 'gsl_vector_const_view_array_with_stride' is
equivalent to 'gsl_vector_view_array_with_stride' but can be used
for arrays which are declared 'const'.
8.3.6 Copying vectors
---------------------
Common operations on vectors such as addition and multiplication are
available in the BLAS part of the library (*note BLAS Support::).
However, it is useful to have a small number of utility functions which
do not require the full BLAS code. The following functions fall into
this category.
-- Function: int gsl_vector_memcpy (gsl_vector * DEST, const gsl_vector
* SRC)
This function copies the elements of the vector SRC into the vector
DEST. The two vectors must have the same length.
-- Function: int gsl_vector_swap (gsl_vector * V, gsl_vector * W)
This function exchanges the elements of the vectors V and W by
copying. The two vectors must have the same length.
8.3.7 Exchanging elements
-------------------------
The following function can be used to exchange, or permute, the elements
of a vector.
-- Function: int gsl_vector_swap_elements (gsl_vector * V, size_t I,
size_t J)
This function exchanges the I-th and J-th elements of the vector V
in-place.
-- Function: int gsl_vector_reverse (gsl_vector * V)
This function reverses the order of the elements of the vector V.
8.3.8 Vector operations
-----------------------
-- Function: int gsl_vector_add (gsl_vector * A, const gsl_vector * B)
This function adds the elements of vector B to the elements of
vector A. The result a_i \leftarrow a_i + b_i is stored in A and B
remains unchanged. The two vectors must have the same length.
-- Function: int gsl_vector_sub (gsl_vector * A, const gsl_vector * B)
This function subtracts the elements of vector B from the elements
of vector A. The result a_i \leftarrow a_i - b_i is stored in A
and B remains unchanged. The two vectors must have the same
length.
-- Function: int gsl_vector_mul (gsl_vector * A, const gsl_vector * B)
This function multiplies the elements of vector A by the elements
of vector B. The result a_i \leftarrow a_i * b_i is stored in A
and B remains unchanged. The two vectors must have the same
length.
-- Function: int gsl_vector_div (gsl_vector * A, const gsl_vector * B)
This function divides the elements of vector A by the elements of
vector B. The result a_i \leftarrow a_i / b_i is stored in A and B
remains unchanged. The two vectors must have the same length.
-- Function: int gsl_vector_scale (gsl_vector * A, const double X)
This function multiplies the elements of vector A by the constant
factor X. The result a_i \leftarrow x a_i is stored in A.
-- Function: int gsl_vector_add_constant (gsl_vector * A, const double
X)
This function adds the constant value X to the elements of the
vector A. The result a_i \leftarrow a_i + x is stored in A.
8.3.9 Finding maximum and minimum elements of vectors
-----------------------------------------------------
The following operations are only defined for real vectors.
-- Function: double gsl_vector_max (const gsl_vector * V)
This function returns the maximum value in the vector V.
-- Function: double gsl_vector_min (const gsl_vector * V)
This function returns the minimum value in the vector V.
-- Function: void gsl_vector_minmax (const gsl_vector * V, double *
MIN_OUT, double * MAX_OUT)
This function returns the minimum and maximum values in the vector
V, storing them in MIN_OUT and MAX_OUT.
-- Function: size_t gsl_vector_max_index (const gsl_vector * V)
This function returns the index of the maximum value in the vector
V. When there are several equal maximum elements then the lowest
index is returned.
-- Function: size_t gsl_vector_min_index (const gsl_vector * V)
This function returns the index of the minimum value in the vector
V. When there are several equal minimum elements then the lowest
index is returned.
-- Function: void gsl_vector_minmax_index (const gsl_vector * V, size_t
* IMIN, size_t * IMAX)
This function returns the indices of the minimum and maximum values
in the vector V, storing them in IMIN and IMAX. When there are
several equal minimum or maximum elements then the lowest indices
are returned.
8.3.10 Vector properties
------------------------
The following functions are defined for real and complex vectors. For
complex vectors both the real and imaginary parts must satisfy the
conditions.
-- Function: int gsl_vector_isnull (const gsl_vector * V)
-- Function: int gsl_vector_ispos (const gsl_vector * V)
-- Function: int gsl_vector_isneg (const gsl_vector * V)
-- Function: int gsl_vector_isnonneg (const gsl_vector * V)
These functions return 1 if all the elements of the vector V are
zero, strictly positive, strictly negative, or non-negative
respectively, and 0 otherwise.
-- Function: int gsl_vector_equal (const gsl_vector * U, const
gsl_vector * V)
This function returns 1 if the vectors U and V are equal (by
comparison of element values) and 0 otherwise.
8.3.11 Example programs for vectors
-----------------------------------
This program shows how to allocate, initialize and read from a vector
using the functions 'gsl_vector_alloc', 'gsl_vector_set' and
'gsl_vector_get'.
#include
#include
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (3);
for (i = 0; i < 3; i++)
{
gsl_vector_set (v, i, 1.23 + i);
}
for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
{
printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
}
gsl_vector_free (v);
return 0;
}
Here is the output from the program. The final loop attempts to read
outside the range of the vector 'v', and the error is trapped by the
range-checking code in 'gsl_vector_get'.
$ ./a.out
v_0 = 1.23
v_1 = 2.23
v_2 = 3.23
gsl: vector_source.c:12: ERROR: index out of range
Default GSL error handler invoked.
Aborted (core dumped)
The next program shows how to write a vector to a file.
#include
#include
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (100);
for (i = 0; i < 100; i++)
{
gsl_vector_set (v, i, 1.23 + i);
}
{
FILE * f = fopen ("test.dat", "w");
gsl_vector_fprintf (f, v, "%.5g");
fclose (f);
}
gsl_vector_free (v);
return 0;
}
After running this program the file 'test.dat' should contain the
elements of 'v', written using the format specifier '%.5g'. The vector
could then be read back in using the function 'gsl_vector_fscanf (f, v)'
as follows:
#include
#include
int
main (void)
{
int i;
gsl_vector * v = gsl_vector_alloc (10);
{
FILE * f = fopen ("test.dat", "r");
gsl_vector_fscanf (f, v);
fclose (f);
}
for (i = 0; i < 10; i++)
{
printf ("%g\n", gsl_vector_get(v, i));
}
gsl_vector_free (v);
return 0;
}
8.4 Matrices
============
Matrices are defined by a 'gsl_matrix' structure which describes a
generalized slice of a block. Like a vector it represents a set of
elements in an area of memory, but uses two indices instead of one.
The 'gsl_matrix' structure contains six components, the two
dimensions of the matrix, a physical dimension, a pointer to the memory
where the elements of the matrix are stored, DATA, a pointer to the
block owned by the matrix BLOCK, if any, and an ownership flag, OWNER.
The physical dimension determines the memory layout and can differ from
the matrix dimension to allow the use of submatrices. The 'gsl_matrix'
structure is very simple and looks like this,
typedef struct
{
size_t size1;
size_t size2;
size_t tda;
double * data;
gsl_block * block;
int owner;
} gsl_matrix;
Matrices are stored in row-major order, meaning that each row of
elements forms a contiguous block in memory. This is the standard
"C-language ordering" of two-dimensional arrays. Note that FORTRAN
stores arrays in column-major order. The number of rows is SIZE1. The
range of valid row indices runs from 0 to 'size1-1'. Similarly SIZE2 is
the number of columns. The range of valid column indices runs from 0 to
'size2-1'. The physical row dimension TDA, or "trailing dimension",
specifies the size of a row of the matrix as laid out in memory.
For example, in the following matrix SIZE1 is 3, SIZE2 is 4, and TDA
is 8. The physical memory layout of the matrix begins in the top left
hand-corner and proceeds from left to right along each row in turn.
00 01 02 03 XX XX XX XX
10 11 12 13 XX XX XX XX
20 21 22 23 XX XX XX XX
Each unused memory location is represented by "'XX'". The pointer DATA
gives the location of the first element of the matrix in memory. The
pointer BLOCK stores the location of the memory block in which the
elements of the matrix are located (if any). If the matrix owns this
block then the OWNER field is set to one and the block will be
deallocated when the matrix is freed. If the matrix is only a slice of
a block owned by another object then the OWNER field is zero and any
underlying block will not be freed.
The functions for allocating and accessing matrices are defined in
'gsl_matrix.h'
8.4.1 Matrix allocation
-----------------------
The functions for allocating memory to a matrix follow the style of
'malloc' and 'free'. They also perform their own error checking. If
there is insufficient memory available to allocate a matrix then the
functions call the GSL error handler (with an error number of
'GSL_ENOMEM') in addition to returning a null pointer. Thus if you use
the library error handler to abort your program then it isn't necessary
to check every 'alloc'.
-- Function: gsl_matrix * gsl_matrix_alloc (size_t N1, size_t N2)
This function creates a matrix of size N1 rows by N2 columns,
returning a pointer to a newly initialized matrix struct. A new
block is allocated for the elements of the matrix, and stored in
the BLOCK component of the matrix struct. The block is "owned" by
the matrix, and will be deallocated when the matrix is deallocated.
-- Function: gsl_matrix * gsl_matrix_calloc (size_t N1, size_t N2)
This function allocates memory for a matrix of size N1 rows by N2
columns and initializes all the elements of the matrix to zero.
-- Function: void gsl_matrix_free (gsl_matrix * M)
This function frees a previously allocated matrix M. If the matrix
was created using 'gsl_matrix_alloc' then the block underlying the
matrix will also be deallocated. If the matrix has been created
from another object then the memory is still owned by that object
and will not be deallocated.
8.4.2 Accessing matrix elements
-------------------------------
The functions for accessing the elements of a matrix use the same range
checking system as vectors. You can turn off range checking by
recompiling your program with the preprocessor definition
'GSL_RANGE_CHECK_OFF'.
The elements of the matrix are stored in "C-order", where the second
index moves continuously through memory. More precisely, the element
accessed by the function 'gsl_matrix_get(m,i,j)' and
'gsl_matrix_set(m,i,j,x)' is
m->data[i * m->tda + j]
where TDA is the physical row-length of the matrix.
-- Function: double gsl_matrix_get (const gsl_matrix * M, const size_t
I, const size_t J)
This function returns the (i,j)-th element of a matrix M. If I or
J lie outside the allowed range of 0 to N1-1 and 0 to N2-1 then the
error handler is invoked and 0 is returned. An inline version of
this function is used when 'HAVE_INLINE' is defined.
-- Function: void gsl_matrix_set (gsl_matrix * M, const size_t I, const
size_t J, double X)
This function sets the value of the (i,j)-th element of a matrix M
to X. If I or J lies outside the allowed range of 0 to N1-1 and 0
to N2-1 then the error handler is invoked. An inline version of
this function is used when 'HAVE_INLINE' is defined.
-- Function: double * gsl_matrix_ptr (gsl_matrix * M, size_t I, size_t
J)
-- Function: const double * gsl_matrix_const_ptr (const gsl_matrix * M,
size_t I, size_t J)
These functions return a pointer to the (i,j)-th element of a
matrix M. If I or J lie outside the allowed range of 0 to N1-1 and
0 to N2-1 then the error handler is invoked and a null pointer is
returned. Inline versions of these functions are used when
'HAVE_INLINE' is defined.
8.4.3 Initializing matrix elements
----------------------------------
-- Function: void gsl_matrix_set_all (gsl_matrix * M, double X)
This function sets all the elements of the matrix M to the value X.
-- Function: void gsl_matrix_set_zero (gsl_matrix * M)
This function sets all the elements of the matrix M to zero.
-- Function: void gsl_matrix_set_identity (gsl_matrix * M)
This function sets the elements of the matrix M to the
corresponding elements of the identity matrix, m(i,j) =
\delta(i,j), i.e. a unit diagonal with all off-diagonal elements
zero. This applies to both square and rectangular matrices.
8.4.4 Reading and writing matrices
----------------------------------
The library provides functions for reading and writing matrices to a
file as binary data or formatted text.
-- Function: int gsl_matrix_fwrite (FILE * STREAM, const gsl_matrix *
M)
This function writes the elements of the matrix M to the stream
STREAM in binary format. The return value is 0 for success and
'GSL_EFAILED' if there was a problem writing to the file. Since
the data is written in the native binary format it may not be
portable between different architectures.
-- Function: int gsl_matrix_fread (FILE * STREAM, gsl_matrix * M)
This function reads into the matrix M from the open stream STREAM
in binary format. The matrix M must be preallocated with the
correct dimensions since the function uses the size of M to
determine how many bytes to read. The return value is 0 for
success and 'GSL_EFAILED' if there was a problem reading from the
file. The data is assumed to have been written in the native
binary format on the same architecture.
-- Function: int gsl_matrix_fprintf (FILE * STREAM, const gsl_matrix *
M, const char * FORMAT)
This function writes the elements of the matrix M line-by-line to
the stream STREAM using the format specifier FORMAT, which should
be one of the '%g', '%e' or '%f' formats for floating point numbers
and '%d' for integers. The function returns 0 for success and
'GSL_EFAILED' if there was a problem writing to the file.
-- Function: int gsl_matrix_fscanf (FILE * STREAM, gsl_matrix * M)
This function reads formatted data from the stream STREAM into the
matrix M. The matrix M must be preallocated with the correct
dimensions since the function uses the size of M to determine how
many numbers to read. The function returns 0 for success and
'GSL_EFAILED' if there was a problem reading from the file.
8.4.5 Matrix views
------------------
A matrix view is a temporary object, stored on the stack, which can be
used to operate on a subset of matrix elements. Matrix views can be
defined for both constant and non-constant matrices using separate types
that preserve constness. A matrix view has the type 'gsl_matrix_view'
and a constant matrix view has the type 'gsl_matrix_const_view'. In
both cases the elements of the view can by accessed using the 'matrix'
component of the view object. A pointer 'gsl_matrix *' or 'const
gsl_matrix *' can be obtained by taking the address of the 'matrix'
component with the '&' operator. In addition to matrix views it is also
possible to create vector views of a matrix, such as row or column
views.
-- Function: gsl_matrix_view gsl_matrix_submatrix (gsl_matrix * M,
size_t K1, size_t K2, size_t N1, size_t N2)
-- Function: gsl_matrix_const_view gsl_matrix_const_submatrix (const
gsl_matrix * M, size_t K1, size_t K2, size_t N1, size_t N2)
These functions return a matrix view of a submatrix of the matrix
M. The upper-left element of the submatrix is the element (K1,K2)
of the original matrix. The submatrix has N1 rows and N2 columns.
The physical number of columns in memory given by TDA is unchanged.
Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = m->data[(k1*m->tda + k2) + i*m->tda + j]
where the index I runs from 0 to 'n1-1' and the index J runs from 0
to 'n2-1'.
The 'data' pointer of the returned matrix struct is set to null if
the combined parameters (I,J,N1,N2,TDA) overrun the ends of the
original matrix.
The new matrix view is only a view of the block underlying the
existing matrix, M. The block containing the elements of M is not
owned by the new matrix view. When the view goes out of scope the
original matrix M and its block will continue to exist. The
original memory can only be deallocated by freeing the original
matrix. Of course, the original matrix should not be deallocated
while the view is still in use.
The function 'gsl_matrix_const_submatrix' is equivalent to
'gsl_matrix_submatrix' but can be used for matrices which are
declared 'const'.
-- Function: gsl_matrix_view gsl_matrix_view_array (double * BASE,
size_t N1, size_t N2)
-- Function: gsl_matrix_const_view gsl_matrix_const_view_array (const
double * BASE, size_t N1, size_t N2)
These functions return a matrix view of the array BASE. The matrix
has N1 rows and N2 columns. The physical number of columns in
memory is also given by N2. Mathematically, the (i,j)-th element
of the new matrix is given by,
m'(i,j) = base[i*n2 + j]
where the index I runs from 0 to 'n1-1' and the index J runs from 0
to 'n2-1'.
The new matrix is only a view of the array BASE. When the view
goes out of scope the original array BASE will continue to exist.
The original memory can only be deallocated by freeing the original
array. Of course, the original array should not be deallocated
while the view is still in use.
The function 'gsl_matrix_const_view_array' is equivalent to
'gsl_matrix_view_array' but can be used for matrices which are
declared 'const'.
-- Function: gsl_matrix_view gsl_matrix_view_array_with_tda (double *
BASE, size_t N1, size_t N2, size_t TDA)
-- Function: gsl_matrix_const_view gsl_matrix_const_view_array_with_tda
(const double * BASE, size_t N1, size_t N2, size_t TDA)
These functions return a matrix view of the array BASE with a
physical number of columns TDA which may differ from the
corresponding dimension of the matrix. The matrix has N1 rows and
N2 columns, and the physical number of columns in memory is given
by TDA. Mathematically, the (i,j)-th element of the new matrix is
given by,
m'(i,j) = base[i*tda + j]
where the index I runs from 0 to 'n1-1' and the index J runs from 0
to 'n2-1'.
The new matrix is only a view of the array BASE. When the view
goes out of scope the original array BASE will continue to exist.
The original memory can only be deallocated by freeing the original
array. Of course, the original array should not be deallocated
while the view is still in use.
The function 'gsl_matrix_const_view_array_with_tda' is equivalent
to 'gsl_matrix_view_array_with_tda' but can be used for matrices
which are declared 'const'.
-- Function: gsl_matrix_view gsl_matrix_view_vector (gsl_vector * V,
size_t N1, size_t N2)
-- Function: gsl_matrix_const_view gsl_matrix_const_view_vector (const
gsl_vector * V, size_t N1, size_t N2)
These functions return a matrix view of the vector V. The matrix
has N1 rows and N2 columns. The vector must have unit stride. The
physical number of columns in memory is also given by N2.
Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*n2 + j]
where the index I runs from 0 to 'n1-1' and the index J runs from 0
to 'n2-1'.
The new matrix is only a view of the vector V. When the view goes
out of scope the original vector V will continue to exist. The
original memory can only be deallocated by freeing the original
vector. Of course, the original vector should not be deallocated
while the view is still in use.
The function 'gsl_matrix_const_view_vector' is equivalent to
'gsl_matrix_view_vector' but can be used for matrices which are
declared 'const'.
-- Function: gsl_matrix_view gsl_matrix_view_vector_with_tda
(gsl_vector * V, size_t N1, size_t N2, size_t TDA)
-- Function: gsl_matrix_const_view
gsl_matrix_const_view_vector_with_tda (const gsl_vector * V,
size_t N1, size_t N2, size_t TDA)
These functions return a matrix view of the vector V with a
physical number of columns TDA which may differ from the
corresponding matrix dimension. The vector must have unit stride.
The matrix has N1 rows and N2 columns, and the physical number of
columns in memory is given by TDA. Mathematically, the (i,j)-th
element of the new matrix is given by,
m'(i,j) = v->data[i*tda + j]
where the index I runs from 0 to 'n1-1' and the index J runs from 0
to 'n2-1'.
The new matrix is only a view of the vector V. When the view goes
out of scope the original vector V will continue to exist. The
original memory can only be deallocated by freeing the original
vector. Of course, the original vector should not be deallocated
while the view is still in use.
The function 'gsl_matrix_const_view_vector_with_tda' is equivalent
to 'gsl_matrix_view_vector_with_tda' but can be used for matrices
which are declared 'const'.
8.4.6 Creating row and column views
-----------------------------------
In general there are two ways to access an object, by reference or by
copying. The functions described in this section create vector views
which allow access to a row or column of a matrix by reference.
Modifying elements of the view is equivalent to modifying the matrix,
since both the vector view and the matrix point to the same memory
block.
-- Function: gsl_vector_view gsl_matrix_row (gsl_matrix * M, size_t I)
-- Function: gsl_vector_const_view gsl_matrix_const_row (const
gsl_matrix * M, size_t I)
These functions return a vector view of the I-th row of the matrix
M. The 'data' pointer of the new vector is set to null if I is out
of range.
The function 'gsl_vector_const_row' is equivalent to
'gsl_matrix_row' but can be used for matrices which are declared
'const'.
-- Function: gsl_vector_view gsl_matrix_column (gsl_matrix * M, size_t
J)
-- Function: gsl_vector_const_view gsl_matrix_const_column (const
gsl_matrix * M, size_t J)
These functions return a vector view of the J-th column of the
matrix M. The 'data' pointer of the new vector is set to null if J
is out of range.
The function 'gsl_vector_const_column' is equivalent to
'gsl_matrix_column' but can be used for matrices which are declared
'const'.
-- Function: gsl_vector_view gsl_matrix_subrow (gsl_matrix * M, size_t
I, size_t OFFSET, size_t N)
-- Function: gsl_vector_const_view gsl_matrix_const_subrow (const
gsl_matrix * M, size_t I, size_t OFFSET, size_t N)
These functions return a vector view of the I-th row of the matrix
M beginning at OFFSET elements past the first column and containing
N elements. The 'data' pointer of the new vector is set to null if
I, OFFSET, or N are out of range.
The function 'gsl_vector_const_subrow' is equivalent to
'gsl_matrix_subrow' but can be used for matrices which are declared
'const'.
-- Function: gsl_vector_view gsl_matrix_subcolumn (gsl_matrix * M,
size_t J, size_t OFFSET, size_t N)
-- Function: gsl_vector_const_view gsl_matrix_const_subcolumn (const
gsl_matrix * M, size_t J, size_t OFFSET, size_t N)
These functions return a vector view of the J-th column of the
matrix M beginning at OFFSET elements past the first row and
containing N elements. The 'data' pointer of the new vector is set
to null if J, OFFSET, or N are out of range.
The function 'gsl_vector_const_subcolumn' is equivalent to
'gsl_matrix_subcolumn' but can be used for matrices which are
declared 'const'.
-- Function: gsl_vector_view gsl_matrix_diagonal (gsl_matrix * M)
-- Function: gsl_vector_const_view gsl_matrix_const_diagonal (const
gsl_matrix * M)
These functions return a vector view of the diagonal of the matrix
M. The matrix M is not required to be square. For a rectangular
matrix the length of the diagonal is the same as the smaller
dimension of the matrix.
The function 'gsl_matrix_const_diagonal' is equivalent to
'gsl_matrix_diagonal' but can be used for matrices which are
declared 'const'.
-- Function: gsl_vector_view gsl_matrix_subdiagonal (gsl_matrix * M,
size_t K)
-- Function: gsl_vector_const_view gsl_matrix_const_subdiagonal (const
gsl_matrix * M, size_t K)
These functions return a vector view of the K-th subdiagonal of the
matrix M. The matrix M is not required to be square. The diagonal
of the matrix corresponds to k = 0.
The function 'gsl_matrix_const_subdiagonal' is equivalent to
'gsl_matrix_subdiagonal' but can be used for matrices which are
declared 'const'.
-- Function: gsl_vector_view gsl_matrix_superdiagonal (gsl_matrix * M,
size_t K)
-- Function: gsl_vector_const_view gsl_matrix_const_superdiagonal
(const gsl_matrix * M, size_t K)
These functions return a vector view of the K-th superdiagonal of
the matrix M. The matrix M is not required to be square. The
diagonal of the matrix corresponds to k = 0.
The function 'gsl_matrix_const_superdiagonal' is equivalent to
'gsl_matrix_superdiagonal' but can be used for matrices which are
declared 'const'.
8.4.7 Copying matrices
----------------------
-- Function: int gsl_matrix_memcpy (gsl_matrix * DEST, const gsl_matrix
* SRC)
This function copies the elements of the matrix SRC into the matrix
DEST. The two matrices must have the same size.
-- Function: int gsl_matrix_swap (gsl_matrix * M1, gsl_matrix * M2)
This function exchanges the elements of the matrices M1 and M2 by
copying. The two matrices must have the same size.
8.4.8 Copying rows and columns
------------------------------
The functions described in this section copy a row or column of a matrix
into a vector. This allows the elements of the vector and the matrix to
be modified independently. Note that if the matrix and the vector point
to overlapping regions of memory then the result will be undefined. The
same effect can be achieved with more generality using
'gsl_vector_memcpy' with vector views of rows and columns.
-- Function: int gsl_matrix_get_row (gsl_vector * V, const gsl_matrix *
M, size_t I)
This function copies the elements of the I-th row of the matrix M
into the vector V. The length of the vector must be the same as
the length of the row.
-- Function: int gsl_matrix_get_col (gsl_vector * V, const gsl_matrix *
M, size_t J)
This function copies the elements of the J-th column of the matrix
M into the vector V. The length of the vector must be the same as
the length of the column.
-- Function: int gsl_matrix_set_row (gsl_matrix * M, size_t I, const
gsl_vector * V)
This function copies the elements of the vector V into the I-th row
of the matrix M. The length of the vector must be the same as the
length of the row.
-- Function: int gsl_matrix_set_col (gsl_matrix * M, size_t J, const
gsl_vector * V)
This function copies the elements of the vector V into the J-th
column of the matrix M. The length of the vector must be the same
as the length of the column.
8.4.9 Exchanging rows and columns
---------------------------------
The following functions can be used to exchange the rows and columns of
a matrix.
-- Function: int gsl_matrix_swap_rows (gsl_matrix * M, size_t I, size_t
J)
This function exchanges the I-th and J-th rows of the matrix M
in-place.
-- Function: int gsl_matrix_swap_columns (gsl_matrix * M, size_t I,
size_t J)
This function exchanges the I-th and J-th columns of the matrix M
in-place.
-- Function: int gsl_matrix_swap_rowcol (gsl_matrix * M, size_t I,
size_t J)
This function exchanges the I-th row and J-th column of the matrix
M in-place. The matrix must be square for this operation to be
possible.
-- Function: int gsl_matrix_transpose_memcpy (gsl_matrix * DEST, const
gsl_matrix * SRC)
This function makes the matrix DEST the transpose of the matrix SRC
by copying the elements of SRC into DEST. This function works for
all matrices provided that the dimensions of the matrix DEST match
the transposed dimensions of the matrix SRC.
-- Function: int gsl_matrix_transpose (gsl_matrix * M)
This function replaces the matrix M by its transpose by copying the
elements of the matrix in-place. The matrix must be square for
this operation to be possible.
8.4.10 Matrix operations
------------------------
The following operations are defined for real and complex matrices.
-- Function: int gsl_matrix_add (gsl_matrix * A, const gsl_matrix * B)
This function adds the elements of matrix B to the elements of
matrix A. The result a(i,j) \leftarrow a(i,j) + b(i,j) is stored
in A and B remains unchanged. The two matrices must have the same
dimensions.
-- Function: int gsl_matrix_sub (gsl_matrix * A, const gsl_matrix * B)
This function subtracts the elements of matrix B from the elements
of matrix A. The result a(i,j) \leftarrow a(i,j) - b(i,j) is
stored in A and B remains unchanged. The two matrices must have
the same dimensions.
-- Function: int gsl_matrix_mul_elements (gsl_matrix * A, const
gsl_matrix * B)
This function multiplies the elements of matrix A by the elements
of matrix B. The result a(i,j) \leftarrow a(i,j) * b(i,j) is
stored in A and B remains unchanged. The two matrices must have
the same dimensions.
-- Function: int gsl_matrix_div_elements (gsl_matrix * A, const
gsl_matrix * B)
This function divides the elements of matrix A by the elements of
matrix B. The result a(i,j) \leftarrow a(i,j) / b(i,j) is stored
in A and B remains unchanged. The two matrices must have the same
dimensions.
-- Function: int gsl_matrix_scale (gsl_matrix * A, const double X)
This function multiplies the elements of matrix A by the constant
factor X. The result a(i,j) \leftarrow x a(i,j) is stored in A.
-- Function: int gsl_matrix_add_constant (gsl_matrix * A, const double
X)
This function adds the constant value X to the elements of the
matrix A. The result a(i,j) \leftarrow a(i,j) + x is stored in A.
8.4.11 Finding maximum and minimum elements of matrices
-------------------------------------------------------
The following operations are only defined for real matrices.
-- Function: double gsl_matrix_max (const gsl_matrix * M)
This function returns the maximum value in the matrix M.
-- Function: double gsl_matrix_min (const gsl_matrix * M)
This function returns the minimum value in the matrix M.
-- Function: void gsl_matrix_minmax (const gsl_matrix * M, double *
MIN_OUT, double * MAX_OUT)
This function returns the minimum and maximum values in the matrix
M, storing them in MIN_OUT and MAX_OUT.
-- Function: void gsl_matrix_max_index (const gsl_matrix * M, size_t *
IMAX, size_t * JMAX)
This function returns the indices of the maximum value in the
matrix M, storing them in IMAX and JMAX. When there are several
equal maximum elements then the first element found is returned,
searching in row-major order.
-- Function: void gsl_matrix_min_index (const gsl_matrix * M, size_t *
IMIN, size_t * JMIN)
This function returns the indices of the minimum value in the
matrix M, storing them in IMIN and JMIN. When there are several
equal minimum elements then the first element found is returned,
searching in row-major order.
-- Function: void gsl_matrix_minmax_index (const gsl_matrix * M, size_t
* IMIN, size_t * JMIN, size_t * IMAX, size_t * JMAX)
This function returns the indices of the minimum and maximum values
in the matrix M, storing them in (IMIN,JMIN) and (IMAX,JMAX). When
there are several equal minimum or maximum elements then the first
elements found are returned, searching in row-major order.
8.4.12 Matrix properties
------------------------
The following functions are defined for real and complex matrices. For
complex matrices both the real and imaginary parts must satisfy the
conditions.
-- Function: int gsl_matrix_isnull (const gsl_matrix * M)
-- Function: int gsl_matrix_ispos (const gsl_matrix * M)
-- Function: int gsl_matrix_isneg (const gsl_matrix * M)
-- Function: int gsl_matrix_isnonneg (const gsl_matrix * M)
These functions return 1 if all the elements of the matrix M are
zero, strictly positive, strictly negative, or non-negative
respectively, and 0 otherwise. To test whether a matrix is
positive-definite, use the Cholesky decomposition (*note Cholesky
Decomposition::).
-- Function: int gsl_matrix_equal (const gsl_matrix * A, const
gsl_matrix * B)
This function returns 1 if the matrices A and B are equal (by
comparison of element values) and 0 otherwise.
8.4.13 Example programs for matrices
------------------------------------
The program below shows how to allocate, initialize and read from a
matrix using the functions 'gsl_matrix_alloc', 'gsl_matrix_set' and
'gsl_matrix_get'.
#include
#include
int
main (void)
{
int i, j;
gsl_matrix * m = gsl_matrix_alloc (10, 3);
for (i = 0; i < 10; i++)
for (j = 0; j < 3; j++)
gsl_matrix_set (m, i, j, 0.23 + 100*i + j);
for (i = 0; i < 100; i++) /* OUT OF RANGE ERROR */
for (j = 0; j < 3; j++)
printf ("m(%d,%d) = %g\n", i, j,
gsl_matrix_get (m, i, j));
gsl_matrix_free (m);
return 0;
}
Here is the output from the program. The final loop attempts to read
outside the range of the matrix 'm', and the error is trapped by the
range-checking code in 'gsl_matrix_get'.
$ ./a.out
m(0,0) = 0.23
m(0,1) = 1.23
m(0,2) = 2.23
m(1,0) = 100.23
m(1,1) = 101.23
m(1,2) = 102.23
...
m(9,2) = 902.23
gsl: matrix_source.c:13: ERROR: first index out of range
Default GSL error handler invoked.
Aborted (core dumped)
The next program shows how to write a matrix to a file.
#include
#include
int
main (void)
{
int i, j, k = 0;
gsl_matrix * m = gsl_matrix_alloc (100, 100);
gsl_matrix * a = gsl_matrix_alloc (100, 100);
for (i = 0; i < 100; i++)
for (j = 0; j < 100; j++)
gsl_matrix_set (m, i, j, 0.23 + i + j);
{
FILE * f = fopen ("test.dat", "wb");
gsl_matrix_fwrite (f, m);
fclose (f);
}
{
FILE * f = fopen ("test.dat", "rb");
gsl_matrix_fread (f, a);
fclose (f);
}
for (i = 0; i < 100; i++)
for (j = 0; j < 100; j++)
{
double mij = gsl_matrix_get (m, i, j);
double aij = gsl_matrix_get (a, i, j);
if (mij != aij) k++;
}
gsl_matrix_free (m);
gsl_matrix_free (a);
printf ("differences = %d (should be zero)\n", k);
return (k > 0);
}
After running this program the file 'test.dat' should contain the
elements of 'm', written in binary format. The matrix which is read
back in using the function 'gsl_matrix_fread' should be exactly equal to
the original matrix.
The following program demonstrates the use of vector views. The
program computes the column norms of a matrix.
#include
#include
#include
#include
int
main (void)
{
size_t i,j;
gsl_matrix *m = gsl_matrix_alloc (10, 10);
for (i = 0; i < 10; i++)
for (j = 0; j < 10; j++)
gsl_matrix_set (m, i, j, sin (i) + cos (j));
for (j = 0; j < 10; j++)
{
gsl_vector_view column = gsl_matrix_column (m, j);
double d;
d = gsl_blas_dnrm2 (&column.vector);
printf ("matrix column %zu, norm = %g\n", j, d);
}
gsl_matrix_free (m);
return 0;
}
Here is the output of the program,
$ ./a.out
matrix column 0, norm = 4.31461
matrix column 1, norm = 3.1205
matrix column 2, norm = 2.19316
matrix column 3, norm = 3.26114
matrix column 4, norm = 2.53416
matrix column 5, norm = 2.57281
matrix column 6, norm = 4.20469
matrix column 7, norm = 3.65202
matrix column 8, norm = 2.08524
matrix column 9, norm = 3.07313
The results can be confirmed using GNU OCTAVE,
$ octave
GNU Octave, version 2.0.16.92
octave> m = sin(0:9)' * ones(1,10)
+ ones(10,1) * cos(0:9);
octave> sqrt(sum(m.^2))
ans =
4.3146 3.1205 2.1932 3.2611 2.5342 2.5728
4.2047 3.6520 2.0852 3.0731
8.5 References and Further Reading
==================================
The block, vector and matrix objects in GSL follow the 'valarray' model
of C++. A description of this model can be found in the following
reference,
B. Stroustrup, 'The C++ Programming Language' (3rd Ed), Section
22.4 Vector Arithmetic. Addison-Wesley 1997, ISBN 0-201-88954-4.
9 Permutations
**************
This chapter describes functions for creating and manipulating
permutations. A permutation p is represented by an array of n integers
in the range 0 to n-1, where each value p_i occurs once and only once.
The application of a permutation p to a vector v yields a new vector v'
where v'_i = v_{p_i}. For example, the array (0,1,3,2) represents a
permutation which exchanges the last two elements of a four element
vector. The corresponding identity permutation is (0,1,2,3).
Note that the permutations produced by the linear algebra routines
correspond to the exchange of matrix columns, and so should be
considered as applying to row-vectors in the form v' = v P rather than
column-vectors, when permuting the elements of a vector.
The functions described in this chapter are defined in the header
file 'gsl_permutation.h'.
9.1 The Permutation struct
==========================
A permutation is defined by a structure containing two components, the
size of the permutation and a pointer to the permutation array. The
elements of the permutation array are all of type 'size_t'. The
'gsl_permutation' structure looks like this,
typedef struct
{
size_t size;
size_t * data;
} gsl_permutation;
9.2 Permutation allocation
==========================
-- Function: gsl_permutation * gsl_permutation_alloc (size_t N)
This function allocates memory for a new permutation of size N.
The permutation is not initialized and its elements are undefined.
Use the function 'gsl_permutation_calloc' if you want to create a
permutation which is initialized to the identity. A null pointer
is returned if insufficient memory is available to create the
permutation.
-- Function: gsl_permutation * gsl_permutation_calloc (size_t N)
This function allocates memory for a new permutation of size N and
initializes it to the identity. A null pointer is returned if
insufficient memory is available to create the permutation.
-- Function: void gsl_permutation_init (gsl_permutation * P)
This function initializes the permutation P to the identity, i.e.
(0,1,2,...,n-1).
-- Function: void gsl_permutation_free (gsl_permutation * P)
This function frees all the memory used by the permutation P.
-- Function: int gsl_permutation_memcpy (gsl_permutation * DEST, const
gsl_permutation * SRC)
This function copies the elements of the permutation SRC into the
permutation DEST. The two permutations must have the same size.
9.3 Accessing permutation elements
==================================
The following functions can be used to access and manipulate
permutations.
-- Function: size_t gsl_permutation_get (const gsl_permutation * P,
const size_t I)
This function returns the value of the I-th element of the
permutation P. If I lies outside the allowed range of 0 to N-1
then the error handler is invoked and 0 is returned. An inline
version of this function is used when 'HAVE_INLINE' is defined.
-- Function: int gsl_permutation_swap (gsl_permutation * P, const
size_t I, const size_t J)
This function exchanges the I-th and J-th elements of the
permutation P.
9.4 Permutation properties
==========================
-- Function: size_t gsl_permutation_size (const gsl_permutation * P)
This function returns the size of the permutation P.
-- Function: size_t * gsl_permutation_data (const gsl_permutation * P)
This function returns a pointer to the array of elements in the
permutation P.
-- Function: int gsl_permutation_valid (const gsl_permutation * P)
This function checks that the permutation P is valid. The N
elements should contain each of the numbers 0 to N-1 once and only
once.
9.5 Permutation functions
=========================
-- Function: void gsl_permutation_reverse (gsl_permutation * P)
This function reverses the elements of the permutation P.
-- Function: int gsl_permutation_inverse (gsl_permutation * INV, const
gsl_permutation * P)
This function computes the inverse of the permutation P, storing
the result in INV.
-- Function: int gsl_permutation_next (gsl_permutation * P)
This function advances the permutation P to the next permutation in
lexicographic order and returns 'GSL_SUCCESS'. If no further
permutations are available it returns 'GSL_FAILURE' and leaves P
unmodified. Starting with the identity permutation and repeatedly
applying this function will iterate through all possible
permutations of a given order.
-- Function: int gsl_permutation_prev (gsl_permutation * P)
This function steps backwards from the permutation P to the
previous permutation in lexicographic order, returning
'GSL_SUCCESS'. If no previous permutation is available it returns
'GSL_FAILURE' and leaves P unmodified.
9.6 Applying Permutations
=========================
-- Function: int gsl_permute (const size_t * P, double * DATA, size_t
STRIDE, size_t N)
This function applies the permutation P to the array DATA of size N
with stride STRIDE.
-- Function: int gsl_permute_inverse (const size_t * P, double * DATA,
size_t STRIDE, size_t N)
This function applies the inverse of the permutation P to the array
DATA of size N with stride STRIDE.
-- Function: int gsl_permute_vector (const gsl_permutation * P,
gsl_vector * V)
This function applies the permutation P to the elements of the
vector V, considered as a row-vector acted on by a permutation
matrix from the right, v' = v P. The j-th column of the
permutation matrix P is given by the P_j-th column of the identity
matrix. The permutation P and the vector V must have the same
length.
-- Function: int gsl_permute_vector_inverse (const gsl_permutation * P,
gsl_vector * V)
This function applies the inverse of the permutation P to the
elements of the vector V, considered as a row-vector acted on by an
inverse permutation matrix from the right, v' = v P^T. Note that
for permutation matrices the inverse is the same as the transpose.
The j-th column of the permutation matrix P is given by the P_j-th
column of the identity matrix. The permutation P and the vector V
must have the same length.
-- Function: int gsl_permute_matrix (const gsl_permutation * P,
gsl_matrix * A)
This function applies the permutation P to the matrix A from the
right, A' = A P. The j-th column of the permutation matrix P is
given by the P_j-th column of the identity matrix. This
effectively permutes the columns of A according to the permutation
P, and so the number of columns of A must equal the size of the
permutation P.
-- Function: int gsl_permutation_mul (gsl_permutation * P, const
gsl_permutation * PA, const gsl_permutation * PB)
This function combines the two permutations PA and PB into a single
permutation P, where P = PA * PB. The permutation P is equivalent
to applying PB first and then PA.
9.7 Reading and writing permutations
====================================
The library provides functions for reading and writing permutations to a
file as binary data or formatted text.
-- Function: int gsl_permutation_fwrite (FILE * STREAM, const
gsl_permutation * P)
This function writes the elements of the permutation P to the
stream STREAM in binary format. The function returns 'GSL_EFAILED'
if there was a problem writing to the file. Since the data is
written in the native binary format it may not be portable between
different architectures.
-- Function: int gsl_permutation_fread (FILE * STREAM, gsl_permutation
* P)
This function reads into the permutation P from the open stream
STREAM in binary format. The permutation P must be preallocated
with the correct length since the function uses the size of P to
determine how many bytes to read. The function returns
'GSL_EFAILED' if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on
the same architecture.
-- Function: int gsl_permutation_fprintf (FILE * STREAM, const
gsl_permutation * P, const char * FORMAT)
This function writes the elements of the permutation P line-by-line
to the stream STREAM using the format specifier FORMAT, which
should be suitable for a type of SIZE_T. In ISO C99 the type
modifier 'z' represents 'size_t', so '"%zu\n"' is a suitable
format.(1) The function returns 'GSL_EFAILED' if there was a
problem writing to the file.
-- Function: int gsl_permutation_fscanf (FILE * STREAM, gsl_permutation
* P)
This function reads formatted data from the stream STREAM into the
permutation P. The permutation P must be preallocated with the
correct length since the function uses the size of P to determine
how many numbers to read. The function returns 'GSL_EFAILED' if
there was a problem reading from the file.
---------- Footnotes ----------
(1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier 'Z' was used instead.
9.8 Permutations in cyclic form
===============================
A permutation can be represented in both "linear" and "cyclic"
notations. The functions described in this section convert between the
two forms. The linear notation is an index mapping, and has already
been described above. The cyclic notation expresses a permutation as a
series of circular rearrangements of groups of elements, or "cycles".
For example, under the cycle (1 2 3), 1 is replaced by 2, 2 is
replaced by 3 and 3 is replaced by 1 in a circular fashion. Cycles of
different sets of elements can be combined independently, for example (1
2 3) (4 5) combines the cycle (1 2 3) with the cycle (4 5), which is an
exchange of elements 4 and 5. A cycle of length one represents an
element which is unchanged by the permutation and is referred to as a
"singleton".
It can be shown that every permutation can be decomposed into
combinations of cycles. The decomposition is not unique, but can always
be rearranged into a standard "canonical form" by a reordering of
elements. The library uses the canonical form defined in Knuth's 'Art
of Computer Programming' (Vol 1, 3rd Ed, 1997) Section 1.3.3, p.178.
The procedure for obtaining the canonical form given by Knuth is,
1. Write all singleton cycles explicitly
2. Within each cycle, put the smallest number first
3. Order the cycles in decreasing order of the first number in the
cycle.
For example, the linear representation (2 4 3 0 1) is represented as (1
4) (0 2 3) in canonical form. The permutation corresponds to an
exchange of elements 1 and 4, and rotation of elements 0, 2 and 3.
The important property of the canonical form is that it can be
reconstructed from the contents of each cycle without the brackets. In
addition, by removing the brackets it can be considered as a linear
representation of a different permutation. In the example given above
the permutation (2 4 3 0 1) would become (1 4 0 2 3). This mapping has
many applications in the theory of permutations.
-- Function: int gsl_permutation_linear_to_canonical (gsl_permutation *
Q, const gsl_permutation * P)
This function computes the canonical form of the permutation P and
stores it in the output argument Q.
-- Function: int gsl_permutation_canonical_to_linear (gsl_permutation *
P, const gsl_permutation * Q)
This function converts a permutation Q in canonical form back into
linear form storing it in the output argument P.
-- Function: size_t gsl_permutation_inversions (const gsl_permutation *
P)
This function counts the number of inversions in the permutation P.
An inversion is any pair of elements that are not in order. For
example, the permutation 2031 has three inversions, corresponding
to the pairs (2,0) (2,1) and (3,1). The identity permutation has
no inversions.
-- Function: size_t gsl_permutation_linear_cycles (const
gsl_permutation * P)
This function counts the number of cycles in the permutation P,
given in linear form.
-- Function: size_t gsl_permutation_canonical_cycles (const
gsl_permutation * Q)
This function counts the number of cycles in the permutation Q,
given in canonical form.
9.9 Examples
============
The example program below creates a random permutation (by shuffling the
elements of the identity) and finds its inverse.
#include
#include
#include
#include
int
main (void)
{
const size_t N = 10;
const gsl_rng_type * T;
gsl_rng * r;
gsl_permutation * p = gsl_permutation_alloc (N);
gsl_permutation * q = gsl_permutation_alloc (N);
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf ("initial permutation:");
gsl_permutation_init (p);
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
printf (" random permutation:");
gsl_ran_shuffle (r, p->data, N, sizeof(size_t));
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
printf ("inverse permutation:");
gsl_permutation_inverse (q, p);
gsl_permutation_fprintf (stdout, q, " %u");
printf ("\n");
gsl_permutation_free (p);
gsl_permutation_free (q);
gsl_rng_free (r);
return 0;
}
Here is the output from the program,
$ ./a.out
initial permutation: 0 1 2 3 4 5 6 7 8 9
random permutation: 1 3 5 2 7 6 0 4 9 8
inverse permutation: 6 0 3 1 7 2 5 4 9 8
The random permutation 'p[i]' and its inverse 'q[i]' are related through
the identity 'p[q[i]] = i', which can be verified from the output.
The next example program steps forwards through all possible third
order permutations, starting from the identity,
#include
#include
int
main (void)
{
gsl_permutation * p = gsl_permutation_alloc (3);
gsl_permutation_init (p);
do
{
gsl_permutation_fprintf (stdout, p, " %u");
printf ("\n");
}
while (gsl_permutation_next(p) == GSL_SUCCESS);
gsl_permutation_free (p);
return 0;
}
Here is the output from the program,
$ ./a.out
0 1 2
0 2 1
1 0 2
1 2 0
2 0 1
2 1 0
The permutations are generated in lexicographic order. To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace 'gsl_permutation_next' with
'gsl_permutation_prev'.
9.10 References and Further Reading
===================================
The subject of permutations is covered extensively in Knuth's 'Sorting
and Searching',
Donald E. Knuth, 'The Art of Computer Programming: Sorting and
Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
For the definition of the "canonical form" see,
Donald E. Knuth, 'The Art of Computer Programming: Fundamental
Algorithms' (Vol 1, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
Section 1.3.3, 'An Unusual Correspondence', p.178-179.
10 Combinations
***************
This chapter describes functions for creating and manipulating
combinations. A combination c is represented by an array of k integers
in the range 0 to n-1, where each value c_i occurs at most once. The
combination c corresponds to indices of k elements chosen from an n
element vector. Combinations are useful for iterating over all
k-element subsets of a set.
The functions described in this chapter are defined in the header
file 'gsl_combination.h'.
10.1 The Combination struct
===========================
A combination is defined by a structure containing three components, the
values of n and k, and a pointer to the combination array. The elements
of the combination array are all of type 'size_t', and are stored in
increasing order. The 'gsl_combination' structure looks like this,
typedef struct
{
size_t n;
size_t k;
size_t *data;
} gsl_combination;
10.2 Combination allocation
===========================
-- Function: gsl_combination * gsl_combination_alloc (size_t N, size_t
K)
This function allocates memory for a new combination with
parameters N, K. The combination is not initialized and its
elements are undefined. Use the function 'gsl_combination_calloc'
if you want to create a combination which is initialized to the
lexicographically first combination. A null pointer is returned if
insufficient memory is available to create the combination.
-- Function: gsl_combination * gsl_combination_calloc (size_t N, size_t
K)
This function allocates memory for a new combination with
parameters N, K and initializes it to the lexicographically first
combination. A null pointer is returned if insufficient memory is
available to create the combination.
-- Function: void gsl_combination_init_first (gsl_combination * C)
This function initializes the combination C to the
lexicographically first combination, i.e. (0,1,2,...,k-1).
-- Function: void gsl_combination_init_last (gsl_combination * C)
This function initializes the combination C to the
lexicographically last combination, i.e. (n-k,n-k+1,...,n-1).
-- Function: void gsl_combination_free (gsl_combination * C)
This function frees all the memory used by the combination C.
-- Function: int gsl_combination_memcpy (gsl_combination * DEST, const
gsl_combination * SRC)
This function copies the elements of the combination SRC into the
combination DEST. The two combinations must have the same size.
10.3 Accessing combination elements
===================================
The following function can be used to access the elements of a
combination.
-- Function: size_t gsl_combination_get (const gsl_combination * C,
const size_t I)
This function returns the value of the I-th element of the
combination C. If I lies outside the allowed range of 0 to K-1
then the error handler is invoked and 0 is returned. An inline
version of this function is used when 'HAVE_INLINE' is defined.
10.4 Combination properties
===========================
-- Function: size_t gsl_combination_n (const gsl_combination * C)
This function returns the range (n) of the combination C.
-- Function: size_t gsl_combination_k (const gsl_combination * C)
This function returns the number of elements (k) in the combination
C.
-- Function: size_t * gsl_combination_data (const gsl_combination * C)
This function returns a pointer to the array of elements in the
combination C.
-- Function: int gsl_combination_valid (gsl_combination * C)
This function checks that the combination C is valid. The K
elements should lie in the range 0 to N-1, with each value
occurring once at most and in increasing order.
10.5 Combination functions
==========================
-- Function: int gsl_combination_next (gsl_combination * C)
This function advances the combination C to the next combination in
lexicographic order and returns 'GSL_SUCCESS'. If no further
combinations are available it returns 'GSL_FAILURE' and leaves C
unmodified. Starting with the first combination and repeatedly
applying this function will iterate through all possible
combinations of a given order.
-- Function: int gsl_combination_prev (gsl_combination * C)
This function steps backwards from the combination C to the
previous combination in lexicographic order, returning
'GSL_SUCCESS'. If no previous combination is available it returns
'GSL_FAILURE' and leaves C unmodified.
10.6 Reading and writing combinations
=====================================
The library provides functions for reading and writing combinations to a
file as binary data or formatted text.
-- Function: int gsl_combination_fwrite (FILE * STREAM, const
gsl_combination * C)
This function writes the elements of the combination C to the
stream STREAM in binary format. The function returns 'GSL_EFAILED'
if there was a problem writing to the file. Since the data is
written in the native binary format it may not be portable between
different architectures.
-- Function: int gsl_combination_fread (FILE * STREAM, gsl_combination
* C)
This function reads elements from the open stream STREAM into the
combination C in binary format. The combination C must be
preallocated with correct values of n and k since the function uses
the size of C to determine how many bytes to read. The function
returns 'GSL_EFAILED' if there was a problem reading from the file.
The data is assumed to have been written in the native binary
format on the same architecture.
-- Function: int gsl_combination_fprintf (FILE * STREAM, const
gsl_combination * C, const char * FORMAT)
This function writes the elements of the combination C line-by-line
to the stream STREAM using the format specifier FORMAT, which
should be suitable for a type of SIZE_T. In ISO C99 the type
modifier 'z' represents 'size_t', so '"%zu\n"' is a suitable
format.(1) The function returns 'GSL_EFAILED' if there was a
problem writing to the file.
-- Function: int gsl_combination_fscanf (FILE * STREAM, gsl_combination
* C)
This function reads formatted data from the stream STREAM into the
combination C. The combination C must be preallocated with correct
values of n and k since the function uses the size of C to
determine how many numbers to read. The function returns
'GSL_EFAILED' if there was a problem reading from the file.
---------- Footnotes ----------
(1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier 'Z' was used instead.
10.7 Examples
=============
The example program below prints all subsets of the set {0,1,2,3}
ordered by size. Subsets of the same size are ordered
lexicographically.
#include
#include
int
main (void)
{
gsl_combination * c;
size_t i;
printf ("All subsets of {0,1,2,3} by size:\n") ;
for (i = 0; i <= 4; i++)
{
c = gsl_combination_calloc (4, i);
do
{
printf ("{");
gsl_combination_fprintf (stdout, c, " %u");
printf (" }\n");
}
while (gsl_combination_next (c) == GSL_SUCCESS);
gsl_combination_free (c);
}
return 0;
}
Here is the output from the program,
$ ./a.out
All subsets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 2 }
{ 1 3 }
{ 2 3 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 3 }
{ 1 2 3 }
{ 0 1 2 3 }
All 16 subsets are generated, and the subsets of each size are sorted
lexicographically.
10.8 References and Further Reading
===================================
Further information on combinations can be found in,
Donald L. Kreher, Douglas R. Stinson, 'Combinatorial Algorithms:
Generation, Enumeration and Search', 1998, CRC Press LLC, ISBN
084933988X
11 Multisets
************
This chapter describes functions for creating and manipulating
multisets. A multiset c is represented by an array of k integers in the
range 0 to n-1, where each value c_i may occur more than once. The
multiset c corresponds to indices of k elements chosen from an n element
vector with replacement. In mathematical terms, n is the cardinality of
the multiset while k is the maximum multiplicity of any value.
Multisets are useful, for example, when iterating over the indices of a
k-th order symmetric tensor in n-space.
The functions described in this chapter are defined in the header
file 'gsl_multiset.h'.
11.1 The Multiset struct
========================
A multiset is defined by a structure containing three components, the
values of n and k, and a pointer to the multiset array. The elements of
the multiset array are all of type 'size_t', and are stored in
increasing order. The 'gsl_multiset' structure looks like this,
typedef struct
{
size_t n;
size_t k;
size_t *data;
} gsl_multiset;
11.2 Multiset allocation
========================
-- Function: gsl_multiset * gsl_multiset_alloc (size_t N, size_t K)
This function allocates memory for a new multiset with parameters
N, K. The multiset is not initialized and its elements are
undefined. Use the function 'gsl_multiset_calloc' if you want to
create a multiset which is initialized to the lexicographically
first multiset element. A null pointer is returned if insufficient
memory is available to create the multiset.
-- Function: gsl_multiset * gsl_multiset_calloc (size_t N, size_t K)
This function allocates memory for a new multiset with parameters
N, K and initializes it to the lexicographically first multiset
element. A null pointer is returned if insufficient memory is
available to create the multiset.
-- Function: void gsl_multiset_init_first (gsl_multiset * C)
This function initializes the multiset C to the lexicographically
first multiset element, i.e. 0 repeated k times.
-- Function: void gsl_multiset_init_last (gsl_multiset * C)
This function initializes the multiset C to the lexicographically
last multiset element, i.e. n-1 repeated k times.
-- Function: void gsl_multiset_free (gsl_multiset * C)
This function frees all the memory used by the multiset C.
-- Function: int gsl_multiset_memcpy (gsl_multiset * DEST, const
gsl_multiset * SRC)
This function copies the elements of the multiset SRC into the
multiset DEST. The two multisets must have the same size.
11.3 Accessing multiset elements
================================
The following function can be used to access the elements of a multiset.
-- Function: size_t gsl_multiset_get (const gsl_multiset * C, const
size_t I)
This function returns the value of the I-th element of the multiset
C. If I lies outside the allowed range of 0 to K-1 then the error
handler is invoked and 0 is returned. An inline version of this
function is used when 'HAVE_INLINE' is defined.
11.4 Multiset properties
========================
-- Function: size_t gsl_multiset_n (const gsl_multiset * C)
This function returns the range (n) of the multiset C.
-- Function: size_t gsl_multiset_k (const gsl_multiset * C)
This function returns the number of elements (k) in the multiset C.
-- Function: size_t * gsl_multiset_data (const gsl_multiset * C)
This function returns a pointer to the array of elements in the
multiset C.
-- Function: int gsl_multiset_valid (gsl_multiset * C)
This function checks that the multiset C is valid. The K elements
should lie in the range 0 to N-1, with each value occurring in
nondecreasing order.
11.5 Multiset functions
=======================
-- Function: int gsl_multiset_next (gsl_multiset * C)
This function advances the multiset C to the next multiset element
in lexicographic order and returns 'GSL_SUCCESS'. If no further
multisets elements are available it returns 'GSL_FAILURE' and
leaves C unmodified. Starting with the first multiset and
repeatedly applying this function will iterate through all possible
multisets of a given order.
-- Function: int gsl_multiset_prev (gsl_multiset * C)
This function steps backwards from the multiset C to the previous
multiset element in lexicographic order, returning 'GSL_SUCCESS'.
If no previous multiset is available it returns 'GSL_FAILURE' and
leaves C unmodified.
11.6 Reading and writing multisets
==================================
The library provides functions for reading and writing multisets to a
file as binary data or formatted text.
-- Function: int gsl_multiset_fwrite (FILE * STREAM, const gsl_multiset
* C)
This function writes the elements of the multiset C to the stream
STREAM in binary format. The function returns 'GSL_EFAILED' if
there was a problem writing to the file. Since the data is written
in the native binary format it may not be portable between
different architectures.
-- Function: int gsl_multiset_fread (FILE * STREAM, gsl_multiset * C)
This function reads elements from the open stream STREAM into the
multiset C in binary format. The multiset C must be preallocated
with correct values of n and k since the function uses the size of
C to determine how many bytes to read. The function returns
'GSL_EFAILED' if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on
the same architecture.
-- Function: int gsl_multiset_fprintf (FILE * STREAM, const
gsl_multiset * C, const char * FORMAT)
This function writes the elements of the multiset C line-by-line to
the stream STREAM using the format specifier FORMAT, which should
be suitable for a type of SIZE_T. In ISO C99 the type modifier 'z'
represents 'size_t', so '"%zu\n"' is a suitable format.(1) The
function returns 'GSL_EFAILED' if there was a problem writing to
the file.
-- Function: int gsl_multiset_fscanf (FILE * STREAM, gsl_multiset * C)
This function reads formatted data from the stream STREAM into the
multiset C. The multiset C must be preallocated with correct
values of n and k since the function uses the size of C to
determine how many numbers to read. The function returns
'GSL_EFAILED' if there was a problem reading from the file.
---------- Footnotes ----------
(1) In versions of the GNU C library prior to the ISO C99 standard,
the type modifier 'Z' was used instead.
11.7 Examples
=============
The example program below prints all multisets elements containing the
values {0,1,2,3} ordered by size. Multiset elements of the same size
are ordered lexicographically.
#include
#include
int
main (void)
{
gsl_multiset * c;
size_t i;
printf ("All multisets of {0,1,2,3} by size:\n") ;
for (i = 0; i <= 4; i++)
{
c = gsl_multiset_calloc (4, i);
do
{
printf ("{");
gsl_multiset_fprintf (stdout, c, " %u");
printf (" }\n");
}
while (gsl_multiset_next (c) == GSL_SUCCESS);
gsl_multiset_free (c);
}
return 0;
}
Here is the output from the program,
$ ./a.out
All multisets of {0,1,2,3} by size:
{ }
{ 0 }
{ 1 }
{ 2 }
{ 3 }
{ 0 0 }
{ 0 1 }
{ 0 2 }
{ 0 3 }
{ 1 1 }
{ 1 2 }
{ 1 3 }
{ 2 2 }
{ 2 3 }
{ 3 3 }
{ 0 0 0 }
{ 0 0 1 }
{ 0 0 2 }
{ 0 0 3 }
{ 0 1 1 }
{ 0 1 2 }
{ 0 1 3 }
{ 0 2 2 }
{ 0 2 3 }
{ 0 3 3 }
{ 1 1 1 }
{ 1 1 2 }
{ 1 1 3 }
{ 1 2 2 }
{ 1 2 3 }
{ 1 3 3 }
{ 2 2 2 }
{ 2 2 3 }
{ 2 3 3 }
{ 3 3 3 }
{ 0 0 0 0 }
{ 0 0 0 1 }
{ 0 0 0 2 }
{ 0 0 0 3 }
{ 0 0 1 1 }
{ 0 0 1 2 }
{ 0 0 1 3 }
{ 0 0 2 2 }
{ 0 0 2 3 }
{ 0 0 3 3 }
{ 0 1 1 1 }
{ 0 1 1 2 }
{ 0 1 1 3 }
{ 0 1 2 2 }
{ 0 1 2 3 }
{ 0 1 3 3 }
{ 0 2 2 2 }
{ 0 2 2 3 }
{ 0 2 3 3 }
{ 0 3 3 3 }
{ 1 1 1 1 }
{ 1 1 1 2 }
{ 1 1 1 3 }
{ 1 1 2 2 }
{ 1 1 2 3 }
{ 1 1 3 3 }
{ 1 2 2 2 }
{ 1 2 2 3 }
{ 1 2 3 3 }
{ 1 3 3 3 }
{ 2 2 2 2 }
{ 2 2 2 3 }
{ 2 2 3 3 }
{ 2 3 3 3 }
{ 3 3 3 3 }
All 70 multisets are generated and sorted lexicographically.
12 Sorting
**********
This chapter describes functions for sorting data, both directly and
indirectly (using an index). All the functions use the "heapsort"
algorithm. Heapsort is an O(N \log N) algorithm which operates in-place
and does not require any additional storage. It also provides
consistent performance, the running time for its worst-case (ordered
data) being not significantly longer than the average and best cases.
Note that the heapsort algorithm does not preserve the relative ordering
of equal elements--it is an "unstable" sort. However the resulting
order of equal elements will be consistent across different platforms
when using these functions.
12.1 Sorting objects
====================
The following function provides a simple alternative to the standard
library function 'qsort'. It is intended for systems lacking 'qsort',
not as a replacement for it. The function 'qsort' should be used
whenever possible, as it will be faster and can provide stable ordering
of equal elements. Documentation for 'qsort' is available in the 'GNU C
Library Reference Manual'.
The functions described in this section are defined in the header
file 'gsl_heapsort.h'.
-- Function: void gsl_heapsort (void * ARRAY, size_t COUNT, size_t
SIZE, gsl_comparison_fn_t COMPARE)
This function sorts the COUNT elements of the array ARRAY, each of
size SIZE, into ascending order using the comparison function
COMPARE. The type of the comparison function is defined by,
int (*gsl_comparison_fn_t) (const void * a,
const void * b)
A comparison function should return a negative integer if the first
argument is less than the second argument, '0' if the two arguments
are equal and a positive integer if the first argument is greater
than the second argument.
For example, the following function can be used to sort doubles
into ascending numerical order.
int
compare_doubles (const double * a,
const double * b)
{
if (*a > *b)
return 1;
else if (*a < *b)
return -1;
else
return 0;
}
The appropriate function call to perform the sort is,
gsl_heapsort (array, count, sizeof(double),
compare_doubles);
Note that unlike 'qsort' the heapsort algorithm cannot be made into
a stable sort by pointer arithmetic. The trick of comparing
pointers for equal elements in the comparison function does not
work for the heapsort algorithm. The heapsort algorithm performs
an internal rearrangement of the data which destroys its initial
ordering.
-- Function: int gsl_heapsort_index (size_t * P, const void * ARRAY,
size_t COUNT, size_t SIZE, gsl_comparison_fn_t COMPARE)
This function indirectly sorts the COUNT elements of the array
ARRAY, each of size SIZE, into ascending order using the comparison
function COMPARE. The resulting permutation is stored in P, an
array of length N. The elements of P give the index of the array
element which would have been stored in that position if the array
had been sorted in place. The first element of P gives the index
of the least element in ARRAY, and the last element of P gives the
index of the greatest element in ARRAY. The array itself is not
changed.
12.2 Sorting vectors
====================
The following functions will sort the elements of an array or vector,
either directly or indirectly. They are defined for all real and
integer types using the normal suffix rules. For example, the 'float'
versions of the array functions are 'gsl_sort_float' and
'gsl_sort_float_index'. The corresponding vector functions are
'gsl_sort_vector_float' and 'gsl_sort_vector_float_index'. The
prototypes are available in the header files 'gsl_sort_float.h'
'gsl_sort_vector_float.h'. The complete set of prototypes can be
included using the header files 'gsl_sort.h' and 'gsl_sort_vector.h'.
There are no functions for sorting complex arrays or vectors, since
the ordering of complex numbers is not uniquely defined. To sort a
complex vector by magnitude compute a real vector containing the
magnitudes of the complex elements, and sort this vector indirectly.
The resulting index gives the appropriate ordering of the original
complex vector.
-- Function: void gsl_sort (double * DATA, const size_t STRIDE, size_t
N)
This function sorts the N elements of the array DATA with stride
STRIDE into ascending numerical order.
-- Function: void gsl_sort2 (double * DATA1, const size_t STRIDE1,
double * DATA2, const size_t STRIDE2, size_t N)
This function sorts the N elements of the array DATA1 with stride
STRIDE1 into ascending numerical order, while making the same
rearrangement of the array DATA2 with stride STRIDE2, also of size
N.
-- Function: void gsl_sort_vector (gsl_vector * V)
This function sorts the elements of the vector V into ascending
numerical order.
-- Function: void gsl_sort_vector2 (gsl_vector * V1, gsl_vector * V2)
This function sorts the elements of the vector V1 into ascending
numerical order, while making the same rearrangement of the vector
V2.
-- Function: void gsl_sort_index (size_t * P, const double * DATA,
size_t STRIDE, size_t N)
This function indirectly sorts the N elements of the array DATA
with stride STRIDE into ascending order, storing the resulting
permutation in P. The array P must be allocated with a sufficient
length to store the N elements of the permutation. The elements of
P give the index of the array element which would have been stored
in that position if the array had been sorted in place. The array
DATA is not changed.
-- Function: int gsl_sort_vector_index (gsl_permutation * P, const
gsl_vector * V)
This function indirectly sorts the elements of the vector V into
ascending order, storing the resulting permutation in P. The
elements of P give the index of the vector element which would have
been stored in that position if the vector had been sorted in
place. The first element of P gives the index of the least element
in V, and the last element of P gives the index of the greatest
element in V. The vector V is not changed.
12.3 Selecting the k smallest or largest elements
=================================================
The functions described in this section select the k smallest or largest
elements of a data set of size N. The routines use an O(kN) direct
insertion algorithm which is suited to subsets that are small compared
with the total size of the dataset. For example, the routines are
useful for selecting the 10 largest values from one million data points,
but not for selecting the largest 100,000 values. If the subset is a
significant part of the total dataset it may be faster to sort all the
elements of the dataset directly with an O(N \log N) algorithm and
obtain the smallest or largest values that way.
-- Function: int gsl_sort_smallest (double * DEST, size_t K, const
double * SRC, size_t STRIDE, size_t N)
This function copies the K smallest elements of the array SRC, of
size N and stride STRIDE, in ascending numerical order into the
array DEST. The size K of the subset must be less than or equal to
N. The data SRC is not modified by this operation.
-- Function: int gsl_sort_largest (double * DEST, size_t K, const
double * SRC, size_t STRIDE, size_t N)
This function copies the K largest elements of the array SRC, of
size N and stride STRIDE, in descending numerical order into the
array DEST. K must be less than or equal to N. The data SRC is
not modified by this operation.
-- Function: int gsl_sort_vector_smallest (double * DEST, size_t K,
const gsl_vector * V)
-- Function: int gsl_sort_vector_largest (double * DEST, size_t K,
const gsl_vector * V)
These functions copy the K smallest or largest elements of the
vector V into the array DEST. K must be less than or equal to the
length of the vector V.
The following functions find the indices of the k smallest or largest
elements of a dataset,
-- Function: int gsl_sort_smallest_index (size_t * P, size_t K, const
double * SRC, size_t STRIDE, size_t N)
This function stores the indices of the K smallest elements of the
array SRC, of size N and stride STRIDE, in the array P. The
indices are chosen so that the corresponding data is in ascending
numerical order. K must be less than or equal to N. The data SRC
is not modified by this operation.
-- Function: int gsl_sort_largest_index (size_t * P, size_t K, const
double * SRC, size_t STRIDE, size_t N)
This function stores the indices of the K largest elements of the
array SRC, of size N and stride STRIDE, in the array P. The
indices are chosen so that the corresponding data is in descending
numerical order. K must be less than or equal to N. The data SRC
is not modified by this operation.
-- Function: int gsl_sort_vector_smallest_index (size_t * P, size_t K,
const gsl_vector * V)
-- Function: int gsl_sort_vector_largest_index (size_t * P, size_t K,
const gsl_vector * V)
These functions store the indices of the K smallest or largest
elements of the vector V in the array P. K must be less than or
equal to the length of the vector V.
12.4 Computing the rank
=======================
The "rank" of an element is its order in the sorted data. The rank is
the inverse of the index permutation, P. It can be computed using the
following algorithm,
for (i = 0; i < p->size; i++)
{
size_t pi = p->data[i];
rank->data[pi] = i;
}
This can be computed directly from the function
'gsl_permutation_inverse(rank,p)'.
The following function will print the rank of each element of the
vector V,
void
print_rank (gsl_vector * v)
{
size_t i;
size_t n = v->size;
gsl_permutation * perm = gsl_permutation_alloc(n);
gsl_permutation * rank = gsl_permutation_alloc(n);
gsl_sort_vector_index (perm, v);
gsl_permutation_inverse (rank, perm);
for (i = 0; i < n; i++)
{
double vi = gsl_vector_get(v, i);
printf ("element = %d, value = %g, rank = %d\n",
i, vi, rank->data[i]);
}
gsl_permutation_free (perm);
gsl_permutation_free (rank);
}
12.5 Examples
=============
The following example shows how to use the permutation P to print the
elements of the vector V in ascending order,
gsl_sort_vector_index (p, v);
for (i = 0; i < v->size; i++)
{
double vpi = gsl_vector_get (v, p->data[i]);
printf ("order = %d, value = %g\n", i, vpi);
}
The next example uses the function 'gsl_sort_smallest' to select the 5
smallest numbers from 100000 uniform random variates stored in an array,
#include
#include
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
size_t i, k = 5, N = 100000;
double * x = malloc (N * sizeof(double));
double * small = malloc (k * sizeof(double));
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < N; i++)
{
x[i] = gsl_rng_uniform(r);
}
gsl_sort_smallest (small, k, x, 1, N);
printf ("%zu smallest values from %zu\n", k, N);
for (i = 0; i < k; i++)
{
printf ("%zu: %.18f\n", i, small[i]);
}
free (x);
free (small);
gsl_rng_free (r);
return 0;
}
The output lists the 5 smallest values, in ascending order,
$ ./a.out
5 smallest values from 100000
0: 0.000003489200025797
1: 0.000008199829608202
2: 0.000008953968062997
3: 0.000010712770745158
4: 0.000033531803637743
12.6 References and Further Reading
===================================
The subject of sorting is covered extensively in Knuth's 'Sorting and
Searching',
Donald E. Knuth, 'The Art of Computer Programming: Sorting and
Searching' (Vol 3, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896850.
The Heapsort algorithm is described in the following book,
Robert Sedgewick, 'Algorithms in C', Addison-Wesley, ISBN
0201514257.
13 BLAS Support
***************
The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental
operations on vectors and matrices which can be used to create optimized
higher-level linear algebra functionality.
The library provides a low-level layer which corresponds directly to
the C-language BLAS standard, referred to here as "CBLAS", and a
higher-level interface for operations on GSL vectors and matrices.
Users who are interested in simple operations on GSL vector and matrix
objects should use the high-level layer described in this chapter. The
functions are declared in the file 'gsl_blas.h' and should satisfy the
needs of most users.
Note that GSL matrices are implemented using dense-storage so the
interface only includes the corresponding dense-storage BLAS functions.
The full BLAS functionality for band-format and packed-format matrices
is available through the low-level CBLAS interface. Similarly, GSL
vectors are restricted to positive strides, whereas the low-level CBLAS
interface supports negative strides as specified in the BLAS
standard.(1)
The interface for the 'gsl_cblas' layer is specified in the file
'gsl_cblas.h'. This interface corresponds to the BLAS Technical Forum's
standard for the C interface to legacy BLAS implementations. Users who
have access to other conforming CBLAS implementations can use these in
place of the version provided by the library. Note that users who have
only a Fortran BLAS library can use a CBLAS conformant wrapper to
convert it into a CBLAS library. A reference CBLAS wrapper for legacy
Fortran implementations exists as part of the CBLAS standard and can be
obtained from Netlib. The complete set of CBLAS functions is listed in
an appendix (*note GSL CBLAS Library::).
There are three levels of BLAS operations,
Level 1
Vector operations, e.g. y = \alpha x + y
Level 2
Matrix-vector operations, e.g. y = \alpha A x + \beta y
Level 3
Matrix-matrix operations, e.g. C = \alpha A B + C
Each routine has a name which specifies the operation, the type of
matrices involved and their precisions. Some of the most common
operations and their names are given below,
DOT
scalar product, x^T y
AXPY
vector sum, \alpha x + y
MV
matrix-vector product, A x
SV
matrix-vector solve, inv(A) x
MM
matrix-matrix product, A B
SM
matrix-matrix solve, inv(A) B
The types of matrices are,
GE
general
GB
general band
SY
symmetric
SB
symmetric band
SP
symmetric packed
HE
hermitian
HB
hermitian band
HP
hermitian packed
TR
triangular
TB
triangular band
TP
triangular packed
Each operation is defined for four precisions,
S
single real
D
double real
C
single complex
Z
double complex
Thus, for example, the name SGEMM stands for "single-precision general
matrix-matrix multiply" and ZGEMM stands for "double-precision complex
matrix-matrix multiply".
Note that the vector and matrix arguments to BLAS functions must not
be aliased, as the results are undefined when the underlying arrays
overlap (*note Aliasing of arrays::).
---------- Footnotes ----------
(1) In the low-level CBLAS interface, a negative stride accesses the
vector elements in reverse order, i.e. the i-th element is given by
(N-i)*|incx| for incx < 0.
13.1 GSL BLAS Interface
=======================
GSL provides dense vector and matrix objects, based on the relevant
built-in types. The library provides an interface to the BLAS
operations which apply to these objects. The interface to this
functionality is given in the file 'gsl_blas.h'.
13.1.1 Level 1
--------------
-- Function: int gsl_blas_sdsdot (float ALPHA, const gsl_vector_float *
X, const gsl_vector_float * Y, float * RESULT)
This function computes the sum \alpha + x^T y for the vectors X and
Y, returning the result in RESULT.
-- Function: int gsl_blas_sdot (const gsl_vector_float * X, const
gsl_vector_float * Y, float * RESULT)
-- Function: int gsl_blas_dsdot (const gsl_vector_float * X, const
gsl_vector_float * Y, double * RESULT)
-- Function: int gsl_blas_ddot (const gsl_vector * X, const gsl_vector
* Y, double * RESULT)
These functions compute the scalar product x^T y for the vectors X
and Y, returning the result in RESULT.
-- Function: int gsl_blas_cdotu (const gsl_vector_complex_float * X,
const gsl_vector_complex_float * Y, gsl_complex_float * DOTU)
-- Function: int gsl_blas_zdotu (const gsl_vector_complex * X, const
gsl_vector_complex * Y, gsl_complex * DOTU)
These functions compute the complex scalar product x^T y for the
vectors X and Y, returning the result in DOTU
-- Function: int gsl_blas_cdotc (const gsl_vector_complex_float * X,
const gsl_vector_complex_float * Y, gsl_complex_float * DOTC)
-- Function: int gsl_blas_zdotc (const gsl_vector_complex * X, const
gsl_vector_complex * Y, gsl_complex * DOTC)
These functions compute the complex conjugate scalar product x^H y
for the vectors X and Y, returning the result in DOTC
-- Function: float gsl_blas_snrm2 (const gsl_vector_float * X)
-- Function: double gsl_blas_dnrm2 (const gsl_vector * X)
These functions compute the Euclidean norm ||x||_2 = \sqrt {\sum
x_i^2} of the vector X.
-- Function: float gsl_blas_scnrm2 (const gsl_vector_complex_float * X)
-- Function: double gsl_blas_dznrm2 (const gsl_vector_complex * X)
These functions compute the Euclidean norm of the complex vector X,
||x||_2 = \sqrt {\sum (\Re(x_i)^2 + \Im(x_i)^2)}.
-- Function: float gsl_blas_sasum (const gsl_vector_float * X)
-- Function: double gsl_blas_dasum (const gsl_vector * X)
These functions compute the absolute sum \sum |x_i| of the elements
of the vector X.
-- Function: float gsl_blas_scasum (const gsl_vector_complex_float * X)
-- Function: double gsl_blas_dzasum (const gsl_vector_complex * X)
These functions compute the sum of the magnitudes of the real and
imaginary parts of the complex vector X, \sum |\Re(x_i)| +
|\Im(x_i)|.
-- Function: CBLAS_INDEX_t gsl_blas_isamax (const gsl_vector_float * X)
-- Function: CBLAS_INDEX_t gsl_blas_idamax (const gsl_vector * X)
-- Function: CBLAS_INDEX_t gsl_blas_icamax (const
gsl_vector_complex_float * X)
-- Function: CBLAS_INDEX_t gsl_blas_izamax (const gsl_vector_complex *
X)
These functions return the index of the largest element of the
vector X. The largest element is determined by its absolute
magnitude for real vectors and by the sum of the magnitudes of the
real and imaginary parts |\Re(x_i)| + |\Im(x_i)| for complex
vectors. If the largest value occurs several times then the index
of the first occurrence is returned.
-- Function: int gsl_blas_sswap (gsl_vector_float * X, gsl_vector_float
* Y)
-- Function: int gsl_blas_dswap (gsl_vector * X, gsl_vector * Y)
-- Function: int gsl_blas_cswap (gsl_vector_complex_float * X,
gsl_vector_complex_float * Y)
-- Function: int gsl_blas_zswap (gsl_vector_complex * X,
gsl_vector_complex * Y)
These functions exchange the elements of the vectors X and Y.
-- Function: int gsl_blas_scopy (const gsl_vector_float * X,
gsl_vector_float * Y)
-- Function: int gsl_blas_dcopy (const gsl_vector * X, gsl_vector * Y)
-- Function: int gsl_blas_ccopy (const gsl_vector_complex_float * X,
gsl_vector_complex_float * Y)
-- Function: int gsl_blas_zcopy (const gsl_vector_complex * X,
gsl_vector_complex * Y)
These functions copy the elements of the vector X into the vector
Y.
-- Function: int gsl_blas_saxpy (float ALPHA, const gsl_vector_float *
X, gsl_vector_float * Y)
-- Function: int gsl_blas_daxpy (double ALPHA, const gsl_vector * X,
gsl_vector * Y)
-- Function: int gsl_blas_caxpy (const gsl_complex_float ALPHA, const
gsl_vector_complex_float * X, gsl_vector_complex_float * Y)
-- Function: int gsl_blas_zaxpy (const gsl_complex ALPHA, const
gsl_vector_complex * X, gsl_vector_complex * Y)
These functions compute the sum y = \alpha x + y for the vectors X
and Y.
-- Function: void gsl_blas_sscal (float ALPHA, gsl_vector_float * X)
-- Function: void gsl_blas_dscal (double ALPHA, gsl_vector * X)
-- Function: void gsl_blas_cscal (const gsl_complex_float ALPHA,
gsl_vector_complex_float * X)
-- Function: void gsl_blas_zscal (const gsl_complex ALPHA,
gsl_vector_complex * X)
-- Function: void gsl_blas_csscal (float ALPHA,
gsl_vector_complex_float * X)
-- Function: void gsl_blas_zdscal (double ALPHA, gsl_vector_complex *
X)
These functions rescale the vector X by the multiplicative factor
ALPHA.
-- Function: int gsl_blas_srotg (float A[], float B[], float C[], float
S[])
-- Function: int gsl_blas_drotg (double A[], double B[], double C[],
double S[])
These functions compute a Givens rotation (c,s) which zeroes the
vector (a,b),
[ c s ] [ a ] = [ r ]
[ -s c ] [ b ] [ 0 ]
The variables A and B are overwritten by the routine.
-- Function: int gsl_blas_srot (gsl_vector_float * X, gsl_vector_float
* Y, float C, float S)
-- Function: int gsl_blas_drot (gsl_vector * X, gsl_vector * Y, const
double C, const double S)
These functions apply a Givens rotation (x', y') = (c x + s y, -s x
+ c y) to the vectors X, Y.
-- Function: int gsl_blas_srotmg (float D1[], float D2[], float B1[],
float B2, float P[])
-- Function: int gsl_blas_drotmg (double D1[], double D2[], double
B1[], double B2, double P[])
These functions compute a modified Givens transformation. The
modified Givens transformation is defined in the original Level-1
BLAS specification, given in the references.
-- Function: int gsl_blas_srotm (gsl_vector_float * X, gsl_vector_float
* Y, const float P[])
-- Function: int gsl_blas_drotm (gsl_vector * X, gsl_vector * Y, const
double P[])
These functions apply a modified Givens transformation.
13.1.2 Level 2
--------------
-- Function: int gsl_blas_sgemv (CBLAS_TRANSPOSE_t TRANSA, float ALPHA,
const gsl_matrix_float * A, const gsl_vector_float * X, float
BETA, gsl_vector_float * Y)
-- Function: int gsl_blas_dgemv (CBLAS_TRANSPOSE_t TRANSA, double
ALPHA, const gsl_matrix * A, const gsl_vector * X, double
BETA, gsl_vector * Y)
-- Function: int gsl_blas_cgemv (CBLAS_TRANSPOSE_t TRANSA, const
gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
const gsl_vector_complex_float * X, const gsl_complex_float
BETA, gsl_vector_complex_float * Y)
-- Function: int gsl_blas_zgemv (CBLAS_TRANSPOSE_t TRANSA, const
gsl_complex ALPHA, const gsl_matrix_complex * A, const
gsl_vector_complex * X, const gsl_complex BETA,
gsl_vector_complex * Y)
These functions compute the matrix-vector product and sum y =
\alpha op(A) x + \beta y, where op(A) = A, A^T, A^H for TRANSA =
'CblasNoTrans', 'CblasTrans', 'CblasConjTrans'.
-- Function: int gsl_blas_strmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_float * A,
gsl_vector_float * X)
-- Function: int gsl_blas_dtrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix * A, gsl_vector *
X)
-- Function: int gsl_blas_ctrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex_float * A,
gsl_vector_complex_float * X)
-- Function: int gsl_blas_ztrmv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex * A,
gsl_vector_complex * X)
These functions compute the matrix-vector product x = op(A) x for
the triangular matrix A, where op(A) = A, A^T, A^H for TRANSA =
'CblasNoTrans', 'CblasTrans', 'CblasConjTrans'. When UPLO is
'CblasUpper' then the upper triangle of A is used, and when UPLO is
'CblasLower' then the lower triangle of A is used. If DIAG is
'CblasNonUnit' then the diagonal of the matrix is used, but if DIAG
is 'CblasUnit' then the diagonal elements of the matrix A are taken
as unity and are not referenced.
-- Function: int gsl_blas_strsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_float * A,
gsl_vector_float * X)
-- Function: int gsl_blas_dtrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix * A, gsl_vector *
X)
-- Function: int gsl_blas_ctrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex_float * A,
gsl_vector_complex_float * X)
-- Function: int gsl_blas_ztrsv (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANSA, CBLAS_DIAG_t DIAG, const gsl_matrix_complex * A,
gsl_vector_complex * X)
These functions compute inv(op(A)) x for X, where op(A) = A, A^T,
A^H for TRANSA = 'CblasNoTrans', 'CblasTrans', 'CblasConjTrans'.
When UPLO is 'CblasUpper' then the upper triangle of A is used, and
when UPLO is 'CblasLower' then the lower triangle of A is used. If
DIAG is 'CblasNonUnit' then the diagonal of the matrix is used, but
if DIAG is 'CblasUnit' then the diagonal elements of the matrix A
are taken as unity and are not referenced.
-- Function: int gsl_blas_ssymv (CBLAS_UPLO_t UPLO, float ALPHA, const
gsl_matrix_float * A, const gsl_vector_float * X, float BETA,
gsl_vector_float * Y)
-- Function: int gsl_blas_dsymv (CBLAS_UPLO_t UPLO, double ALPHA, const
gsl_matrix * A, const gsl_vector * X, double BETA, gsl_vector
* Y)
These functions compute the matrix-vector product and sum y =
\alpha A x + \beta y for the symmetric matrix A. Since the matrix
A is symmetric only its upper half or lower half need to be stored.
When UPLO is 'CblasUpper' then the upper triangle and diagonal of A
are used, and when UPLO is 'CblasLower' then the lower triangle and
diagonal of A are used.
-- Function: int gsl_blas_chemv (CBLAS_UPLO_t UPLO, const
gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
const gsl_vector_complex_float * X, const gsl_complex_float
BETA, gsl_vector_complex_float * Y)
-- Function: int gsl_blas_zhemv (CBLAS_UPLO_t UPLO, const gsl_complex
ALPHA, const gsl_matrix_complex * A, const gsl_vector_complex
* X, const gsl_complex BETA, gsl_vector_complex * Y)
These functions compute the matrix-vector product and sum y =
\alpha A x + \beta y for the hermitian matrix A. Since the matrix
A is hermitian only its upper half or lower half need to be stored.
When UPLO is 'CblasUpper' then the upper triangle and diagonal of A
are used, and when UPLO is 'CblasLower' then the lower triangle and
diagonal of A are used. The imaginary elements of the diagonal are
automatically assumed to be zero and are not referenced.
-- Function: int gsl_blas_sger (float ALPHA, const gsl_vector_float *
X, const gsl_vector_float * Y, gsl_matrix_float * A)
-- Function: int gsl_blas_dger (double ALPHA, const gsl_vector * X,
const gsl_vector * Y, gsl_matrix * A)
-- Function: int gsl_blas_cgeru (const gsl_complex_float ALPHA, const
gsl_vector_complex_float * X, const gsl_vector_complex_float *
Y, gsl_matrix_complex_float * A)
-- Function: int gsl_blas_zgeru (const gsl_complex ALPHA, const
gsl_vector_complex * X, const gsl_vector_complex * Y,
gsl_matrix_complex * A)
These functions compute the rank-1 update A = \alpha x y^T + A of
the matrix A.
-- Function: int gsl_blas_cgerc (const gsl_complex_float ALPHA, const
gsl_vector_complex_float * X, const gsl_vector_complex_float *
Y, gsl_matrix_complex_float * A)
-- Function: int gsl_blas_zgerc (const gsl_complex ALPHA, const
gsl_vector_complex * X, const gsl_vector_complex * Y,
gsl_matrix_complex * A)
These functions compute the conjugate rank-1 update A = \alpha x
y^H + A of the matrix A.
-- Function: int gsl_blas_ssyr (CBLAS_UPLO_t UPLO, float ALPHA, const
gsl_vector_float * X, gsl_matrix_float * A)
-- Function: int gsl_blas_dsyr (CBLAS_UPLO_t UPLO, double ALPHA, const
gsl_vector * X, gsl_matrix * A)
These functions compute the symmetric rank-1 update A = \alpha x
x^T + A of the symmetric matrix A. Since the matrix A is symmetric
only its upper half or lower half need to be stored. When UPLO is
'CblasUpper' then the upper triangle and diagonal of A are used,
and when UPLO is 'CblasLower' then the lower triangle and diagonal
of A are used.
-- Function: int gsl_blas_cher (CBLAS_UPLO_t UPLO, float ALPHA, const
gsl_vector_complex_float * X, gsl_matrix_complex_float * A)
-- Function: int gsl_blas_zher (CBLAS_UPLO_t UPLO, double ALPHA, const
gsl_vector_complex * X, gsl_matrix_complex * A)
These functions compute the hermitian rank-1 update A = \alpha x
x^H + A of the hermitian matrix A. Since the matrix A is hermitian
only its upper half or lower half need to be stored. When UPLO is
'CblasUpper' then the upper triangle and diagonal of A are used,
and when UPLO is 'CblasLower' then the lower triangle and diagonal
of A are used. The imaginary elements of the diagonal are
automatically set to zero.
-- Function: int gsl_blas_ssyr2 (CBLAS_UPLO_t UPLO, float ALPHA, const
gsl_vector_float * X, const gsl_vector_float * Y,
gsl_matrix_float * A)
-- Function: int gsl_blas_dsyr2 (CBLAS_UPLO_t UPLO, double ALPHA, const
gsl_vector * X, const gsl_vector * Y, gsl_matrix * A)
These functions compute the symmetric rank-2 update A = \alpha x
y^T + \alpha y x^T + A of the symmetric matrix A. Since the matrix
A is symmetric only its upper half or lower half need to be stored.
When UPLO is 'CblasUpper' then the upper triangle and diagonal of A
are used, and when UPLO is 'CblasLower' then the lower triangle and
diagonal of A are used.
-- Function: int gsl_blas_cher2 (CBLAS_UPLO_t UPLO, const
gsl_complex_float ALPHA, const gsl_vector_complex_float * X,
const gsl_vector_complex_float * Y, gsl_matrix_complex_float *
A)
-- Function: int gsl_blas_zher2 (CBLAS_UPLO_t UPLO, const gsl_complex
ALPHA, const gsl_vector_complex * X, const gsl_vector_complex
* Y, gsl_matrix_complex * A)
These functions compute the hermitian rank-2 update A = \alpha x
y^H + \alpha^* y x^H + A of the hermitian matrix A. Since the
matrix A is hermitian only its upper half or lower half need to be
stored. When UPLO is 'CblasUpper' then the upper triangle and
diagonal of A are used, and when UPLO is 'CblasLower' then the
lower triangle and diagonal of A are used. The imaginary elements
of the diagonal are automatically set to zero.
13.1.3 Level 3
--------------
-- Function: int gsl_blas_sgemm (CBLAS_TRANSPOSE_t TRANSA,
CBLAS_TRANSPOSE_t TRANSB, float ALPHA, const gsl_matrix_float
* A, const gsl_matrix_float * B, float BETA, gsl_matrix_float
* C)
-- Function: int gsl_blas_dgemm (CBLAS_TRANSPOSE_t TRANSA,
CBLAS_TRANSPOSE_t TRANSB, double ALPHA, const gsl_matrix * A,
const gsl_matrix * B, double BETA, gsl_matrix * C)
-- Function: int gsl_blas_cgemm (CBLAS_TRANSPOSE_t TRANSA,
CBLAS_TRANSPOSE_t TRANSB, const gsl_complex_float ALPHA, const
gsl_matrix_complex_float * A, const gsl_matrix_complex_float *
B, const gsl_complex_float BETA, gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zgemm (CBLAS_TRANSPOSE_t TRANSA,
CBLAS_TRANSPOSE_t TRANSB, const gsl_complex ALPHA, const
gsl_matrix_complex * A, const gsl_matrix_complex * B, const
gsl_complex BETA, gsl_matrix_complex * C)
These functions compute the matrix-matrix product and sum C =
\alpha op(A) op(B) + \beta C where op(A) = A, A^T, A^H for TRANSA =
'CblasNoTrans', 'CblasTrans', 'CblasConjTrans' and similarly for
the parameter TRANSB.
-- Function: int gsl_blas_ssymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
float ALPHA, const gsl_matrix_float * A, const
gsl_matrix_float * B, float BETA, gsl_matrix_float * C)
-- Function: int gsl_blas_dsymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
double ALPHA, const gsl_matrix * A, const gsl_matrix * B,
double BETA, gsl_matrix * C)
-- Function: int gsl_blas_csymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
const gsl_complex_float ALPHA, const gsl_matrix_complex_float
* A, const gsl_matrix_complex_float * B, const
gsl_complex_float BETA, gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zsymm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
const gsl_complex ALPHA, const gsl_matrix_complex * A, const
gsl_matrix_complex * B, const gsl_complex BETA,
gsl_matrix_complex * C)
These functions compute the matrix-matrix product and sum C =
\alpha A B + \beta C for SIDE is 'CblasLeft' and C = \alpha B A +
\beta C for SIDE is 'CblasRight', where the matrix A is symmetric.
When UPLO is 'CblasUpper' then the upper triangle and diagonal of A
are used, and when UPLO is 'CblasLower' then the lower triangle and
diagonal of A are used.
-- Function: int gsl_blas_chemm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
const gsl_complex_float ALPHA, const gsl_matrix_complex_float
* A, const gsl_matrix_complex_float * B, const
gsl_complex_float BETA, gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zhemm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
const gsl_complex ALPHA, const gsl_matrix_complex * A, const
gsl_matrix_complex * B, const gsl_complex BETA,
gsl_matrix_complex * C)
These functions compute the matrix-matrix product and sum C =
\alpha A B + \beta C for SIDE is 'CblasLeft' and C = \alpha B A +
\beta C for SIDE is 'CblasRight', where the matrix A is hermitian.
When UPLO is 'CblasUpper' then the upper triangle and diagonal of A
are used, and when UPLO is 'CblasLower' then the lower triangle and
diagonal of A are used. The imaginary elements of the diagonal are
automatically set to zero.
-- Function: int gsl_blas_strmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, float ALPHA,
const gsl_matrix_float * A, gsl_matrix_float * B)
-- Function: int gsl_blas_dtrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, double ALPHA,
const gsl_matrix * A, gsl_matrix * B)
-- Function: int gsl_blas_ctrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
gsl_matrix_complex_float * B)
-- Function: int gsl_blas_ztrmm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const gsl_complex
ALPHA, const gsl_matrix_complex * A, gsl_matrix_complex * B)
These functions compute the matrix-matrix product B = \alpha op(A)
B for SIDE is 'CblasLeft' and B = \alpha B op(A) for SIDE is
'CblasRight'. The matrix A is triangular and op(A) = A, A^T, A^H
for TRANSA = 'CblasNoTrans', 'CblasTrans', 'CblasConjTrans'. When
UPLO is 'CblasUpper' then the upper triangle of A is used, and when
UPLO is 'CblasLower' then the lower triangle of A is used. If DIAG
is 'CblasNonUnit' then the diagonal of A is used, but if DIAG is
'CblasUnit' then the diagonal elements of the matrix A are taken as
unity and are not referenced.
-- Function: int gsl_blas_strsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, float ALPHA,
const gsl_matrix_float * A, gsl_matrix_float * B)
-- Function: int gsl_blas_dtrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, double ALPHA,
const gsl_matrix * A, gsl_matrix * B)
-- Function: int gsl_blas_ctrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const
gsl_complex_float ALPHA, const gsl_matrix_complex_float * A,
gsl_matrix_complex_float * B)
-- Function: int gsl_blas_ztrsm (CBLAS_SIDE_t SIDE, CBLAS_UPLO_t UPLO,
CBLAS_TRANSPOSE_t TRANSA, CBLAS_DIAG_t DIAG, const gsl_complex
ALPHA, const gsl_matrix_complex * A, gsl_matrix_complex * B)
These functions compute the inverse-matrix matrix product B =
\alpha op(inv(A))B for SIDE is 'CblasLeft' and B = \alpha B
op(inv(A)) for SIDE is 'CblasRight'. The matrix A is triangular
and op(A) = A, A^T, A^H for TRANSA = 'CblasNoTrans', 'CblasTrans',
'CblasConjTrans'. When UPLO is 'CblasUpper' then the upper
triangle of A is used, and when UPLO is 'CblasLower' then the lower
triangle of A is used. If DIAG is 'CblasNonUnit' then the diagonal
of A is used, but if DIAG is 'CblasUnit' then the diagonal elements
of the matrix A are taken as unity and are not referenced.
-- Function: int gsl_blas_ssyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, float ALPHA, const gsl_matrix_float * A, float BETA,
gsl_matrix_float * C)
-- Function: int gsl_blas_dsyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, double ALPHA, const gsl_matrix * A, double BETA,
gsl_matrix * C)
-- Function: int gsl_blas_csyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, const gsl_complex_float ALPHA, const
gsl_matrix_complex_float * A, const gsl_complex_float BETA,
gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zsyrk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
const gsl_complex BETA, gsl_matrix_complex * C)
These functions compute a rank-k update of the symmetric matrix C,
C = \alpha A A^T + \beta C when TRANS is 'CblasNoTrans' and C =
\alpha A^T A + \beta C when TRANS is 'CblasTrans'. Since the
matrix C is symmetric only its upper half or lower half need to be
stored. When UPLO is 'CblasUpper' then the upper triangle and
diagonal of C are used, and when UPLO is 'CblasLower' then the
lower triangle and diagonal of C are used.
-- Function: int gsl_blas_cherk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, float ALPHA, const gsl_matrix_complex_float * A, float
BETA, gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zherk (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, double ALPHA, const gsl_matrix_complex * A, double
BETA, gsl_matrix_complex * C)
These functions compute a rank-k update of the hermitian matrix C,
C = \alpha A A^H + \beta C when TRANS is 'CblasNoTrans' and C =
\alpha A^H A + \beta C when TRANS is 'CblasConjTrans'. Since the
matrix C is hermitian only its upper half or lower half need to be
stored. When UPLO is 'CblasUpper' then the upper triangle and
diagonal of C are used, and when UPLO is 'CblasLower' then the
lower triangle and diagonal of C are used. The imaginary elements
of the diagonal are automatically set to zero.
-- Function: int gsl_blas_ssyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, float ALPHA, const gsl_matrix_float * A, const
gsl_matrix_float * B, float BETA, gsl_matrix_float * C)
-- Function: int gsl_blas_dsyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, double ALPHA, const gsl_matrix * A, const gsl_matrix *
B, double BETA, gsl_matrix * C)
-- Function: int gsl_blas_csyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, const gsl_complex_float ALPHA, const
gsl_matrix_complex_float * A, const gsl_matrix_complex_float *
B, const gsl_complex_float BETA, gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zsyr2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
const gsl_matrix_complex * B, const gsl_complex BETA,
gsl_matrix_complex * C)
These functions compute a rank-2k update of the symmetric matrix C,
C = \alpha A B^T + \alpha B A^T + \beta C when TRANS is
'CblasNoTrans' and C = \alpha A^T B + \alpha B^T A + \beta C when
TRANS is 'CblasTrans'. Since the matrix C is symmetric only its
upper half or lower half need to be stored. When UPLO is
'CblasUpper' then the upper triangle and diagonal of C are used,
and when UPLO is 'CblasLower' then the lower triangle and diagonal
of C are used.
-- Function: int gsl_blas_cher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, const gsl_complex_float ALPHA, const
gsl_matrix_complex_float * A, const gsl_matrix_complex_float *
B, float BETA, gsl_matrix_complex_float * C)
-- Function: int gsl_blas_zher2k (CBLAS_UPLO_t UPLO, CBLAS_TRANSPOSE_t
TRANS, const gsl_complex ALPHA, const gsl_matrix_complex * A,
const gsl_matrix_complex * B, double BETA, gsl_matrix_complex
* C)
These functions compute a rank-2k update of the hermitian matrix C,
C = \alpha A B^H + \alpha^* B A^H + \beta C when TRANS is
'CblasNoTrans' and C = \alpha A^H B + \alpha^* B^H A + \beta C when
TRANS is 'CblasConjTrans'. Since the matrix C is hermitian only
its upper half or lower half need to be stored. When UPLO is
'CblasUpper' then the upper triangle and diagonal of C are used,
and when UPLO is 'CblasLower' then the lower triangle and diagonal
of C are used. The imaginary elements of the diagonal are
automatically set to zero.
13.2 Examples
=============
The following program computes the product of two matrices using the
Level-3 BLAS function DGEMM,
[ 0.11 0.12 0.13 ] [ 1011 1012 ] [ 367.76 368.12 ]
[ 0.21 0.22 0.23 ] [ 1021 1022 ] = [ 674.06 674.72 ]
[ 1031 1032 ]
The matrices are stored in row major order, according to the C
convention for arrays.
#include
#include
int
main (void)
{
double a[] = { 0.11, 0.12, 0.13,
0.21, 0.22, 0.23 };
double b[] = { 1011, 1012,
1021, 1022,
1031, 1032 };
double c[] = { 0.00, 0.00,
0.00, 0.00 };
gsl_matrix_view A = gsl_matrix_view_array(a, 2, 3);
gsl_matrix_view B = gsl_matrix_view_array(b, 3, 2);
gsl_matrix_view C = gsl_matrix_view_array(c, 2, 2);
/* Compute C = A B */
gsl_blas_dgemm (CblasNoTrans, CblasNoTrans,
1.0, &A.matrix, &B.matrix,
0.0, &C.matrix);
printf ("[ %g, %g\n", c[0], c[1]);
printf (" %g, %g ]\n", c[2], c[3]);
return 0;
}
Here is the output from the program,
$ ./a.out
[ 367.76, 368.12
674.06, 674.72 ]
13.3 References and Further Reading
===================================
Information on the BLAS standards, including both the legacy and updated
interface standards, is available online from the BLAS Homepage and BLAS
Technical Forum web-site.
'BLAS Homepage'
'BLAS Technical Forum'
The following papers contain the specifications for Level 1, Level 2 and
Level 3 BLAS.
C. Lawson, R. Hanson, D. Kincaid, F. Krogh, "Basic Linear Algebra
Subprograms for Fortran Usage", 'ACM Transactions on Mathematical
Software', Vol. 5 (1979), Pages 308-325.
J.J. Dongarra, J. DuCroz, S. Hammarling, R. Hanson, "An Extended
Set of Fortran Basic Linear Algebra Subprograms", 'ACM Transactions
on Mathematical Software', Vol. 14, No. 1 (1988), Pages 1-32.
J.J. Dongarra, I. Duff, J. DuCroz, S. Hammarling, "A Set of Level 3
Basic Linear Algebra Subprograms", 'ACM Transactions on
Mathematical Software', Vol. 16 (1990), Pages 1-28.
Postscript versions of the latter two papers are available from
. A CBLAS wrapper for Fortran BLAS
libraries is available from the same location.
14 Linear Algebra
*****************
This chapter describes functions for solving linear systems. The
library provides linear algebra operations which operate directly on the
'gsl_vector' and 'gsl_matrix' objects. These routines use the standard
algorithms from Golub & Van Loan's 'Matrix Computations' with Level-1
and Level-2 BLAS calls for efficiency.
The functions described in this chapter are declared in the header
file 'gsl_linalg.h'.
14.1 LU Decomposition
=====================
A general N-by-N square matrix A has an LU decomposition into upper and
lower triangular matrices,
P A = L U
where P is a permutation matrix, L is unit lower triangular matrix and U
is upper triangular matrix. For square matrices this decomposition can
be used to convert the linear system A x = b into a pair of triangular
systems (L y = P b, U x = y), which can be solved by forward and
back-substitution. Note that the LU decomposition is valid for singular
matrices.
-- Function: int gsl_linalg_LU_decomp (gsl_matrix * A, gsl_permutation
* P, int * SIGNUM)
-- Function: int gsl_linalg_complex_LU_decomp (gsl_matrix_complex * A,
gsl_permutation * P, int * SIGNUM)
These functions factorize the square matrix A into the LU
decomposition PA = LU. On output the diagonal and upper triangular
part of the input matrix A contain the matrix U. The lower
triangular part of the input matrix (excluding the diagonal)
contains L. The diagonal elements of L are unity, and are not
stored.
The permutation matrix P is encoded in the permutation P on output.
The j-th column of the matrix P is given by the k-th column of the
identity matrix, where k = p_j the j-th element of the permutation
vector. The sign of the permutation is given by SIGNUM. It has
the value (-1)^n, where n is the number of interchanges in the
permutation.
The algorithm used in the decomposition is Gaussian Elimination
with partial pivoting (Golub & Van Loan, 'Matrix Computations',
Algorithm 3.4.1).
-- Function: int gsl_linalg_LU_solve (const gsl_matrix * LU, const
gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
-- Function: int gsl_linalg_complex_LU_solve (const gsl_matrix_complex
* LU, const gsl_permutation * P, const gsl_vector_complex * B,
gsl_vector_complex * X)
These functions solve the square system A x = b using the LU
decomposition of A into (LU, P) given by 'gsl_linalg_LU_decomp' or
'gsl_linalg_complex_LU_decomp' as input.
-- Function: int gsl_linalg_LU_svx (const gsl_matrix * LU, const
gsl_permutation * P, gsl_vector * X)
-- Function: int gsl_linalg_complex_LU_svx (const gsl_matrix_complex *
LU, const gsl_permutation * P, gsl_vector_complex * X)
These functions solve the square system A x = b in-place using the
precomputed LU decomposition of A into (LU,P). On input X should
contain the right-hand side b, which is replaced by the solution on
output.
-- Function: int gsl_linalg_LU_refine (const gsl_matrix * A, const
gsl_matrix * LU, const gsl_permutation * P, const gsl_vector *
B, gsl_vector * X, gsl_vector * WORK)
-- Function: int gsl_linalg_complex_LU_refine (const gsl_matrix_complex
* A, const gsl_matrix_complex * LU, const gsl_permutation * P,
const gsl_vector_complex * B, gsl_vector_complex * X,
gsl_vector_complex * WORK)
These functions apply an iterative improvement to X, the solution
of A x = b, from the precomputed LU decomposition of A into (LU,P).
Additional workspace of length N is required in WORK.
-- Function: int gsl_linalg_LU_invert (const gsl_matrix * LU, const
gsl_permutation * P, gsl_matrix * INVERSE)
-- Function: int gsl_linalg_complex_LU_invert (const gsl_matrix_complex
* LU, const gsl_permutation * P, gsl_matrix_complex * INVERSE)
These functions compute the inverse of a matrix A from its LU
decomposition (LU,P), storing the result in the matrix INVERSE.
The inverse is computed by solving the system A x = b for each
column of the identity matrix. It is preferable to avoid direct
use of the inverse whenever possible, as the linear solver
functions can obtain the same result more efficiently and reliably
(consult any introductory textbook on numerical linear algebra for
details).
-- Function: double gsl_linalg_LU_det (gsl_matrix * LU, int SIGNUM)
-- Function: gsl_complex gsl_linalg_complex_LU_det (gsl_matrix_complex
* LU, int SIGNUM)
These functions compute the determinant of a matrix A from its LU
decomposition, LU. The determinant is computed as the product of
the diagonal elements of U and the sign of the row permutation
SIGNUM.
-- Function: double gsl_linalg_LU_lndet (gsl_matrix * LU)
-- Function: double gsl_linalg_complex_LU_lndet (gsl_matrix_complex *
LU)
These functions compute the logarithm of the absolute value of the
determinant of a matrix A, \ln|\det(A)|, from its LU decomposition,
LU. This function may be useful if the direct computation of the
determinant would overflow or underflow.
-- Function: int gsl_linalg_LU_sgndet (gsl_matrix * LU, int SIGNUM)
-- Function: gsl_complex gsl_linalg_complex_LU_sgndet
(gsl_matrix_complex * LU, int SIGNUM)
These functions compute the sign or phase factor of the determinant
of a matrix A, \det(A)/|\det(A)|, from its LU decomposition, LU.
14.2 QR Decomposition
=====================
A general rectangular M-by-N matrix A has a QR decomposition into the
product of an orthogonal M-by-M square matrix Q (where Q^T Q = I) and an
M-by-N right-triangular matrix R,
A = Q R
This decomposition can be used to convert the linear system A x = b into
the triangular system R x = Q^T b, which can be solved by
back-substitution. Another use of the QR decomposition is to compute an
orthonormal basis for a set of vectors. The first N columns of Q form
an orthonormal basis for the range of A, ran(A), when A has full column
rank.
-- Function: int gsl_linalg_QR_decomp (gsl_matrix * A, gsl_vector *
TAU)
This function factorizes the M-by-N matrix A into the QR
decomposition A = Q R. On output the diagonal and upper triangular
part of the input matrix contain the matrix R. The vector TAU and
the columns of the lower triangular part of the matrix A contain
the Householder coefficients and Householder vectors which encode
the orthogonal matrix Q. The vector TAU must be of length
k=\min(M,N). The matrix Q is related to these components by, Q =
Q_k ... Q_2 Q_1 where Q_i = I - \tau_i v_i v_i^T and v_i is the
Householder vector v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)).
This is the same storage scheme as used by LAPACK.
The algorithm used to perform the decomposition is Householder QR
(Golub & Van Loan, 'Matrix Computations', Algorithm 5.2.1).
-- Function: int gsl_linalg_QR_solve (const gsl_matrix * QR, const
gsl_vector * TAU, const gsl_vector * B, gsl_vector * X)
This function solves the square system A x = b using the QR
decomposition of A held in (QR, TAU) which must have been computed
previously with 'gsl_linalg_QR_decomp'. The least-squares solution
for rectangular systems can be found using 'gsl_linalg_QR_lssolve'.
-- Function: int gsl_linalg_QR_svx (const gsl_matrix * QR, const
gsl_vector * TAU, gsl_vector * X)
This function solves the square system A x = b in-place using the
QR decomposition of A held in (QR,TAU) which must have been
computed previously by 'gsl_linalg_QR_decomp'. On input X should
contain the right-hand side b, which is replaced by the solution on
output.
-- Function: int gsl_linalg_QR_lssolve (const gsl_matrix * QR, const
gsl_vector * TAU, const gsl_vector * B, gsl_vector * X,
gsl_vector * RESIDUAL)
This function finds the least squares solution to the
overdetermined system A x = b where the matrix A has more rows than
columns. The least squares solution minimizes the Euclidean norm
of the residual, ||Ax - b||.The routine requires as input the QR
decomposition of A into (QR, TAU) given by 'gsl_linalg_QR_decomp'.
The solution is returned in X. The residual is computed as a
by-product and stored in RESIDUAL.
-- Function: int gsl_linalg_QR_QTvec (const gsl_matrix * QR, const
gsl_vector * TAU, gsl_vector * V)
This function applies the matrix Q^T encoded in the decomposition
(QR,TAU) to the vector V, storing the result Q^T v in V. The
matrix multiplication is carried out directly using the encoding of
the Householder vectors without needing to form the full matrix
Q^T.
-- Function: int gsl_linalg_QR_Qvec (const gsl_matrix * QR, const
gsl_vector * TAU, gsl_vector * V)
This function applies the matrix Q encoded in the decomposition
(QR,TAU) to the vector V, storing the result Q v in V. The matrix
multiplication is carried out directly using the encoding of the
Householder vectors without needing to form the full matrix Q.
-- Function: int gsl_linalg_QR_QTmat (const gsl_matrix * QR, const
gsl_vector * TAU, gsl_matrix * A)
This function applies the matrix Q^T encoded in the decomposition
(QR,TAU) to the matrix A, storing the result Q^T A in A. The
matrix multiplication is carried out directly using the encoding of
the Householder vectors without needing to form the full matrix
Q^T.
-- Function: int gsl_linalg_QR_Rsolve (const gsl_matrix * QR, const
gsl_vector * B, gsl_vector * X)
This function solves the triangular system R x = b for X. It may
be useful if the product b' = Q^T b has already been computed using
'gsl_linalg_QR_QTvec'.
-- Function: int gsl_linalg_QR_Rsvx (const gsl_matrix * QR, gsl_vector
* X)
This function solves the triangular system R x = b for X in-place.
On input X should contain the right-hand side b and is replaced by
the solution on output. This function may be useful if the product
b' = Q^T b has already been computed using 'gsl_linalg_QR_QTvec'.
-- Function: int gsl_linalg_QR_unpack (const gsl_matrix * QR, const
gsl_vector * TAU, gsl_matrix * Q, gsl_matrix * R)
This function unpacks the encoded QR decomposition (QR,TAU) into
the matrices Q and R, where Q is M-by-M and R is M-by-N.
-- Function: int gsl_linalg_QR_QRsolve (gsl_matrix * Q, gsl_matrix * R,
const gsl_vector * B, gsl_vector * X)
This function solves the system R x = Q^T b for X. It can be used
when the QR decomposition of a matrix is available in unpacked form
as (Q, R).
-- Function: int gsl_linalg_QR_update (gsl_matrix * Q, gsl_matrix * R,
gsl_vector * W, const gsl_vector * V)
This function performs a rank-1 update w v^T of the QR
decomposition (Q, R). The update is given by Q'R' = Q (R + w v^T)
where the output matrices Q' and R' are also orthogonal and right
triangular. Note that W is destroyed by the update.
-- Function: int gsl_linalg_R_solve (const gsl_matrix * R, const
gsl_vector * B, gsl_vector * X)
This function solves the triangular system R x = b for the N-by-N
matrix R.
-- Function: int gsl_linalg_R_svx (const gsl_matrix * R, gsl_vector *
X)
This function solves the triangular system R x = b in-place. On
input X should contain the right-hand side b, which is replaced by
the solution on output.
14.3 QR Decomposition with Column Pivoting
==========================================
The QR decomposition of an M-by-N matrix A can be extended to the rank
deficient case by introducing a column permutation P,
A P = Q R
The first r columns of Q form an orthonormal basis for the range of A
for a matrix with column rank r. This decomposition can also be used to
convert the linear system A x = b into the triangular system R y = Q^T
b, x = P y, which can be solved by back-substitution and permutation.
We denote the QR decomposition with column pivoting by QRP^T since A = Q
R P^T. When A is rank deficient with r = {\rm rank}(A), the matrix R
can be partitioned as
R = [ R11 R12; 0 R22 ] =~ [ R11 R12; 0 0 ]
where R_{11} is r-by-r and nonsingular. In this case, a "basic"
least squares solution for the overdetermined system A x = b can be
obtained as
x = P [ R11^-1 c1 ; 0 ]
where c_1 consists of the first r elements of Q^T b. This basic
solution is not guaranteed to be the minimum norm solution unless R_{12}
= 0 (see *note Complete Orthogonal Decomposition::).
-- Function: int gsl_linalg_QRPT_decomp (gsl_matrix * A, gsl_vector *
TAU, gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
This function factorizes the M-by-N matrix A into the QRP^T
decomposition A = Q R P^T. On output the diagonal and upper
triangular part of the input matrix contain the matrix R. The
permutation matrix P is stored in the permutation P. The sign of
the permutation is given by SIGNUM. It has the value (-1)^n, where
n is the number of interchanges in the permutation. The vector TAU
and the columns of the lower triangular part of the matrix A
contain the Householder coefficients and vectors which encode the
orthogonal matrix Q. The vector TAU must be of length k=\min(M,N).
The matrix Q is related to these components by, Q = Q_k ... Q_2 Q_1
where Q_i = I - \tau_i v_i v_i^T and v_i is the Householder vector
v_i = (0,...,1,A(i+1,i),A(i+2,i),...,A(m,i)). This is the same
storage scheme as used by LAPACK. The vector NORM is a workspace
of length N used for column pivoting.
The algorithm used to perform the decomposition is Householder QR
with column pivoting (Golub & Van Loan, 'Matrix Computations',
Algorithm 5.4.1).
-- Function: int gsl_linalg_QRPT_decomp2 (const gsl_matrix * A,
gsl_matrix * Q, gsl_matrix * R, gsl_vector * TAU,
gsl_permutation * P, int * SIGNUM, gsl_vector * NORM)
This function factorizes the matrix A into the decomposition A = Q
R P^T without modifying A itself and storing the output in the
separate matrices Q and R.
-- Function: int gsl_linalg_QRPT_solve (const gsl_matrix * QR, const
gsl_vector * TAU, const gsl_permutation * P, const gsl_vector
* B, gsl_vector * X)
This function solves the square system A x = b using the QRP^T
decomposition of A held in (QR, TAU, P) which must have been
computed previously by 'gsl_linalg_QRPT_decomp'.
-- Function: int gsl_linalg_QRPT_svx (const gsl_matrix * QR, const
gsl_vector * TAU, const gsl_permutation * P, gsl_vector * X)
This function solves the square system A x = b in-place using the
QRP^T decomposition of A held in (QR,TAU,P). On input X should
contain the right-hand side b, which is replaced by the solution on
output.
-- Function: int gsl_linalg_QRPT_lssolve (const gsl_matrix * QR, const
gsl_vector * TAU, const gsl_permutation * P, const gsl_vector
* B, gsl_vector * X, gsl_vector * RESIDUAL)
This function finds the least squares solution to the
overdetermined system A x = b where the matrix A has more rows than
columns and is assumed to have full rank. The least squares
solution minimizes the Euclidean norm of the residual, ||b - A x||.
The routine requires as input the QR decomposition of A into (QR,
TAU, P) given by 'gsl_linalg_QRPT_decomp'. The solution is
returned in X. The residual is computed as a by-product and stored
in RESIDUAL. For rank deficient matrices,
'gsl_linalg_QRPT_lssolve2' should be used instead.
-- Function: int gsl_linalg_QRPT_lssolve2 (const gsl_matrix * QR, const
gsl_vector * TAU, const gsl_permutation * P, const gsl_vector
* B, const size_t RANK, gsl_vector * X, gsl_vector * RESIDUAL)
This function finds the least squares solution to the
overdetermined system A x = b where the matrix A has more rows than
columns and has rank given by the input RANK. If the user does not
know the rank of A, the routine 'gsl_linalg_QRPT_rank' can be
called to estimate it. The least squares solution is the so-called
"basic" solution discussed above and may not be the minimum norm
solution. The routine requires as input the QR decomposition of A
into (QR, TAU, P) given by 'gsl_linalg_QRPT_decomp'. The solution
is returned in X. The residual is computed as a by-product and
stored in RESIDUAL.
-- Function: int gsl_linalg_QRPT_QRsolve (const gsl_matrix * Q, const
gsl_matrix * R, const gsl_permutation * P, const gsl_vector *
B, gsl_vector * X)
This function solves the square system R P^T x = Q^T b for X. It
can be used when the QR decomposition of a matrix is available in
unpacked form as (Q, R).
-- Function: int gsl_linalg_QRPT_update (gsl_matrix * Q, gsl_matrix *
R, const gsl_permutation * P, gsl_vector * W, const gsl_vector
* V)
This function performs a rank-1 update w v^T of the QRP^T
decomposition (Q, R, P). The update is given by Q'R' = Q (R + w
v^T P) where the output matrices Q' and R' are also orthogonal and
right triangular. Note that W is destroyed by the update. The
permutation P is not changed.
-- Function: int gsl_linalg_QRPT_Rsolve (const gsl_matrix * QR, const
gsl_permutation * P, const gsl_vector * B, gsl_vector * X)
This function solves the triangular system R P^T x = b for the
N-by-N matrix R contained in QR.
-- Function: int gsl_linalg_QRPT_Rsvx (const gsl_matrix * QR, const
gsl_permutation * P, gsl_vector * X)
This function solves the triangular system R P^T x = b in-place for
the N-by-N matrix R contained in QR. On input X should contain the
right-hand side b, which is replaced by the solution on output.
-- Function: size_t gsl_linalg_QRPT_rank (const gsl_matrix * QR, const
double TOL)
This function estimates the rank of the triangular matrix R
contained in QR. The algorithm simply counts the number of
diagonal elements of R whose absolute value is greater than the
specified tolerance TOL. If the input TOL is negative, a default
value of 20 (M + N) eps(max(|diag(R)|)) is used.
-- Function: int gsl_linalg_QRPT_rcond (const gsl_matrix * QR, double *
RCOND, gsl_vector * WORK)
This function estimates the reciprocal condition number (using the
1-norm) of the R factor, stored in the upper triangle of QR. The
reciprocal condition number estimate, defined as 1 / (||R||_1 \cdot
||R^{-1}||_1), is stored in RCOND. Additional workspace of size 3
N is required in WORK.
14.4 Complete Orthogonal Decomposition
======================================
The complete orthogonal decomposition of a M-by-N matrix A is a
generalization of the QR decomposition with column pivoting, given by
A P = Q [ R11 0 ] Z
[ 0 0 ]
where P is a N-by-N permutation matrix, Q is M-by-M orthogonal, R_{11}
is r-by-r upper triangular, with r = {\rm rank}(A), and Z is N-by-N
orthogonal. If A has full rank, then R_{11} = R, Z = I and this reduces
to the QR decomposition with column pivoting. The advantage of using
the complete orthogonal decomposition for rank deficient matrices is the
ability to compute the minimum norm solution to the linear least squares
problem Ax = b, which is given by
x = P Z^T [ R11^-1 c1 ]
[ 0 ]
and the vector c_1 is the first r elements of Q^T b.
-- Function: int gsl_linalg_COD_decomp (gsl_matrix * A, gsl_vector *
TAU_Q, gsl_vector * TAU_Z, gsl_permutation * P, size_t * RANK,
gsl_vector * WORK)
-- Function: int gsl_linalg_COD_decomp_e (gsl_matrix * A, gsl_vector *
TAU_Q, gsl_vector * TAU_Z, gsl_permutation * P, double TOL,
size_t * RANK, gsl_vector * WORK)
These functions factor the M-by-N matrix A into the decomposition A
= Q R Z P^T. The rank of A is computed as the number of diagonal
elements of R greater than the tolerance TOL and output in RANK.
If TOL is not specified, a default value is used (see
'gsl_linalg_QRPT_rank'). On output, the permutation matrix P is
stored in P. The matrix R_{11} is stored in the upper RANK-by-RANK
block of A. The matrices Q and Z are encoded in packed storage in
A on output. The vectors TAU_Q and TAU_Z contain the Householder
scalars corresponding to the matrices Q and Z respectively and must
be of length k = \min(M,N). The vector WORK is additional
workspace of length N.
-- Function: int gsl_linalg_COD_lssolve (const gsl_matrix * QRZ, const
gsl_vector * TAU_Q, const gsl_vector * TAU_Z, const
gsl_permutation * P, const size_t RANK, const gsl_vector * B,
gsl_vector * X, gsl_vector * RESIDUAL)
This function finds the least squares solution to the
overdetermined system A x = b where the matrix A has more rows than
columns. The least squares solution minimizes the Euclidean norm
of the residual, ||b - A x||. The routine requires as input the
QRZ decomposition of A into (QRZ, TAU_Q, TAU_Z, P, RANK) given by
'gsl_linalg_COD_decomp'. The solution is returned in X. The
residual is computed as a by-product and stored in RESIDUAL.
-- Function: int gsl_linalg_COD_unpack (const gsl_matrix * QRZ, const
gsl_vector * TAU_Q, const gsl_vector * TAU_Z, const size_t
RANK, gsl_matrix * Q, gsl_matrix * R, gsl_matrix * Z)
This function unpacks the encoded QRZ decomposition (QRZ, TAU_Q,
TAU_Z, RANK) into the matrices Q, R, and Z, where Q is M-by-M, R is
M-by-N, and Z is N-by-N.
-- Function: int gsl_linalg_COD_matZ (const gsl_matrix * QRZ, const
gsl_vector * TAU_Z, const size_t RANK, gsl_matrix * A,
gsl_vector * WORK)
This function multiplies the input matrix A on the right by Z, A' =
A Z using the encoded QRZ decomposition (QRZ, TAU_Z, RANK). A must
have N columns but may have any number of rows. Additional
workspace of length M is provided in WORK.
14.5 Singular Value Decomposition
=================================
A general rectangular M-by-N matrix A has a singular value decomposition
(SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N
diagonal matrix of singular values S and the transpose of an N-by-N
orthogonal square matrix V,
A = U S V^T
The singular values \sigma_i = S_{ii} are all non-negative and are
generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2
>= ... >= \sigma_N >= 0.
The singular value decomposition of a matrix has many practical uses.
The condition number of the matrix is given by the ratio of the largest
singular value to the smallest singular value. The presence of a zero
singular value indicates that the matrix is singular. The number of
non-zero singular values indicates the rank of the matrix. In practice
singular value decomposition of a rank-deficient matrix will not produce
exact zeroes for singular values, due to finite numerical precision.
Small singular values should be edited by choosing a suitable tolerance.
For a rank-deficient matrix, the null space of A is given by the
columns of V corresponding to the zero singular values. Similarly, the
range of A is given by columns of U corresponding to the non-zero
singular values.
Note that the routines here compute the "thin" version of the SVD
with U as M-by-N orthogonal matrix. This allows in-place computation
and is the most commonly-used form in practice. Mathematically, the
"full" SVD is defined with U as an M-by-M orthogonal matrix and S as an
M-by-N diagonal matrix (with additional rows of zeros).
-- Function: int gsl_linalg_SV_decomp (gsl_matrix * A, gsl_matrix * V,
gsl_vector * S, gsl_vector * WORK)
This function factorizes the M-by-N matrix A into the singular
value decomposition A = U S V^T for M >= N. On output the matrix A
is replaced by U. The diagonal elements of the singular value
matrix S are stored in the vector S. The singular values are
non-negative and form a non-increasing sequence from S_1 to S_N.
The matrix V contains the elements of V in untransposed form. To
form the product U S V^T it is necessary to take the transpose of
V. A workspace of length N is required in WORK.
This routine uses the Golub-Reinsch SVD algorithm.
-- Function: int gsl_linalg_SV_decomp_mod (gsl_matrix * A, gsl_matrix *
X, gsl_matrix * V, gsl_vector * S, gsl_vector * WORK)
This function computes the SVD using the modified Golub-Reinsch
algorithm, which is faster for M>>N. It requires the vector WORK
of length N and the N-by-N matrix X as additional working space.
-- Function: int gsl_linalg_SV_decomp_jacobi (gsl_matrix * A,
gsl_matrix * V, gsl_vector * S)
This function computes the SVD of the M-by-N matrix A using
one-sided Jacobi orthogonalization for M >= N. The Jacobi method
can compute singular values to higher relative accuracy than
Golub-Reinsch algorithms (see references for details).
-- Function: int gsl_linalg_SV_solve (const gsl_matrix * U, const
gsl_matrix * V, const gsl_vector * S, const gsl_vector * B,
gsl_vector * X)
This function solves the system A x = b using the singular value
decomposition (U, S, V) of A which must have been computed
previously with 'gsl_linalg_SV_decomp'.
Only non-zero singular values are used in computing the solution.
The parts of the solution corresponding to singular values of zero
are ignored. Other singular values can be edited out by setting
them to zero before calling this function.
In the over-determined case where A has more rows than columns the
system is solved in the least squares sense, returning the solution
X which minimizes ||A x - b||_2.
-- Function: int gsl_linalg_SV_leverage (const gsl_matrix * U,
gsl_vector * H)
This function computes the statistical leverage values h_i of a
matrix A using its singular value decomposition (U, S, V)
previously computed with 'gsl_linalg_SV_decomp'. h_i are the
diagonal values of the matrix A (A^T A)^{-1} A^T and depend only on
the matrix U which is the input to this function.
14.6 Cholesky Decomposition
===========================
A symmetric, positive definite square matrix A has a Cholesky
decomposition into a product of a lower triangular matrix L and its
transpose L^T,
A = L L^T
This is sometimes referred to as taking the square-root of a matrix.
The Cholesky decomposition can only be carried out when all the
eigenvalues of the matrix are positive. This decomposition can be used
to convert the linear system A x = b into a pair of triangular systems
(L y = b, L^T x = y), which can be solved by forward and
back-substitution.
If the matrix A is near singular, it is sometimes possible to reduce
the condition number and recover a more accurate solution vector x by
scaling as
( S A S ) ( S^(-1) x ) = S b
where S is a diagonal matrix whose elements are given by S_{ii} =
1/\sqrt{A_{ii}}. This scaling is also known as Jacobi preconditioning.
There are routines below to solve both the scaled and unscaled systems.
-- Function: int gsl_linalg_cholesky_decomp1 (gsl_matrix * A)
-- Function: int gsl_linalg_complex_cholesky_decomp (gsl_matrix_complex
* A)
These functions factorize the symmetric, positive-definite square
matrix A into the Cholesky decomposition A = L L^T (or A = L L^H
for the complex case). On input, the values from the diagonal and
lower-triangular part of the matrix A are used (the upper
triangular part is ignored). On output the diagonal and lower
triangular part of the input matrix A contain the matrix L, while
the upper triangular part is unmodified. If the matrix is not
positive-definite then the decomposition will fail, returning the
error code 'GSL_EDOM'.
When testing whether a matrix is positive-definite, disable the
error handler first to avoid triggering an error.
-- Function: int gsl_linalg_cholesky_decomp (gsl_matrix * A)
This function is now deprecated and is provided only for backward
compatibility.
-- Function: int gsl_linalg_cholesky_solve (const gsl_matrix *
CHOLESKY, const gsl_vector * B, gsl_vector * X)
-- Function: int gsl_linalg_complex_cholesky_solve (const
gsl_matrix_complex * CHOLESKY, const gsl_vector_complex * B,
gsl_vector_complex * X)
These functions solve the system A x = b using the Cholesky
decomposition of A held in the matrix CHOLESKY which must have been
previously computed by 'gsl_linalg_cholesky_decomp' or
'gsl_linalg_complex_cholesky_decomp'.
-- Function: int gsl_linalg_cholesky_svx (const gsl_matrix * CHOLESKY,
gsl_vector * X)
-- Function: int gsl_linalg_complex_cholesky_svx (const
gsl_matrix_complex * CHOLESKY, gsl_vector_complex * X)
These functions solve the system A x = b in-place using the
Cholesky decomposition of A held in the matrix CHOLESKY which must
have been previously computed by 'gsl_linalg_cholesky_decomp' or
'gsl_linalg_complex_cholesky_decomp'. On input X should contain
the right-hand side b, which is replaced by the solution on output.
-- Function: int gsl_linalg_cholesky_invert (gsl_matrix * CHOLESKY)
-- Function: int gsl_linalg_complex_cholesky_invert (gsl_matrix_complex
* CHOLESKY)
These functions compute the inverse of a matrix from its Cholesky
decomposition CHOLESKY, which must have been previously computed by
'gsl_linalg_cholesky_decomp' or
'gsl_linalg_complex_cholesky_decomp'. On output, the inverse is
stored in-place in CHOLESKY.
-- Function: int gsl_linalg_cholesky_decomp2 (gsl_matrix * A,
gsl_vector * S)
This function calculates a diagonal scaling transformation S for
the symmetric, positive-definite square matrix A, and then computes
the Cholesky decomposition S A S = L L^T. On input, the values
from the diagonal and lower-triangular part of the matrix A are
used (the upper triangular part is ignored). On output the
diagonal and lower triangular part of the input matrix A contain
the matrix L, while the upper triangular part of the input matrix
is overwritten with L^T (the diagonal terms being identical for
both L and L^T). If the matrix is not positive-definite then the
decomposition will fail, returning the error code 'GSL_EDOM'. The
diagonal scale factors are stored in S on output.
When testing whether a matrix is positive-definite, disable the
error handler first to avoid triggering an error.
-- Function: int gsl_linalg_cholesky_solve2 (const gsl_matrix *
CHOLESKY, const gsl_vector * S, const gsl_vector * B,
gsl_vector * X)
This function solves the system (S A S) (S^{-1} x) = S b using the
Cholesky decomposition of S A S held in the matrix CHOLESKY which
must have been previously computed by
'gsl_linalg_cholesky_decomp2'.
-- Function: int gsl_linalg_cholesky_svx2 (const gsl_matrix * CHOLESKY,
const gsl_vector * S, gsl_vector * X)
This function solves the system (S A S) (S^{-1} x) = S b in-place
using the Cholesky decomposition of S A S held in the matrix
CHOLESKY which must have been previously computed by
'gsl_linalg_cholesky_decomp2'. On input X should contain the
right-hand side b, which is replaced by the solution on output.
-- Function: int gsl_linalg_cholesky_scale (const gsl_matrix * A,
gsl_vector * S)
This function calculates a diagonal scaling transformation of the
symmetric, positive definite matrix A, such that S A S has a
condition number within a factor of N of the matrix of smallest
possible condition number over all possible diagonal scalings. On
output, S contains the scale factors, given by S_i =
1/\sqrt{A_{ii}}. For any A_{ii} \le 0, the corresponding scale
factor S_i is set to 1.
-- Function: int gsl_linalg_cholesky_scale_apply (gsl_matrix * A, const
gsl_vector * S)
This function applies the scaling transformation S to the matrix A.
On output, A is replaced by S A S.
-- Function: int gsl_linalg_cholesky_rcond (const gsl_matrix *
CHOLESKY, double * RCOND, gsl_vector * WORK)
This function estimates the reciprocal condition number (using the
1-norm) of the symmetric positive definite matrix A, using its
Cholesky decomposition provided in CHOLESKY. The reciprocal
condition number estimate, defined as 1 / (||A||_1 \cdot
||A^{-1}||_1), is stored in RCOND. Additional workspace of size 3
N is required in WORK.
14.7 Pivoted Cholesky Decomposition
===================================
A symmetric, positive definite square matrix A has an alternate Cholesky
decomposition into a product of a lower unit triangular matrix L, a
diagonal matrix D and L^T, given by L D L^T. This is equivalent to the
Cholesky formulation discussed above, with the standard Cholesky lower
triangular factor given by L D^{1 \over 2}. For ill-conditioned
matrices, it can help to use a pivoting strategy to prevent the entries
of D and L from growing too large, and also ensure D_1 \ge D_2 \ge
\cdots \ge D_n > 0, where D_i are the diagonal entries of D. The final
decomposition is given by
P A P^T = L D L^T
where P is a permutation matrix.
-- Function: int gsl_linalg_pcholesky_decomp (gsl_matrix * A,
gsl_permutation * P)
This function factors the symmetric, positive-definite square
matrix A into the Pivoted Cholesky decomposition P A P^T = L D L^T.
On input, the values from the diagonal and lower-triangular part of
the matrix A are used to construct the factorization. On output
the diagonal of the input matrix A stores the diagonal elements of
D, and the lower triangular portion of A contains the matrix L.
Since L has ones on its diagonal these do not need to be
explicitely stored. The upper triangular portion of A is
unmodified. The permutation matrix P is stored in P on output.
-- Function: int gsl_linalg_pcholesky_solve (const gsl_matrix * LDLT,
const gsl_permutation * P, const gsl_vector * B, gsl_vector *
X)
This function solves the system A x = b using the Pivoted Cholesky
decomposition of A held in the matrix LDLT and permutation P which
must have been previously computed by
'gsl_linalg_pcholesky_decomp'.
-- Function: int gsl_linalg_pcholesky_svx (const gsl_matrix * LDLT,
const gsl_permutation * P, gsl_vector * X)
This function solves the system A x = b in-place using the Pivoted
Cholesky decomposition of A held in the matrix LDLT and permutation
P which must have been previously computed by
'gsl_linalg_pcholesky_decomp'. On input, X contains the right hand
side vector b which is replaced by the solution vector on output.
-- Function: int gsl_linalg_pcholesky_decomp2 (gsl_matrix * A,
gsl_permutation * P, gsl_vector * S)
This function computes the pivoted Cholesky factorization of the
matrix S A S, where the input matrix A is symmetric and positive
definite, and the diagonal scaling matrix S is computed to reduce
the condition number of A as much as possible. See *note Cholesky
Decomposition:: for more information on the matrix S. The Pivoted
Cholesky decomposition satisfies P S A S P^T = L D L^T. On input,
the values from the diagonal and lower-triangular part of the
matrix A are used to construct the factorization. On output the
diagonal of the input matrix A stores the diagonal elements of D,
and the lower triangular portion of A contains the matrix L. Since
L has ones on its diagonal these do not need to be explicitely
stored. The upper triangular portion of A is unmodified. The
permutation matrix P is stored in P on output. The diagonal
scaling transformation is stored in S on output.
-- Function: int gsl_linalg_pcholesky_solve2 (const gsl_matrix * LDLT,
const gsl_permutation * P, const gsl_vector * S, const
gsl_vector * B, gsl_vector * X)
This function solves the system (S A S) (S^{-1} x) = S b using the
Pivoted Cholesky decomposition of S A S held in the matrix LDLT,
permutation P, and vector S, which must have been previously
computed by 'gsl_linalg_pcholesky_decomp2'.
-- Function: int gsl_linalg_pcholesky_svx2 (const gsl_matrix * LDLT,
const gsl_permutation * P, const gsl_vector * S, gsl_vector *
X)
This function solves the system (S A S) (S^{-1} x) = S b in-place
using the Pivoted Cholesky decomposition of S A S held in the
matrix LDLT, permutation P and vector S, which must have been
previously computed by 'gsl_linalg_pcholesky_decomp2'. On input, X
contains the right hand side vector b which is replaced by the
solution vector on output.
-- Function: int gsl_linalg_pcholesky_invert (const gsl_matrix * LDLT,
const gsl_permutation * P, gsl_matrix * AINV)
This function computes the inverse of the matrix A, using the
Pivoted Cholesky decomposition stored in LDLT and P. On output,
the matrix AINV contains A^{-1}.
-- Function: int gsl_linalg_pcholesky_rcond (const gsl_matrix * LDLT,
const gsl_permutation * P, double * RCOND, gsl_vector * WORK)
This function estimates the reciprocal condition number (using the
1-norm) of the symmetric positive definite matrix A, using its
pivoted Cholesky decomposition provided in LDLT. The reciprocal
condition number estimate, defined as 1 / (||A||_1 \cdot
||A^{-1}||_1), is stored in RCOND. Additional workspace of size 3
N is required in WORK.
14.8 Modified Cholesky Decomposition
====================================
The modified Cholesky decomposition is suitable for solving systems A x
= b where A is a symmetric indefinite matrix. Such matrices arise in
nonlinear optimization algorithms. The standard Cholesky decomposition
requires a positive definite matrix and would fail in this case.
Instead of resorting to a method like QR or SVD, which do not take into
account the symmetry of the matrix, we can instead introduce a small
perturbation to the matrix A to make it positive definite, and then use
a Cholesky decomposition on the perturbed matrix. The resulting
decomposition satisfies
P (A + E) P^T = L D L^T
where P is a permutation matrix, E is a diagonal perturbation matrix,
L is unit lower triangular, and D is diagonal. If A is sufficiently
positive definite, then the perturbation matrix E will be zero and this
method is equivalent to the pivoted Cholesky algorithm. For indefinite
matrices, the perturbation matrix E is computed to ensure that A + E is
positive definite and well conditioned.
-- Function: int gsl_linalg_mcholesky_decomp (gsl_matrix * A,
gsl_permutation * P, gsl_vector * E)
This function factors the symmetric, indefinite square matrix A
into the Modified Cholesky decomposition P (A + E) P^T = L D L^T.
On input, the values from the diagonal and lower-triangular part of
the matrix A are used to construct the factorization. On output
the diagonal of the input matrix A stores the diagonal elements of
D, and the lower triangular portion of A contains the matrix L.
Since L has ones on its diagonal these do not need to be
explicitely stored. The upper triangular portion of A is
unmodified. The permutation matrix P is stored in P on output.
The diagonal perturbation matrix is stored in E on output. The
parameter E may be set to NULL if it is not required.
-- Function: int gsl_linalg_mcholesky_solve (const gsl_matrix * LDLT,
const gsl_permutation * P, const gsl_vector * B, gsl_vector *
X)
This function solves the perturbed system (A + E) x = b using the
Cholesky decomposition of A + E held in the matrix LDLT and
permutation P which must have been previously computed by
'gsl_linalg_mcholesky_decomp'.
-- Function: int gsl_linalg_mcholesky_svx (const gsl_matrix * LDLT,
const gsl_permutation * P, gsl_vector * X)
This function solves the perturbed system (A + E) x = b in-place
using the Cholesky decomposition of A + E held in the matrix LDLT
and permutation P which must have been previously computed by
'gsl_linalg_mcholesky_decomp'. On input, X contains the right hand
side vector b which is replaced by the solution vector on output.
-- Function: int gsl_linalg_mcholesky_rcond (const gsl_matrix * LDLT,
const gsl_permutation * P, double * RCOND, gsl_vector * WORK)
This function estimates the reciprocal condition number (using the
1-norm) of the perturbed matrix A + E, using its pivoted Cholesky
decomposition provided in LDLT. The reciprocal condition number
estimate, defined as 1 / (||A + E||_1 \cdot ||(A + E)^{-1}||_1), is
stored in RCOND. Additional workspace of size 3 N is required in
WORK.
14.9 Tridiagonal Decomposition of Real Symmetric Matrices
=========================================================
A symmetric matrix A can be factorized by similarity transformations
into the form,
A = Q T Q^T
where Q is an orthogonal matrix and T is a symmetric tridiagonal matrix.
-- Function: int gsl_linalg_symmtd_decomp (gsl_matrix * A, gsl_vector *
TAU)
This function factorizes the symmetric square matrix A into the
symmetric tridiagonal decomposition Q T Q^T. On output the
diagonal and subdiagonal part of the input matrix A contain the
tridiagonal matrix T. The remaining lower triangular part of the
input matrix contains the Householder vectors which, together with
the Householder coefficients TAU, encode the orthogonal matrix Q.
This storage scheme is the same as used by LAPACK. The upper
triangular part of A is not referenced.
-- Function: int gsl_linalg_symmtd_unpack (const gsl_matrix * A, const
gsl_vector * TAU, gsl_matrix * Q, gsl_vector * DIAG,
gsl_vector * SUBDIAG)
This function unpacks the encoded symmetric tridiagonal
decomposition (A, TAU) obtained from 'gsl_linalg_symmtd_decomp'
into the orthogonal matrix Q, the vector of diagonal elements DIAG
and the vector of subdiagonal elements SUBDIAG.
-- Function: int gsl_linalg_symmtd_unpack_T (const gsl_matrix * A,
gsl_vector * DIAG, gsl_vector * SUBDIAG)
This function unpacks the diagonal and subdiagonal of the encoded
symmetric tridiagonal decomposition (A, TAU) obtained from
'gsl_linalg_symmtd_decomp' into the vectors DIAG and SUBDIAG.
14.10 Tridiagonal Decomposition of Hermitian Matrices
=====================================================
A hermitian matrix A can be factorized by similarity transformations
into the form,
A = U T U^T
where U is a unitary matrix and T is a real symmetric tridiagonal
matrix.
-- Function: int gsl_linalg_hermtd_decomp (gsl_matrix_complex * A,
gsl_vector_complex * TAU)
This function factorizes the hermitian matrix A into the symmetric
tridiagonal decomposition U T U^T. On output the real parts of the
diagonal and subdiagonal part of the input matrix A contain the
tridiagonal matrix T. The remaining lower triangular part of the
input matrix contains the Householder vectors which, together with
the Householder coefficients TAU, encode the unitary matrix U.
This storage scheme is the same as used by LAPACK. The upper
triangular part of A and imaginary parts of the diagonal are not
referenced.
-- Function: int gsl_linalg_hermtd_unpack (const gsl_matrix_complex *
A, const gsl_vector_complex * TAU, gsl_matrix_complex * U,
gsl_vector * DIAG, gsl_vector * SUBDIAG)
This function unpacks the encoded tridiagonal decomposition (A,
TAU) obtained from 'gsl_linalg_hermtd_decomp' into the unitary
matrix U, the real vector of diagonal elements DIAG and the real
vector of subdiagonal elements SUBDIAG.
-- Function: int gsl_linalg_hermtd_unpack_T (const gsl_matrix_complex *
A, gsl_vector * DIAG, gsl_vector * SUBDIAG)
This function unpacks the diagonal and subdiagonal of the encoded
tridiagonal decomposition (A, TAU) obtained from the
'gsl_linalg_hermtd_decomp' into the real vectors DIAG and SUBDIAG.
14.11 Hessenberg Decomposition of Real Matrices
===============================================
A general real matrix A can be decomposed by orthogonal similarity
transformations into the form
A = U H U^T
where U is orthogonal and H is an upper Hessenberg matrix, meaning
that it has zeros below the first subdiagonal. The Hessenberg reduction
is the first step in the Schur decomposition for the nonsymmetric
eigenvalue problem, but has applications in other areas as well.
-- Function: int gsl_linalg_hessenberg_decomp (gsl_matrix * A,
gsl_vector * TAU)
This function computes the Hessenberg decomposition of the matrix A
by applying the similarity transformation H = U^T A U. On output,
H is stored in the upper portion of A. The information required to
construct the matrix U is stored in the lower triangular portion of
A. U is a product of N - 2 Householder matrices. The Householder
vectors are stored in the lower portion of A (below the
subdiagonal) and the Householder coefficients are stored in the
vector TAU. TAU must be of length N.
-- Function: int gsl_linalg_hessenberg_unpack (gsl_matrix * H,
gsl_vector * TAU, gsl_matrix * U)
This function constructs the orthogonal matrix U from the
information stored in the Hessenberg matrix H along with the vector
TAU. H and TAU are outputs from 'gsl_linalg_hessenberg_decomp'.
-- Function: int gsl_linalg_hessenberg_unpack_accum (gsl_matrix * H,
gsl_vector * TAU, gsl_matrix * V)
This function is similar to 'gsl_linalg_hessenberg_unpack', except
it accumulates the matrix U into V, so that V' = VU. The matrix V
must be initialized prior to calling this function. Setting V to
the identity matrix provides the same result as
'gsl_linalg_hessenberg_unpack'. If H is order N, then V must have
N columns but may have any number of rows.
-- Function: int gsl_linalg_hessenberg_set_zero (gsl_matrix * H)
This function sets the lower triangular portion of H, below the
subdiagonal, to zero. It is useful for clearing out the
Householder vectors after calling 'gsl_linalg_hessenberg_decomp'.
14.12 Hessenberg-Triangular Decomposition of Real Matrices
==========================================================
A general real matrix pair (A, B) can be decomposed by orthogonal
similarity transformations into the form
A = U H V^T
B = U R V^T
where U and V are orthogonal, H is an upper Hessenberg matrix, and R
is upper triangular. The Hessenberg-Triangular reduction is the first
step in the generalized Schur decomposition for the generalized
eigenvalue problem.
-- Function: int gsl_linalg_hesstri_decomp (gsl_matrix * A, gsl_matrix
* B, gsl_matrix * U, gsl_matrix * V, gsl_vector * WORK)
This function computes the Hessenberg-Triangular decomposition of
the matrix pair (A, B). On output, H is stored in A, and R is
stored in B. If U and V are provided (they may be null), the
similarity transformations are stored in them. Additional
workspace of length N is needed in WORK.
14.13 Bidiagonalization
=======================
A general matrix A can be factorized by similarity transformations into
the form,
A = U B V^T
where U and V are orthogonal matrices and B is a N-by-N bidiagonal
matrix with non-zero entries only on the diagonal and superdiagonal.
The size of U is M-by-N and the size of V is N-by-N.
-- Function: int gsl_linalg_bidiag_decomp (gsl_matrix * A, gsl_vector *
TAU_U, gsl_vector * TAU_V)
This function factorizes the M-by-N matrix A into bidiagonal form U
B V^T. The diagonal and superdiagonal of the matrix B are stored
in the diagonal and superdiagonal of A. The orthogonal matrices U
and V are stored as compressed Householder vectors in the remaining
elements of A. The Householder coefficients are stored in the
vectors TAU_U and TAU_V. The length of TAU_U must equal the number
of elements in the diagonal of A and the length of TAU_V should be
one element shorter.
-- Function: int gsl_linalg_bidiag_unpack (const gsl_matrix * A, const
gsl_vector * TAU_U, gsl_matrix * U, const gsl_vector * TAU_V,
gsl_matrix * V, gsl_vector * DIAG, gsl_vector * SUPERDIAG)
This function unpacks the bidiagonal decomposition of A produced by
'gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
orthogonal matrices U, V and the diagonal vector DIAG and
superdiagonal SUPERDIAG. Note that U is stored as a compact M-by-N
orthogonal matrix satisfying U^T U = I for efficiency.
-- Function: int gsl_linalg_bidiag_unpack2 (gsl_matrix * A, gsl_vector
* TAU_U, gsl_vector * TAU_V, gsl_matrix * V)
This function unpacks the bidiagonal decomposition of A produced by
'gsl_linalg_bidiag_decomp', (A, TAU_U, TAU_V) into the separate
orthogonal matrices U, V and the diagonal vector DIAG and
superdiagonal SUPERDIAG. The matrix U is stored in-place in A.
-- Function: int gsl_linalg_bidiag_unpack_B (const gsl_matrix * A,
gsl_vector * DIAG, gsl_vector * SUPERDIAG)
This function unpacks the diagonal and superdiagonal of the
bidiagonal decomposition of A from 'gsl_linalg_bidiag_decomp', into
the diagonal vector DIAG and superdiagonal vector SUPERDIAG.
14.14 Givens Rotations
======================
A Givens rotation is a rotation in the plane acting on two elements of a
given vector. It can be represented in matrix form as
where the \cos{\theta} and \sin{\theta} appear at the intersection of
the ith and jth rows and columns. When acting on a vector x,
G(i,j,\theta) x performs a rotation of the (i,j) elements of x. Givens
rotations are typically used to introduce zeros in vectors, such as
during the QR decomposition of a matrix. In this case, it is typically
desired to find c and s such that
with r = \sqrt{a^2 + b^2}.
-- Function: void gsl_linalg_givens (const double A, const double B,
double * C, double * S)
This function computes c = \cos{\theta} and s = \sin{\theta} so
that the Givens matrix G(\theta) acting on the vector (a,b)
produces (r, 0), with r = \sqrt{a^2 + b^2}.
-- Function: void gsl_linalg_givens_gv (gsl_vector * V, const size_t I,
const size_t J, const double C, const double S)
This function applies the Givens rotation defined by c =
\cos{\theta} and s = \sin{\theta} to the I and J elements of V. On
output, (v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)).
14.15 Householder Transformations
=================================
A Householder transformation is a rank-1 modification of the identity
matrix which can be used to zero out selected elements of a vector. A
Householder matrix P takes the form,
P = I - \tau v v^T
where v is a vector (called the "Householder vector") and \tau = 2/(v^T
v). The functions described in this section use the rank-1 structure of
the Householder matrix to create and apply Householder transformations
efficiently.
-- Function: double gsl_linalg_householder_transform (gsl_vector * W)
-- Function: gsl_complex gsl_linalg_complex_householder_transform
(gsl_vector_complex * W)
This function prepares a Householder transformation P = I - \tau v
v^T which can be used to zero all the elements of the input vector
W except the first. On output the Householder vector V is stored
in W and the scalar \tau is returned. The householder vector V is
normalized so that V[0] = 1, however this 1 is not stored in the
output vector. Instead, W[0] is set to the first element of the
transformed vector, so that if u = P w, W[0] = u[0] on output and
the remainder of u is zero.
-- Function: int gsl_linalg_householder_hm (double TAU, const
gsl_vector * V, gsl_matrix * A)
-- Function: int gsl_linalg_complex_householder_hm (gsl_complex TAU,
const gsl_vector_complex * V, gsl_matrix_complex * A)
This function applies the Householder matrix P defined by the
scalar TAU and the vector V to the left-hand side of the matrix A.
On output the result P A is stored in A.
-- Function: int gsl_linalg_householder_mh (double TAU, const
gsl_vector * V, gsl_matrix * A)
-- Function: int gsl_linalg_complex_householder_mh (gsl_complex TAU,
const gsl_vector_complex * V, gsl_matrix_complex * A)
This function applies the Householder matrix P defined by the
scalar TAU and the vector V to the right-hand side of the matrix A.
On output the result A P is stored in A.
-- Function: int gsl_linalg_householder_hv (double TAU, const
gsl_vector * V, gsl_vector * W)
-- Function: int gsl_linalg_complex_householder_hv (gsl_complex TAU,
const gsl_vector_complex * V, gsl_vector_complex * W)
This function applies the Householder transformation P defined by
the scalar TAU and the vector V to the vector W. On output the
result P w is stored in W.
14.16 Householder solver for linear systems
===========================================
-- Function: int gsl_linalg_HH_solve (gsl_matrix * A, const gsl_vector
* B, gsl_vector * X)
This function solves the system A x = b directly using Householder
transformations. On output the solution is stored in X and B is
not modified. The matrix A is destroyed by the Householder
transformations.
-- Function: int gsl_linalg_HH_svx (gsl_matrix * A, gsl_vector * X)
This function solves the system A x = b in-place using Householder
transformations. On input X should contain the right-hand side b,
which is replaced by the solution on output. The matrix A is
destroyed by the Householder transformations.
14.17 Tridiagonal Systems
=========================
The functions described in this section efficiently solve symmetric,
non-symmetric and cyclic tridiagonal systems with minimal storage. Note
that the current implementations of these functions use a variant of
Cholesky decomposition, so the tridiagonal matrix must be positive
definite. For non-positive definite matrices, the functions return the
error code 'GSL_ESING'.
-- Function: int gsl_linalg_solve_tridiag (const gsl_vector * DIAG,
const gsl_vector * E, const gsl_vector * F, const gsl_vector *
B, gsl_vector * X)
This function solves the general N-by-N system A x = b where A is
tridiagonal (N >= 2). The super-diagonal and sub-diagonal vectors
E and F must be one element shorter than the diagonal vector DIAG.
The form of A for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 0 )
( f_0 d_1 e_1 0 )
( 0 f_1 d_2 e_2 )
( 0 0 f_2 d_3 )
-- Function: int gsl_linalg_solve_symm_tridiag (const gsl_vector *
DIAG, const gsl_vector * E, const gsl_vector * B, gsl_vector *
X)
This function solves the general N-by-N system A x = b where A is
symmetric tridiagonal (N >= 2). The off-diagonal vector E must be
one element shorter than the diagonal vector DIAG. The form of A
for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 0 )
( e_0 d_1 e_1 0 )
( 0 e_1 d_2 e_2 )
( 0 0 e_2 d_3 )
-- Function: int gsl_linalg_solve_cyc_tridiag (const gsl_vector * DIAG,
const gsl_vector * E, const gsl_vector * F, const gsl_vector *
B, gsl_vector * X)
This function solves the general N-by-N system A x = b where A is
cyclic tridiagonal (N >= 3). The cyclic super-diagonal and
sub-diagonal vectors E and F must have the same number of elements
as the diagonal vector DIAG. The form of A for the 4-by-4 case is
shown below,
A = ( d_0 e_0 0 f_3 )
( f_0 d_1 e_1 0 )
( 0 f_1 d_2 e_2 )
( e_3 0 f_2 d_3 )
-- Function: int gsl_linalg_solve_symm_cyc_tridiag (const gsl_vector *
DIAG, const gsl_vector * E, const gsl_vector * B, gsl_vector *
X)
This function solves the general N-by-N system A x = b where A is
symmetric cyclic tridiagonal (N >= 3). The cyclic off-diagonal
vector E must have the same number of elements as the diagonal
vector DIAG. The form of A for the 4-by-4 case is shown below,
A = ( d_0 e_0 0 e_3 )
( e_0 d_1 e_1 0 )
( 0 e_1 d_2 e_2 )
( e_3 0 e_2 d_3 )
14.18 Triangular Systems
========================
-- Function: int gsl_linalg_tri_upper_invert (gsl_matrix * T)
-- Function: int gsl_linalg_tri_lower_invert (gsl_matrix * T)
-- Function: int gsl_linalg_tri_upper_unit_invert (gsl_matrix * T)
-- Function: int gsl_linalg_tri_lower_unit_invert (gsl_matrix * T)
These functions calculate the in-place inverse of the triangular
matrix T. When the 'upper' prefix is specified, then the upper
triangle of T is used, and when the 'lower' prefix is specified,
the lower triangle is used. If the 'unit' prefix is specified,
then the diagonal elements of the matrix T are taken as unity and
are not referenced. Otherwise the diagonal elements are used in
the inversion.
-- Function: int gsl_linalg_tri_upper_rcond (const gsl_matrix * T,
double * RCOND, gsl_vector * WORK)
-- Function: int gsl_linalg_tri_lower_rcond (const gsl_matrix * T,
double * RCOND, gsl_vector * WORK)
These functions estimate the reciprocal condition number, in the
1-norm, of the upper or lower N-by-N triangular matrix T. The
reciprocal condition number is stored in RCOND on output, and is
defined by 1 / (||T||_1 \cdot ||T^{-1}||_1). Additional workspace
of size 3 N is required in WORK.
14.19 Balancing
===============
The process of balancing a matrix applies similarity transformations to
make the rows and columns have comparable norms. This is useful, for
example, to reduce roundoff errors in the solution of eigenvalue
problems. Balancing a matrix A consists of replacing A with a similar
matrix
A' = D^(-1) A D
where D is a diagonal matrix whose entries are powers of the floating
point radix.
-- Function: int gsl_linalg_balance_matrix (gsl_matrix * A, gsl_vector
* D)
This function replaces the matrix A with its balanced counterpart
and stores the diagonal elements of the similarity transformation
into the vector D.
14.20 Examples
==============
The following program solves the linear system A x = b. The system to
be solved is,
[ 0.18 0.60 0.57 0.96 ] [x0] [1.0]
[ 0.41 0.24 0.99 0.58 ] [x1] = [2.0]
[ 0.14 0.30 0.97 0.66 ] [x2] [3.0]
[ 0.51 0.13 0.19 0.85 ] [x3] [4.0]
and the solution is found using LU decomposition of the matrix A.
#include
#include
int
main (void)
{
double a_data[] = { 0.18, 0.60, 0.57, 0.96,
0.41, 0.24, 0.99, 0.58,
0.14, 0.30, 0.97, 0.66,
0.51, 0.13, 0.19, 0.85 };
double b_data[] = { 1.0, 2.0, 3.0, 4.0 };
gsl_matrix_view m
= gsl_matrix_view_array (a_data, 4, 4);
gsl_vector_view b
= gsl_vector_view_array (b_data, 4);
gsl_vector *x = gsl_vector_alloc (4);
int s;
gsl_permutation * p = gsl_permutation_alloc (4);
gsl_linalg_LU_decomp (&m.matrix, p, &s);
gsl_linalg_LU_solve (&m.matrix, p, &b.vector, x);
printf ("x = \n");
gsl_vector_fprintf (stdout, x, "%g");
gsl_permutation_free (p);
gsl_vector_free (x);
return 0;
}
Here is the output from the program,
x =
-4.05205
-12.6056
1.66091
8.69377
This can be verified by multiplying the solution x by the original
matrix A using GNU OCTAVE,
octave> A = [ 0.18, 0.60, 0.57, 0.96;
0.41, 0.24, 0.99, 0.58;
0.14, 0.30, 0.97, 0.66;
0.51, 0.13, 0.19, 0.85 ];
octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377];
octave> A * x
ans =
1.0000
2.0000
3.0000
4.0000
This reproduces the original right-hand side vector, b, in accordance
with the equation A x = b.
14.21 References and Further Reading
====================================
Further information on the algorithms described in this section can be
found in the following book,
G. H. Golub, C. F. Van Loan, 'Matrix Computations' (3rd Ed, 1996),
Johns Hopkins University Press, ISBN 0-8018-5414-8.
The LAPACK library is described in the following manual,
'LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
ISBN 0-89871-447-8.
The LAPACK source code can be found at the website above, along with an
online copy of the users guide.
The Modified Golub-Reinsch algorithm is described in the following
paper,
T.F. Chan, "An Improved Algorithm for Computing the Singular Value
Decomposition", 'ACM Transactions on Mathematical Software', 8
(1982), pp 72-83.
The Jacobi algorithm for singular value decomposition is described in
the following papers,
J.C. Nash, "A one-sided transformation method for the singular
value decomposition and algebraic eigenproblem", 'Computer
Journal', Volume 18, Number 1 (1975), p 74-76
J.C. Nash and S. Shlien "Simple algorithms for the partial singular
value decomposition", 'Computer Journal', Volume 30 (1987), p
268-275.
James Demmel, Kres in the
'lawns' or 'lawnspdf' directories.
The algorithm for estimating a matrix condition number is described in
the following paper,
N. J. Higham, "FORTRAN codes for estimating the one-norm of a real
or complex matrix, with applications to condition estimation", ACM
Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December
1988.
15 Eigensystems
***************
This chapter describes functions for computing eigenvalues and
eigenvectors of matrices. There are routines for real symmetric, real
nonsymmetric, complex hermitian, real generalized symmetric-definite,
complex generalized hermitian-definite, and real generalized
nonsymmetric eigensystems. Eigenvalues can be computed with or without
eigenvectors. The hermitian and real symmetric matrix algorithms are
symmetric bidiagonalization followed by QR reduction. The nonsymmetric
algorithm is the Francis QR double-shift. The generalized nonsymmetric
algorithm is the QZ method due to Moler and Stewart.
The functions described in this chapter are declared in the header
file 'gsl_eigen.h'.
15.1 Real Symmetric Matrices
============================
For real symmetric matrices, the library uses the symmetric
bidiagonalization and QR reduction method. This is described in Golub &
van Loan, section 8.3. The computed eigenvalues are accurate to an
absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine
precision.
-- Function: gsl_eigen_symm_workspace * gsl_eigen_symm_alloc (const
size_t N)
This function allocates a workspace for computing eigenvalues of
N-by-N real symmetric matrices. The size of the workspace is
O(2n).
-- Function: void gsl_eigen_symm_free (gsl_eigen_symm_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_symm (gsl_matrix * A, gsl_vector * EVAL,
gsl_eigen_symm_workspace * W)
This function computes the eigenvalues of the real symmetric matrix
A. Additional workspace of the appropriate size must be provided
in W. The diagonal and lower triangular part of A are destroyed
during the computation, but the strict upper triangular part is not
referenced. The eigenvalues are stored in the vector EVAL and are
unordered.
-- Function: gsl_eigen_symmv_workspace * gsl_eigen_symmv_alloc (const
size_t N)
This function allocates a workspace for computing eigenvalues and
eigenvectors of N-by-N real symmetric matrices. The size of the
workspace is O(4n).
-- Function: void gsl_eigen_symmv_free (gsl_eigen_symmv_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_symmv (gsl_matrix * A, gsl_vector * EVAL,
gsl_matrix * EVEC, gsl_eigen_symmv_workspace * W)
This function computes the eigenvalues and eigenvectors of the real
symmetric matrix A. Additional workspace of the appropriate size
must be provided in W. The diagonal and lower triangular part of A
are destroyed during the computation, but the strict upper
triangular part is not referenced. The eigenvalues are stored in
the vector EVAL and are unordered. The corresponding eigenvectors
are stored in the columns of the matrix EVEC. For example, the
eigenvector in the first column corresponds to the first
eigenvalue. The eigenvectors are guaranteed to be mutually
orthogonal and normalised to unit magnitude.
15.2 Complex Hermitian Matrices
===============================
For hermitian matrices, the library uses the complex form of the
symmetric bidiagonalization and QR reduction method.
-- Function: gsl_eigen_herm_workspace * gsl_eigen_herm_alloc (const
size_t N)
This function allocates a workspace for computing eigenvalues of
N-by-N complex hermitian matrices. The size of the workspace is
O(3n).
-- Function: void gsl_eigen_herm_free (gsl_eigen_herm_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_herm (gsl_matrix_complex * A, gsl_vector *
EVAL, gsl_eigen_herm_workspace * W)
This function computes the eigenvalues of the complex hermitian
matrix A. Additional workspace of the appropriate size must be
provided in W. The diagonal and lower triangular part of A are
destroyed during the computation, but the strict upper triangular
part is not referenced. The imaginary parts of the diagonal are
assumed to be zero and are not referenced. The eigenvalues are
stored in the vector EVAL and are unordered.
-- Function: gsl_eigen_hermv_workspace * gsl_eigen_hermv_alloc (const
size_t N)
This function allocates a workspace for computing eigenvalues and
eigenvectors of N-by-N complex hermitian matrices. The size of the
workspace is O(5n).
-- Function: void gsl_eigen_hermv_free (gsl_eigen_hermv_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_hermv (gsl_matrix_complex * A, gsl_vector *
EVAL, gsl_matrix_complex * EVEC, gsl_eigen_hermv_workspace *
W)
This function computes the eigenvalues and eigenvectors of the
complex hermitian matrix A. Additional workspace of the
appropriate size must be provided in W. The diagonal and lower
triangular part of A are destroyed during the computation, but the
strict upper triangular part is not referenced. The imaginary
parts of the diagonal are assumed to be zero and are not
referenced. The eigenvalues are stored in the vector EVAL and are
unordered. The corresponding complex eigenvectors are stored in
the columns of the matrix EVEC. For example, the eigenvector in
the first column corresponds to the first eigenvalue. The
eigenvectors are guaranteed to be mutually orthogonal and
normalised to unit magnitude.
15.3 Real Nonsymmetric Matrices
===============================
The solution of the real nonsymmetric eigensystem problem for a matrix A
involves computing the Schur decomposition
A = Z T Z^T
where Z is an orthogonal matrix of Schur vectors and T, the Schur
form, is quasi upper triangular with diagonal 1-by-1 blocks which are
real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are
complex conjugate eigenvalues of A. The algorithm used is the
double-shift Francis method.
-- Function: gsl_eigen_nonsymm_workspace * gsl_eigen_nonsymm_alloc
(const size_t N)
This function allocates a workspace for computing eigenvalues of
N-by-N real nonsymmetric matrices. The size of the workspace is
O(2n).
-- Function: void gsl_eigen_nonsymm_free (gsl_eigen_nonsymm_workspace *
W)
This function frees the memory associated with the workspace W.
-- Function: void gsl_eigen_nonsymm_params (const int COMPUTE_T, const
int BALANCE, gsl_eigen_nonsymm_workspace * W)
This function sets some parameters which determine how the
eigenvalue problem is solved in subsequent calls to
'gsl_eigen_nonsymm'.
If COMPUTE_T is set to 1, the full Schur form T will be computed by
'gsl_eigen_nonsymm'. If it is set to 0, T will not be computed
(this is the default setting). Computing the full Schur form T
requires approximately 1.5-2 times the number of flops.
If BALANCE is set to 1, a balancing transformation is applied to
the matrix prior to computing eigenvalues. This transformation is
designed to make the rows and columns of the matrix have comparable
norms, and can result in more accurate eigenvalues for matrices
whose entries vary widely in magnitude. See *note Balancing:: for
more information. Note that the balancing transformation does not
preserve the orthogonality of the Schur vectors, so if you wish to
compute the Schur vectors with 'gsl_eigen_nonsymm_Z' you will
obtain the Schur vectors of the balanced matrix instead of the
original matrix. The relationship will be
T = Q^t D^(-1) A D Q
where Q is the matrix of Schur vectors for the balanced matrix, and
D is the balancing transformation. Then 'gsl_eigen_nonsymm_Z' will
compute a matrix Z which satisfies
T = Z^(-1) A Z
with Z = D Q. Note that Z will not be orthogonal. For this
reason, balancing is not performed by default.
-- Function: int gsl_eigen_nonsymm (gsl_matrix * A, gsl_vector_complex
* EVAL, gsl_eigen_nonsymm_workspace * W)
This function computes the eigenvalues of the real nonsymmetric
matrix A and stores them in the vector EVAL. If T is desired, it
is stored in the upper portion of A on output. Otherwise, on
output, the diagonal of A will contain the 1-by-1 real eigenvalues
and 2-by-2 complex conjugate eigenvalue systems, and the rest of A
is destroyed. In rare cases, this function may fail to find all
eigenvalues. If this happens, an error code is returned and the
number of converged eigenvalues is stored in 'w->n_evals'. The
converged eigenvalues are stored in the beginning of EVAL.
-- Function: int gsl_eigen_nonsymm_Z (gsl_matrix * A,
gsl_vector_complex * EVAL, gsl_matrix * Z,
gsl_eigen_nonsymm_workspace * W)
This function is identical to 'gsl_eigen_nonsymm' except that it
also computes the Schur vectors and stores them into Z.
-- Function: gsl_eigen_nonsymmv_workspace * gsl_eigen_nonsymmv_alloc
(const size_t N)
This function allocates a workspace for computing eigenvalues and
eigenvectors of N-by-N real nonsymmetric matrices. The size of the
workspace is O(5n).
-- Function: void gsl_eigen_nonsymmv_free (gsl_eigen_nonsymmv_workspace
* W)
This function frees the memory associated with the workspace W.
-- Function: void gsl_eigen_nonsymmv_params (const int BALANCE,
gsl_eigen_nonsymm_workspace * W)
This function sets parameters which determine how the eigenvalue
problem is solved in subsequent calls to 'gsl_eigen_nonsymmv'. If
BALANCE is set to 1, a balancing transformation is applied to the
matrix. See 'gsl_eigen_nonsymm_params' for more information.
Balancing is turned off by default since it does not preserve the
orthogonality of the Schur vectors.
-- Function: int gsl_eigen_nonsymmv (gsl_matrix * A, gsl_vector_complex
* EVAL, gsl_matrix_complex * EVEC,
gsl_eigen_nonsymmv_workspace * W)
This function computes eigenvalues and right eigenvectors of the
N-by-N real nonsymmetric matrix A. It first calls
'gsl_eigen_nonsymm' to compute the eigenvalues, Schur form T, and
Schur vectors. Then it finds eigenvectors of T and backtransforms
them using the Schur vectors. The Schur vectors are destroyed in
the process, but can be saved by using 'gsl_eigen_nonsymmv_Z'. The
computed eigenvectors are normalized to have unit magnitude. On
output, the upper portion of A contains the Schur form T. If
'gsl_eigen_nonsymm' fails, no eigenvectors are computed, and an
error code is returned.
-- Function: int gsl_eigen_nonsymmv_Z (gsl_matrix * A,
gsl_vector_complex * EVAL, gsl_matrix_complex * EVEC,
gsl_matrix * Z, gsl_eigen_nonsymmv_workspace * W)
This function is identical to 'gsl_eigen_nonsymmv' except that it
also saves the Schur vectors into Z.
15.4 Real Generalized Symmetric-Definite Eigensystems
=====================================================
The real generalized symmetric-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
where A and B are symmetric matrices, and B is positive-definite.
This problem reduces to the standard symmetric eigenvalue problem by
applying the Cholesky decomposition to B:
A x = \lambda B x
A x = \lambda L L^t x
( L^{-1} A L^{-t} ) L^t x = \lambda L^t x
Therefore, the problem becomes C y = \lambda y where C = L^{-1} A
L^{-t} is symmetric, and y = L^t x. The standard symmetric eigensolver
can be applied to the matrix C. The resulting eigenvectors are
backtransformed to find the vectors of the original problem. The
eigenvalues and eigenvectors of the generalized symmetric-definite
eigenproblem are always real.
-- Function: gsl_eigen_gensymm_workspace * gsl_eigen_gensymm_alloc
(const size_t N)
This function allocates a workspace for computing eigenvalues of
N-by-N real generalized symmetric-definite eigensystems. The size
of the workspace is O(2n).
-- Function: void gsl_eigen_gensymm_free (gsl_eigen_gensymm_workspace *
W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_gensymm (gsl_matrix * A, gsl_matrix * B,
gsl_vector * EVAL, gsl_eigen_gensymm_workspace * W)
This function computes the eigenvalues of the real generalized
symmetric-definite matrix pair (A, B), and stores them in EVAL,
using the method outlined above. On output, B contains its
Cholesky decomposition and A is destroyed.
-- Function: gsl_eigen_gensymmv_workspace * gsl_eigen_gensymmv_alloc
(const size_t N)
This function allocates a workspace for computing eigenvalues and
eigenvectors of N-by-N real generalized symmetric-definite
eigensystems. The size of the workspace is O(4n).
-- Function: void gsl_eigen_gensymmv_free (gsl_eigen_gensymmv_workspace
* W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_gensymmv (gsl_matrix * A, gsl_matrix * B,
gsl_vector * EVAL, gsl_matrix * EVEC,
gsl_eigen_gensymmv_workspace * W)
This function computes the eigenvalues and eigenvectors of the real
generalized symmetric-definite matrix pair (A, B), and stores them
in EVAL and EVEC respectively. The computed eigenvectors are
normalized to have unit magnitude. On output, B contains its
Cholesky decomposition and A is destroyed.
15.5 Complex Generalized Hermitian-Definite Eigensystems
========================================================
The complex generalized hermitian-definite eigenvalue problem is to find
eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
where A and B are hermitian matrices, and B is positive-definite.
Similarly to the real case, this can be reduced to C y = \lambda y where
C = L^{-1} A L^{-H} is hermitian, and y = L^H x. The standard hermitian
eigensolver can be applied to the matrix C. The resulting eigenvectors
are backtransformed to find the vectors of the original problem. The
eigenvalues of the generalized hermitian-definite eigenproblem are
always real.
-- Function: gsl_eigen_genherm_workspace * gsl_eigen_genherm_alloc
(const size_t N)
This function allocates a workspace for computing eigenvalues of
N-by-N complex generalized hermitian-definite eigensystems. The
size of the workspace is O(3n).
-- Function: void gsl_eigen_genherm_free (gsl_eigen_genherm_workspace *
W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_genherm (gsl_matrix_complex * A,
gsl_matrix_complex * B, gsl_vector * EVAL,
gsl_eigen_genherm_workspace * W)
This function computes the eigenvalues of the complex generalized
hermitian-definite matrix pair (A, B), and stores them in EVAL,
using the method outlined above. On output, B contains its
Cholesky decomposition and A is destroyed.
-- Function: gsl_eigen_genhermv_workspace * gsl_eigen_genhermv_alloc
(const size_t N)
This function allocates a workspace for computing eigenvalues and
eigenvectors of N-by-N complex generalized hermitian-definite
eigensystems. The size of the workspace is O(5n).
-- Function: void gsl_eigen_genhermv_free (gsl_eigen_genhermv_workspace
* W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_genhermv (gsl_matrix_complex * A,
gsl_matrix_complex * B, gsl_vector * EVAL, gsl_matrix_complex
* EVEC, gsl_eigen_genhermv_workspace * W)
This function computes the eigenvalues and eigenvectors of the
complex generalized hermitian-definite matrix pair (A, B), and
stores them in EVAL and EVEC respectively. The computed
eigenvectors are normalized to have unit magnitude. On output, B
contains its Cholesky decomposition and A is destroyed.
15.6 Real Generalized Nonsymmetric Eigensystems
===============================================
Given two square matrices (A, B), the generalized nonsymmetric
eigenvalue problem is to find eigenvalues \lambda and eigenvectors x
such that
A x = \lambda B x
We may also define the problem as finding eigenvalues \mu and
eigenvectors y such that
\mu A y = B y
Note that these two problems are equivalent (with \lambda = 1/\mu) if
neither \lambda nor \mu is zero. If say, \lambda is zero, then it is
still a well defined eigenproblem, but its alternate problem involving
\mu is not. Therefore, to allow for zero (and infinite) eigenvalues,
the problem which is actually solved is
\beta A x = \alpha B x
The eigensolver routines below will return two values \alpha and
\beta and leave it to the user to perform the divisions \lambda = \alpha
/ \beta and \mu = \beta / \alpha.
If the determinant of the matrix pencil A - \lambda B is zero for all
\lambda, the problem is said to be singular; otherwise it is called
regular. Singularity normally leads to some \alpha = \beta = 0 which
means the eigenproblem is ill-conditioned and generally does not have
well defined eigenvalue solutions. The routines below are intended for
regular matrix pencils and could yield unpredictable results when
applied to singular pencils.
The solution of the real generalized nonsymmetric eigensystem problem
for a matrix pair (A, B) involves computing the generalized Schur
decomposition
A = Q S Z^T
B = Q T Z^T
where Q and Z are orthogonal matrices of left and right Schur vectors
respectively, and (S, T) is the generalized Schur form whose diagonal
elements give the \alpha and \beta values. The algorithm used is the QZ
method due to Moler and Stewart (see references).
-- Function: gsl_eigen_gen_workspace * gsl_eigen_gen_alloc (const
size_t N)
This function allocates a workspace for computing eigenvalues of
N-by-N real generalized nonsymmetric eigensystems. The size of the
workspace is O(n).
-- Function: void gsl_eigen_gen_free (gsl_eigen_gen_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: void gsl_eigen_gen_params (const int COMPUTE_S, const int
COMPUTE_T, const int BALANCE, gsl_eigen_gen_workspace * W)
This function sets some parameters which determine how the
eigenvalue problem is solved in subsequent calls to
'gsl_eigen_gen'.
If COMPUTE_S is set to 1, the full Schur form S will be computed by
'gsl_eigen_gen'. If it is set to 0, S will not be computed (this
is the default setting). S is a quasi upper triangular matrix with
1-by-1 and 2-by-2 blocks on its diagonal. 1-by-1 blocks correspond
to real eigenvalues, and 2-by-2 blocks correspond to complex
eigenvalues.
If COMPUTE_T is set to 1, the full Schur form T will be computed by
'gsl_eigen_gen'. If it is set to 0, T will not be computed (this
is the default setting). T is an upper triangular matrix with
non-negative elements on its diagonal. Any 2-by-2 blocks in S will
correspond to a 2-by-2 diagonal block in T.
The BALANCE parameter is currently ignored, since generalized
balancing is not yet implemented.
-- Function: int gsl_eigen_gen (gsl_matrix * A, gsl_matrix * B,
gsl_vector_complex * ALPHA, gsl_vector * BETA,
gsl_eigen_gen_workspace * W)
This function computes the eigenvalues of the real generalized
nonsymmetric matrix pair (A, B), and stores them as pairs in
(ALPHA, BETA), where ALPHA is complex and BETA is real. If \beta_i
is non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue.
Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i is
an eigenvalue of the alternate problem \mu A y = B y. The elements
of BETA are normalized to be non-negative.
If S is desired, it is stored in A on output. If T is desired, it
is stored in B on output. The ordering of eigenvalues in (ALPHA,
BETA) follows the ordering of the diagonal blocks in the Schur
forms S and T. In rare cases, this function may fail to find all
eigenvalues. If this occurs, an error code is returned.
-- Function: int gsl_eigen_gen_QZ (gsl_matrix * A, gsl_matrix * B,
gsl_vector_complex * ALPHA, gsl_vector * BETA, gsl_matrix * Q,
gsl_matrix * Z, gsl_eigen_gen_workspace * W)
This function is identical to 'gsl_eigen_gen' except that it also
computes the left and right Schur vectors and stores them into Q
and Z respectively.
-- Function: gsl_eigen_genv_workspace * gsl_eigen_genv_alloc (const
size_t N)
This function allocates a workspace for computing eigenvalues and
eigenvectors of N-by-N real generalized nonsymmetric eigensystems.
The size of the workspace is O(7n).
-- Function: void gsl_eigen_genv_free (gsl_eigen_genv_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_eigen_genv (gsl_matrix * A, gsl_matrix * B,
gsl_vector_complex * ALPHA, gsl_vector * BETA,
gsl_matrix_complex * EVEC, gsl_eigen_genv_workspace * W)
This function computes eigenvalues and right eigenvectors of the
N-by-N real generalized nonsymmetric matrix pair (A, B). The
eigenvalues are stored in (ALPHA, BETA) and the eigenvectors are
stored in EVEC. It first calls 'gsl_eigen_gen' to compute the
eigenvalues, Schur forms, and Schur vectors. Then it finds
eigenvectors of the Schur forms and backtransforms them using the
Schur vectors. The Schur vectors are destroyed in the process, but
can be saved by using 'gsl_eigen_genv_QZ'. The computed
eigenvectors are normalized to have unit magnitude. On output, (A,
B) contains the generalized Schur form (S, T). If 'gsl_eigen_gen'
fails, no eigenvectors are computed, and an error code is returned.
-- Function: int gsl_eigen_genv_QZ (gsl_matrix * A, gsl_matrix * B,
gsl_vector_complex * ALPHA, gsl_vector * BETA,
gsl_matrix_complex * EVEC, gsl_matrix * Q, gsl_matrix * Z,
gsl_eigen_genv_workspace * W)
This function is identical to 'gsl_eigen_genv' except that it also
computes the left and right Schur vectors and stores them into Q
and Z respectively.
15.7 Sorting Eigenvalues and Eigenvectors
=========================================
-- Function: int gsl_eigen_symmv_sort (gsl_vector * EVAL, gsl_matrix *
EVEC, gsl_eigen_sort_t SORT_TYPE)
This function simultaneously sorts the eigenvalues stored in the
vector EVAL and the corresponding real eigenvectors stored in the
columns of the matrix EVEC into ascending or descending order
according to the value of the parameter SORT_TYPE,
'GSL_EIGEN_SORT_VAL_ASC'
ascending order in numerical value
'GSL_EIGEN_SORT_VAL_DESC'
descending order in numerical value
'GSL_EIGEN_SORT_ABS_ASC'
ascending order in magnitude
'GSL_EIGEN_SORT_ABS_DESC'
descending order in magnitude
-- Function: int gsl_eigen_hermv_sort (gsl_vector * EVAL,
gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
This function simultaneously sorts the eigenvalues stored in the
vector EVAL and the corresponding complex eigenvectors stored in
the columns of the matrix EVEC into ascending or descending order
according to the value of the parameter SORT_TYPE as shown above.
-- Function: int gsl_eigen_nonsymmv_sort (gsl_vector_complex * EVAL,
gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
This function simultaneously sorts the eigenvalues stored in the
vector EVAL and the corresponding complex eigenvectors stored in
the columns of the matrix EVEC into ascending or descending order
according to the value of the parameter SORT_TYPE as shown above.
Only 'GSL_EIGEN_SORT_ABS_ASC' and 'GSL_EIGEN_SORT_ABS_DESC' are
supported due to the eigenvalues being complex.
-- Function: int gsl_eigen_gensymmv_sort (gsl_vector * EVAL, gsl_matrix
* EVEC, gsl_eigen_sort_t SORT_TYPE)
This function simultaneously sorts the eigenvalues stored in the
vector EVAL and the corresponding real eigenvectors stored in the
columns of the matrix EVEC into ascending or descending order
according to the value of the parameter SORT_TYPE as shown above.
-- Function: int gsl_eigen_genhermv_sort (gsl_vector * EVAL,
gsl_matrix_complex * EVEC, gsl_eigen_sort_t SORT_TYPE)
This function simultaneously sorts the eigenvalues stored in the
vector EVAL and the corresponding complex eigenvectors stored in
the columns of the matrix EVEC into ascending or descending order
according to the value of the parameter SORT_TYPE as shown above.
-- Function: int gsl_eigen_genv_sort (gsl_vector_complex * ALPHA,
gsl_vector * BETA, gsl_matrix_complex * EVEC, gsl_eigen_sort_t
SORT_TYPE)
This function simultaneously sorts the eigenvalues stored in the
vectors (ALPHA, BETA) and the corresponding complex eigenvectors
stored in the columns of the matrix EVEC into ascending or
descending order according to the value of the parameter SORT_TYPE
as shown above. Only 'GSL_EIGEN_SORT_ABS_ASC' and
'GSL_EIGEN_SORT_ABS_DESC' are supported due to the eigenvalues
being complex.
15.8 Examples
=============
The following program computes the eigenvalues and eigenvectors of the
4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).
#include
#include
#include
int
main (void)
{
double data[] = { 1.0 , 1/2.0, 1/3.0, 1/4.0,
1/2.0, 1/3.0, 1/4.0, 1/5.0,
1/3.0, 1/4.0, 1/5.0, 1/6.0,
1/4.0, 1/5.0, 1/6.0, 1/7.0 };
gsl_matrix_view m
= gsl_matrix_view_array (data, 4, 4);
gsl_vector *eval = gsl_vector_alloc (4);
gsl_matrix *evec = gsl_matrix_alloc (4, 4);
gsl_eigen_symmv_workspace * w =
gsl_eigen_symmv_alloc (4);
gsl_eigen_symmv (&m.matrix, eval, evec, w);
gsl_eigen_symmv_free (w);
gsl_eigen_symmv_sort (eval, evec,
GSL_EIGEN_SORT_ABS_ASC);
{
int i;
for (i = 0; i < 4; i++)
{
double eval_i
= gsl_vector_get (eval, i);
gsl_vector_view evec_i
= gsl_matrix_column (evec, i);
printf ("eigenvalue = %g\n", eval_i);
printf ("eigenvector = \n");
gsl_vector_fprintf (stdout,
&evec_i.vector, "%g");
}
}
gsl_vector_free (eval);
gsl_matrix_free (evec);
return 0;
}
Here is the beginning of the output from the program,
$ ./a.out
eigenvalue = 9.67023e-05
eigenvector =
-0.0291933
0.328712
-0.791411
0.514553
...
This can be compared with the corresponding output from GNU OCTAVE,
octave> [v,d] = eig(hilb(4));
octave> diag(d)
ans =
9.6702e-05
6.7383e-03
1.6914e-01
1.5002e+00
octave> v
v =
0.029193 0.179186 -0.582076 0.792608
-0.328712 -0.741918 0.370502 0.451923
0.791411 0.100228 0.509579 0.322416
-0.514553 0.638283 0.514048 0.252161
Note that the eigenvectors can differ by a change of sign, since the
sign of an eigenvector is arbitrary.
The following program illustrates the use of the nonsymmetric
eigensolver, by computing the eigenvalues and eigenvectors of the
Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).
#include
#include
#include
int
main (void)
{
double data[] = { -1.0, 1.0, -1.0, 1.0,
-8.0, 4.0, -2.0, 1.0,
27.0, 9.0, 3.0, 1.0,
64.0, 16.0, 4.0, 1.0 };
gsl_matrix_view m
= gsl_matrix_view_array (data, 4, 4);
gsl_vector_complex *eval = gsl_vector_complex_alloc (4);
gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4);
gsl_eigen_nonsymmv_workspace * w =
gsl_eigen_nonsymmv_alloc (4);
gsl_eigen_nonsymmv (&m.matrix, eval, evec, w);
gsl_eigen_nonsymmv_free (w);
gsl_eigen_nonsymmv_sort (eval, evec,
GSL_EIGEN_SORT_ABS_DESC);
{
int i, j;
for (i = 0; i < 4; i++)
{
gsl_complex eval_i
= gsl_vector_complex_get (eval, i);
gsl_vector_complex_view evec_i
= gsl_matrix_complex_column (evec, i);
printf ("eigenvalue = %g + %gi\n",
GSL_REAL(eval_i), GSL_IMAG(eval_i));
printf ("eigenvector = \n");
for (j = 0; j < 4; ++j)
{
gsl_complex z =
gsl_vector_complex_get(&evec_i.vector, j);
printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z));
}
}
}
gsl_vector_complex_free(eval);
gsl_matrix_complex_free(evec);
return 0;
}
Here is the beginning of the output from the program,
$ ./a.out
eigenvalue = -6.41391 + 0i
eigenvector =
-0.0998822 + 0i
-0.111251 + 0i
0.292501 + 0i
0.944505 + 0i
eigenvalue = 5.54555 + 3.08545i
eigenvector =
-0.043487 + -0.0076308i
0.0642377 + -0.142127i
-0.515253 + 0.0405118i
-0.840592 + -0.00148565i
...
This can be compared with the corresponding output from GNU OCTAVE,
octave> [v,d] = eig(vander([-1 -2 3 4]));
octave> diag(d)
ans =
-6.4139 + 0.0000i
5.5456 + 3.0854i
5.5456 - 3.0854i
2.3228 + 0.0000i
octave> v
v =
Columns 1 through 3:
-0.09988 + 0.00000i -0.04350 - 0.00755i -0.04350 + 0.00755i
-0.11125 + 0.00000i 0.06399 - 0.14224i 0.06399 + 0.14224i
0.29250 + 0.00000i -0.51518 + 0.04142i -0.51518 - 0.04142i
0.94451 + 0.00000i -0.84059 + 0.00000i -0.84059 - 0.00000i
Column 4:
-0.14493 + 0.00000i
0.35660 + 0.00000i
0.91937 + 0.00000i
0.08118 + 0.00000i
Note that the eigenvectors corresponding to the eigenvalue 5.54555 +
3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i
which is an arbitrary phase factor of magnitude 1.
15.9 References and Further Reading
===================================
Further information on the algorithms described in this section can be
found in the following book,
G. H. Golub, C. F. Van Loan, 'Matrix Computations' (3rd Ed, 1996),
Johns Hopkins University Press, ISBN 0-8018-5414-8.
Further information on the generalized eigensystems QZ algorithm can be
found in this paper,
C. Moler, G. Stewart, "An Algorithm for Generalized Matrix
Eigenvalue Problems", SIAM J. Numer. Anal., Vol 10, No 2, 1973.
Eigensystem routines for very large matrices can be found in the Fortran
library LAPACK. The LAPACK library is described in,
'LAPACK Users' Guide' (Third Edition, 1999), Published by SIAM,
ISBN 0-89871-447-8.
The LAPACK source code can be found at the website above along with an
online copy of the users guide.
16 Fast Fourier Transforms (FFTs)
*********************************
This chapter describes functions for performing Fast Fourier Transforms
(FFTs). The library includes radix-2 routines (for lengths which are a
power of two) and mixed-radix routines (which work for any length). For
efficiency there are separate versions of the routines for real data and
for complex data. The mixed-radix routines are a reimplementation of
the FFTPACK library of Paul Swarztrauber. Fortran code for FFTPACK is
available on Netlib (FFTPACK also includes some routines for sine and
cosine transforms but these are currently not available in GSL). For
details and derivations of the underlying algorithms consult the
document 'GSL FFT Algorithms' (*note FFT References and Further
Reading::)
16.1 Mathematical Definitions
=============================
Fast Fourier Transforms are efficient algorithms for calculating the
discrete Fourier transform (DFT),
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
The DFT usually arises as an approximation to the continuous Fourier
transform when functions are sampled at discrete intervals in space or
time. The naive evaluation of the discrete Fourier transform is a
matrix-vector multiplication W\vec{z}. A general matrix-vector
multiplication takes O(n^2) operations for n data-points. Fast Fourier
transform algorithms use a divide-and-conquer strategy to factorize the
matrix W into smaller sub-matrices, corresponding to the integer factors
of the length n. If n can be factorized into a product of integers f_1
f_2 ... f_m then the DFT can be computed in O(n \sum f_i) operations.
For a radix-2 FFT this gives an operation count of O(n \log_2 n).
All the FFT functions offer three types of transform: forwards,
inverse and backwards, based on the same mathematical definitions. The
definition of the "forward Fourier transform", x = FFT(z), is,
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
and the definition of the "inverse Fourier transform", x = IFFT(z), is,
z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
The factor of 1/n makes this a true inverse. For example, a call to
'gsl_fft_complex_forward' followed by a call to
'gsl_fft_complex_inverse' should return the original data (within
numerical errors).
In general there are two possible choices for the sign of the
exponential in the transform/ inverse-transform pair. GSL follows the
same convention as FFTPACK, using a negative exponential for the forward
transform. The advantage of this convention is that the inverse
transform recreates the original function with simple Fourier synthesis.
Numerical Recipes uses the opposite convention, a positive exponential
in the forward transform.
The "backwards FFT" is simply our terminology for an unscaled version
of the inverse FFT,
z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
When the overall scale of the result is unimportant it is often
convenient to use the backwards FFT instead of the inverse to save
unnecessary divisions.
16.2 Overview of complex data FFTs
==================================
The inputs and outputs for the complex FFT routines are "packed arrays"
of floating point numbers. In a packed array the real and imaginary
parts of each complex number are placed in alternate neighboring
elements. For example, the following definition of a packed array of
length 6,
double x[3*2];
gsl_complex_packed_array data = x;
can be used to hold an array of three complex numbers, 'z[3]', in the
following way,
data[0] = Re(z[0])
data[1] = Im(z[0])
data[2] = Re(z[1])
data[3] = Im(z[1])
data[4] = Re(z[2])
data[5] = Im(z[2])
The array indices for the data have the same ordering as those in the
definition of the DFT--i.e. there are no index transformations or
permutations of the data.
A "stride" parameter allows the user to perform transforms on the
elements 'z[stride*i]' instead of 'z[i]'. A stride greater than 1 can
be used to take an in-place FFT of the column of a matrix. A stride of
1 accesses the array without any additional spacing between elements.
To perform an FFT on a vector argument, such as 'gsl_vector_complex *
v', use the following definitions (or their equivalents) when calling
the functions described in this chapter:
gsl_complex_packed_array data = v->data;
size_t stride = v->stride;
size_t n = v->size;
For physical applications it is important to remember that the index
appearing in the DFT does not correspond directly to a physical
frequency. If the time-step of the DFT is \Delta then the
frequency-domain includes both positive and negative frequencies,
ranging from -1/(2\Delta) through 0 to +1/(2\Delta). The positive
frequencies are stored from the beginning of the array up to the middle,
and the negative frequencies are stored backwards from the end of the
array.
Here is a table which shows the layout of the array DATA, and the
correspondence between the time-domain data z, and the frequency-domain
data x.
index z x = FFT(z)
0 z(t = 0) x(f = 0)
1 z(t = 1) x(f = 1/(n Delta))
2 z(t = 2) x(f = 2/(n Delta))
. ........ ..................
n/2 z(t = n/2) x(f = +1/(2 Delta),
-1/(2 Delta))
. ........ ..................
n-3 z(t = n-3) x(f = -3/(n Delta))
n-2 z(t = n-2) x(f = -2/(n Delta))
n-1 z(t = n-1) x(f = -1/(n Delta))
When n is even the location n/2 contains the most positive and negative
frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent. If n
is odd then general structure of the table above still applies, but n/2
does not appear.
16.3 Radix-2 FFT routines for complex data
==========================================
The radix-2 algorithms described in this section are simple and compact,
although not necessarily the most efficient. They use the Cooley-Tukey
algorithm to compute in-place complex FFTs for lengths which are a power
of 2--no additional storage is required. The corresponding self-sorting
mixed-radix routines offer better performance at the expense of
requiring additional working space.
All the functions described in this section are declared in the
header file 'gsl_fft_complex.h'.
-- Function: int gsl_fft_complex_radix2_forward
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
-- Function: int gsl_fft_complex_radix2_transform
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N,
gsl_fft_direction SIGN)
-- Function: int gsl_fft_complex_radix2_backward
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
-- Function: int gsl_fft_complex_radix2_inverse
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
These functions compute forward, backward and inverse FFTs of
length N with stride STRIDE, on the packed complex array DATA using
an in-place radix-2 decimation-in-time algorithm. The length of
the transform N is restricted to powers of two. For the
'transform' version of the function the SIGN argument can be either
'forward' (-1) or 'backward' (+1).
The functions return a value of 'GSL_SUCCESS' if no errors were
detected, or 'GSL_EDOM' if the length of the data N is not a power
of two.
-- Function: int gsl_fft_complex_radix2_dif_forward
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
-- Function: int gsl_fft_complex_radix2_dif_transform
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N,
gsl_fft_direction SIGN)
-- Function: int gsl_fft_complex_radix2_dif_backward
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
-- Function: int gsl_fft_complex_radix2_dif_inverse
(gsl_complex_packed_array DATA, size_t STRIDE, size_t N)
These are decimation-in-frequency versions of the radix-2 FFT
functions.
Here is an example program which computes the FFT of a short pulse in
a sample of length 128. To make the resulting Fourier transform real
the pulse is defined for equal positive and negative times (-10 ... 10),
where the negative times wrap around the end of the array.
#include
#include
#include
#include
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
int i; double data[2*128];
for (i = 0; i < 128; i++)
{
REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
}
REAL(data,0) = 1.0;
for (i = 1; i <= 10; i++)
{
REAL(data,i) = REAL(data,128-i) = 1.0;
}
for (i = 0; i < 128; i++)
{
printf ("%d %e %e\n", i,
REAL(data,i), IMAG(data,i));
}
printf ("\n");
gsl_fft_complex_radix2_forward (data, 1, 128);
for (i = 0; i < 128; i++)
{
printf ("%d %e %e\n", i,
REAL(data,i)/sqrt(128),
IMAG(data,i)/sqrt(128));
}
return 0;
}
Note that we have assumed that the program is using the default error
handler (which calls 'abort' for any errors). If you are not using a
safe error handler you would need to check the return status of
'gsl_fft_complex_radix2_forward'.
The transformed data is rescaled by 1/\sqrt n so that it fits on the
same plot as the input. Only the real part is shown, by the choice of
the input data the imaginary part is zero. Allowing for the wrap-around
of negative times at t=128, and working in units of k/n, the DFT
approximates the continuum Fourier transform, giving a modulated sine
function.
16.4 Mixed-radix FFT routines for complex data
==============================================
This section describes mixed-radix FFT algorithms for complex data. The
mixed-radix functions work for FFTs of any length. They are a
reimplementation of Paul Swarztrauber's Fortran FFTPACK library. The
theory is explained in the review article 'Self-sorting Mixed-radix
FFTs' by Clive Temperton. The routines here use the same indexing
scheme and basic algorithms as FFTPACK.
The mixed-radix algorithm is based on sub-transform modules--highly
optimized small length FFTs which are combined to create larger FFTs.
There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The
modules for the composite factors of 4 and 6 are faster than combining
the modules for 2*2 and 2*3.
For factors which are not implemented as modules there is a fall-back
to a general length-n module which uses Singleton's method for
efficiently computing a DFT. This module is O(n^2), and slower than a
dedicated module would be but works for any length n. Of course,
lengths which use the general length-n module will still be factorized
as much as possible. For example, a length of 143 will be factorized
into 11*13. Large prime factors are the worst case scenario, e.g. as
found in n=2*3*99991, and should be avoided because their O(n^2) scaling
will dominate the run-time (consult the document 'GSL FFT Algorithms'
included in the GSL distribution if you encounter this problem).
The mixed-radix initialization function
'gsl_fft_complex_wavetable_alloc' returns the list of factors chosen by
the library for a given length n. It can be used to check how well the
length has been factorized, and estimate the run-time. To a first
approximation the run-time scales as n \sum f_i, where the f_i are the
factors of n. For programs under user control you may wish to issue a
warning that the transform will be slow when the length is poorly
factorized. If you frequently encounter data lengths which cannot be
factorized using the existing small-prime modules consult 'GSL FFT
Algorithms' for details on adding support for other factors.
All the functions described in this section are declared in the
header file 'gsl_fft_complex.h'.
-- Function: gsl_fft_complex_wavetable *
gsl_fft_complex_wavetable_alloc (size_t N)
This function prepares a trigonometric lookup table for a complex
FFT of length N. The function returns a pointer to the newly
allocated 'gsl_fft_complex_wavetable' if no errors were detected,
and a null pointer in the case of error. The length N is
factorized into a product of subtransforms, and the factors and
their trigonometric coefficients are stored in the wavetable. The
trigonometric coefficients are computed using direct calls to 'sin'
and 'cos', for accuracy. Recursion relations could be used to
compute the lookup table faster, but if an application performs
many FFTs of the same length then this computation is a one-off
overhead which does not affect the final throughput.
The wavetable structure can be used repeatedly for any transform of
the same length. The table is not modified by calls to any of the
other FFT functions. The same wavetable can be used for both
forward and backward (or inverse) transforms of a given length.
-- Function: void gsl_fft_complex_wavetable_free
(gsl_fft_complex_wavetable * WAVETABLE)
This function frees the memory associated with the wavetable
WAVETABLE. The wavetable can be freed if no further FFTs of the
same length will be needed.
These functions operate on a 'gsl_fft_complex_wavetable' structure which
contains internal parameters for the FFT. It is not necessary to set any
of the components directly but it can sometimes be useful to examine
them. For example, the chosen factorization of the FFT length is given
and can be used to provide an estimate of the run-time or numerical
error. The wavetable structure is declared in the header file
'gsl_fft_complex.h'.
-- Data Type: gsl_fft_complex_wavetable
This is a structure that holds the factorization and trigonometric
lookup tables for the mixed radix fft algorithm. It has the
following components:
'size_t n'
This is the number of complex data points
'size_t nf'
This is the number of factors that the length 'n' was
decomposed into.
'size_t factor[64]'
This is the array of factors. Only the first 'nf' elements
are used.
'gsl_complex * trig'
This is a pointer to a preallocated trigonometric lookup table
of 'n' complex elements.
'gsl_complex * twiddle[64]'
This is an array of pointers into 'trig', giving the twiddle
factors for each pass.
The mixed radix algorithms require additional working space to hold the
intermediate steps of the transform.
-- Function: gsl_fft_complex_workspace *
gsl_fft_complex_workspace_alloc (size_t N)
This function allocates a workspace for a complex transform of
length N.
-- Function: void gsl_fft_complex_workspace_free
(gsl_fft_complex_workspace * WORKSPACE)
This function frees the memory associated with the workspace
WORKSPACE. The workspace can be freed if no further FFTs of the
same length will be needed.
The following functions compute the transform,
-- Function: int gsl_fft_complex_forward (gsl_complex_packed_array
DATA, size_t STRIDE, size_t N, const gsl_fft_complex_wavetable
* WAVETABLE, gsl_fft_complex_workspace * WORK)
-- Function: int gsl_fft_complex_transform (gsl_complex_packed_array
DATA, size_t STRIDE, size_t N, const gsl_fft_complex_wavetable
* WAVETABLE, gsl_fft_complex_workspace * WORK,
gsl_fft_direction SIGN)
-- Function: int gsl_fft_complex_backward (gsl_complex_packed_array
DATA, size_t STRIDE, size_t N, const gsl_fft_complex_wavetable
* WAVETABLE, gsl_fft_complex_workspace * WORK)
-- Function: int gsl_fft_complex_inverse (gsl_complex_packed_array
DATA, size_t STRIDE, size_t N, const gsl_fft_complex_wavetable
* WAVETABLE, gsl_fft_complex_workspace * WORK)
These functions compute forward, backward and inverse FFTs of
length N with stride STRIDE, on the packed complex array DATA,
using a mixed radix decimation-in-frequency algorithm. There is no
restriction on the length N. Efficient modules are provided for
subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors
are computed with a slow, O(n^2), general-n module. The caller
must supply a WAVETABLE containing the trigonometric lookup tables
and a workspace WORK. For the 'transform' version of the function
the SIGN argument can be either 'forward' (-1) or 'backward' (+1).
The functions return a value of '0' if no errors were detected.
The following 'gsl_errno' conditions are defined for these
functions:
'GSL_EDOM'
The length of the data N is not a positive integer (i.e. N is
zero).
'GSL_EINVAL'
The length of the data N and the length used to compute the
given WAVETABLE do not match.
Here is an example program which computes the FFT of a short pulse in
a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.
#include
#include
#include
#include
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
int i;
const int n = 630;
double data[2*n];
gsl_fft_complex_wavetable * wavetable;
gsl_fft_complex_workspace * workspace;
for (i = 0; i < n; i++)
{
REAL(data,i) = 0.0;
IMAG(data,i) = 0.0;
}
data[0] = 1.0;
for (i = 1; i <= 10; i++)
{
REAL(data,i) = REAL(data,n-i) = 1.0;
}
for (i = 0; i < n; i++)
{
printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));
}
printf ("\n");
wavetable = gsl_fft_complex_wavetable_alloc (n);
workspace = gsl_fft_complex_workspace_alloc (n);
for (i = 0; i < (int) wavetable->nf; i++)
{
printf ("# factor %d: %zu\n", i,
wavetable->factor[i]);
}
gsl_fft_complex_forward (data, 1, n,
wavetable, workspace);
for (i = 0; i < n; i++)
{
printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));
}
gsl_fft_complex_wavetable_free (wavetable);
gsl_fft_complex_workspace_free (workspace);
return 0;
}
Note that we have assumed that the program is using the default 'gsl'
error handler (which calls 'abort' for any errors). If you are not
using a safe error handler you would need to check the return status of
all the 'gsl' routines.
16.5 Overview of real data FFTs
===============================
The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms. The Fourier transform of a real sequence is not real. It
is a complex sequence with a special symmetry:
z_k = z_{n-k}^*
A sequence with this symmetry is called "conjugate-complex" or
"half-complex". This different structure requires different storage
layouts for the forward transform (from real to half-complex) and
inverse transform (from half-complex back to real). As a consequence
the routines are divided into two sets: functions in 'gsl_fft_real'
which operate on real sequences and functions in 'gsl_fft_halfcomplex'
which operate on half-complex sequences.
Functions in 'gsl_fft_real' compute the frequency coefficients of a
real sequence. The half-complex coefficients c of a real sequence x are
given by Fourier analysis,
c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)
Functions in 'gsl_fft_halfcomplex' compute inverse or backwards
transforms. They reconstruct real sequences by Fourier synthesis from
their half-complex frequency coefficients, c,
x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)
The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored. The remaining half can
be reconstructed using the half-complex symmetry condition. This works
for all lengths, even and odd--when the length is even the middle value
where k=n/2 is also real. Thus only N real numbers are required to
store the half-complex sequence, and the transform of a real sequence
can be stored in the same size array as the original data.
The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines. The radix-2 function
operates in-place, which constrains the locations where each element can
be stored. The restriction forces real and imaginary parts to be stored
far apart. The mixed-radix algorithm does not have this restriction,
and it stores the real and imaginary parts of a given term in
neighboring locations (which is desirable for better locality of memory
accesses).
16.6 Radix-2 FFT routines for real data
=======================================
This section describes radix-2 FFT algorithms for real data. They use
the Cooley-Tukey algorithm to compute in-place FFTs for lengths which
are a power of 2.
The radix-2 FFT functions for real data are declared in the header
files 'gsl_fft_real.h'
-- Function: int gsl_fft_real_radix2_transform (double DATA[], size_t
STRIDE, size_t N)
This function computes an in-place radix-2 FFT of length N and
stride STRIDE on the real array DATA. The output is a half-complex
sequence, which is stored in-place. The arrangement of the
half-complex terms uses the following scheme: for k < n/2 the real
part of the k-th term is stored in location k, and the
corresponding imaginary part is stored in location n-k. Terms with
k > n/2 can be reconstructed using the symmetry z_k = z^*_{n-k}.
The terms for k=0 and k=n/2 are both purely real, and count as a
special case. Their real parts are stored in locations 0 and n/2
respectively, while their imaginary parts which are zero are not
stored.
The following table shows the correspondence between the output
DATA and the equivalent results obtained by considering the input
data as a complex sequence with zero imaginary part (assuming
STRIDE=1),
complex[0].real = data[0]
complex[0].imag = 0
complex[1].real = data[1]
complex[1].imag = data[n-1]
............... ................
complex[k].real = data[k]
complex[k].imag = data[n-k]
............... ................
complex[n/2].real = data[n/2]
complex[n/2].imag = 0
............... ................
complex[k'].real = data[k] k' = n - k
complex[k'].imag = -data[n-k]
............... ................
complex[n-1].real = data[1]
complex[n-1].imag = -data[n-1]
Note that the output data can be converted into the full complex
sequence using the function 'gsl_fft_halfcomplex_radix2_unpack'
described below.
The radix-2 FFT functions for halfcomplex data are declared in the
header file 'gsl_fft_halfcomplex.h'.
-- Function: int gsl_fft_halfcomplex_radix2_inverse (double DATA[],
size_t STRIDE, size_t N)
-- Function: int gsl_fft_halfcomplex_radix2_backward (double DATA[],
size_t STRIDE, size_t N)
These functions compute the inverse or backwards in-place radix-2
FFT of length N and stride STRIDE on the half-complex sequence DATA
stored according the output scheme used by 'gsl_fft_real_radix2'.
The result is a real array stored in natural order.
-- Function: int gsl_fft_halfcomplex_radix2_unpack (const double
HALFCOMPLEX_COEFFICIENT[], gsl_complex_packed_array
COMPLEX_COEFFICIENT, size_t STRIDE, size_t N)
This function converts HALFCOMPLEX_COEFFICIENT, an array of
half-complex coefficients as returned by
'gsl_fft_real_radix2_transform', into an ordinary complex array,
COMPLEX_COEFFICIENT. It fills in the complex array using the
symmetry z_k = z_{n-k}^* to reconstruct the redundant elements.
The algorithm for the conversion is,
complex_coefficient[0].real
= halfcomplex_coefficient[0];
complex_coefficient[0].imag
= 0.0;
for (i = 1; i < n - i; i++)
{
double hc_real
= halfcomplex_coefficient[i*stride];
double hc_imag
= halfcomplex_coefficient[(n-i)*stride];
complex_coefficient[i*stride].real = hc_real;
complex_coefficient[i*stride].imag = hc_imag;
complex_coefficient[(n - i)*stride].real = hc_real;
complex_coefficient[(n - i)*stride].imag = -hc_imag;
}
if (i == n - i)
{
complex_coefficient[i*stride].real
= halfcomplex_coefficient[(n - 1)*stride];
complex_coefficient[i*stride].imag
= 0.0;
}
16.7 Mixed-radix FFT routines for real data
===========================================
This section describes mixed-radix FFT algorithms for real data. The
mixed-radix functions work for FFTs of any length. They are a
reimplementation of the real-FFT routines in the Fortran FFTPACK library
by Paul Swarztrauber. The theory behind the algorithm is explained in
the article 'Fast Mixed-Radix Real Fourier Transforms' by Clive
Temperton. The routines here use the same indexing scheme and basic
algorithms as FFTPACK.
The functions use the FFTPACK storage convention for half-complex
sequences. In this convention the half-complex transform of a real
sequence is stored with frequencies in increasing order, starting at
zero, with the real and imaginary parts of each frequency in neighboring
locations. When a value is known to be real the imaginary part is not
stored. The imaginary part of the zero-frequency component is never
stored. It is known to be zero (since the zero frequency component is
simply the sum of the input data (all real)). For a sequence of even
length the imaginary part of the frequency n/2 is not stored either,
since the symmetry z_k = z_{n-k}^* implies that this is purely real too.
The storage scheme is best shown by some examples. The table below
shows the output for an odd-length sequence, n=5. The two columns give
the correspondence between the 5 values in the half-complex sequence
returned by 'gsl_fft_real_transform', HALFCOMPLEX[] and the values
COMPLEX[] that would be returned if the same real input sequence were
passed to 'gsl_fft_complex_backward' as a complex sequence (with
imaginary parts set to '0'),
complex[0].real = halfcomplex[0]
complex[0].imag = 0
complex[1].real = halfcomplex[1]
complex[1].imag = halfcomplex[2]
complex[2].real = halfcomplex[3]
complex[2].imag = halfcomplex[4]
complex[3].real = halfcomplex[3]
complex[3].imag = -halfcomplex[4]
complex[4].real = halfcomplex[1]
complex[4].imag = -halfcomplex[2]
The upper elements of the COMPLEX array, 'complex[3]' and 'complex[4]'
are filled in using the symmetry condition. The imaginary part of the
zero-frequency term 'complex[0].imag' is known to be zero by the
symmetry.
The next table shows the output for an even-length sequence, n=6. In
the even case there are two values which are purely real,
complex[0].real = halfcomplex[0]
complex[0].imag = 0
complex[1].real = halfcomplex[1]
complex[1].imag = halfcomplex[2]
complex[2].real = halfcomplex[3]
complex[2].imag = halfcomplex[4]
complex[3].real = halfcomplex[5]
complex[3].imag = 0
complex[4].real = halfcomplex[3]
complex[4].imag = -halfcomplex[4]
complex[5].real = halfcomplex[1]
complex[5].imag = -halfcomplex[2]
The upper elements of the COMPLEX array, 'complex[4]' and 'complex[5]'
are filled in using the symmetry condition. Both 'complex[0].imag' and
'complex[3].imag' are known to be zero.
All these functions are declared in the header files 'gsl_fft_real.h'
and 'gsl_fft_halfcomplex.h'.
-- Function: gsl_fft_real_wavetable * gsl_fft_real_wavetable_alloc
(size_t N)
-- Function: gsl_fft_halfcomplex_wavetable *
gsl_fft_halfcomplex_wavetable_alloc (size_t N)
These functions prepare trigonometric lookup tables for an FFT of
size n real elements. The functions return a pointer to the newly
allocated struct if no errors were detected, and a null pointer in
the case of error. The length N is factorized into a product of
subtransforms, and the factors and their trigonometric coefficients
are stored in the wavetable. The trigonometric coefficients are
computed using direct calls to 'sin' and 'cos', for accuracy.
Recursion relations could be used to compute the lookup table
faster, but if an application performs many FFTs of the same length
then computing the wavetable is a one-off overhead which does not
affect the final throughput.
The wavetable structure can be used repeatedly for any transform of
the same length. The table is not modified by calls to any of the
other FFT functions. The appropriate type of wavetable must be
used for forward real or inverse half-complex transforms.
-- Function: void gsl_fft_real_wavetable_free (gsl_fft_real_wavetable *
WAVETABLE)
-- Function: void gsl_fft_halfcomplex_wavetable_free
(gsl_fft_halfcomplex_wavetable * WAVETABLE)
These functions free the memory associated with the wavetable
WAVETABLE. The wavetable can be freed if no further FFTs of the
same length will be needed.
The mixed radix algorithms require additional working space to hold the
intermediate steps of the transform,
-- Function: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc
(size_t N)
This function allocates a workspace for a real transform of length
N. The same workspace can be used for both forward real and
inverse halfcomplex transforms.
-- Function: void gsl_fft_real_workspace_free (gsl_fft_real_workspace *
WORKSPACE)
This function frees the memory associated with the workspace
WORKSPACE. The workspace can be freed if no further FFTs of the
same length will be needed.
The following functions compute the transforms of real and half-complex
data,
-- Function: int gsl_fft_real_transform (double DATA[], size_t STRIDE,
size_t N, const gsl_fft_real_wavetable * WAVETABLE,
gsl_fft_real_workspace * WORK)
-- Function: int gsl_fft_halfcomplex_transform (double DATA[], size_t
STRIDE, size_t N, const gsl_fft_halfcomplex_wavetable *
WAVETABLE, gsl_fft_real_workspace * WORK)
These functions compute the FFT of DATA, a real or half-complex
array of length N, using a mixed radix decimation-in-frequency
algorithm. For 'gsl_fft_real_transform' DATA is an array of
time-ordered real data. For 'gsl_fft_halfcomplex_transform' DATA
contains Fourier coefficients in the half-complex ordering
described above. There is no restriction on the length N.
Efficient modules are provided for subtransforms of length 2, 3, 4
and 5. Any remaining factors are computed with a slow, O(n^2),
general-n module. The caller must supply a WAVETABLE containing
trigonometric lookup tables and a workspace WORK.
-- Function: int gsl_fft_real_unpack (const double REAL_COEFFICIENT[],
gsl_complex_packed_array COMPLEX_COEFFICIENT, size_t STRIDE,
size_t N)
This function converts a single real array, REAL_COEFFICIENT into
an equivalent complex array, COMPLEX_COEFFICIENT, (with imaginary
part set to zero), suitable for 'gsl_fft_complex' routines. The
algorithm for the conversion is simply,
for (i = 0; i < n; i++)
{
complex_coefficient[i*stride].real
= real_coefficient[i*stride];
complex_coefficient[i*stride].imag
= 0.0;
}
-- Function: int gsl_fft_halfcomplex_unpack (const double
HALFCOMPLEX_COEFFICIENT[], gsl_complex_packed_array
COMPLEX_COEFFICIENT, size_t STRIDE, size_t N)
This function converts HALFCOMPLEX_COEFFICIENT, an array of
half-complex coefficients as returned by 'gsl_fft_real_transform',
into an ordinary complex array, COMPLEX_COEFFICIENT. It fills in
the complex array using the symmetry z_k = z_{n-k}^* to reconstruct
the redundant elements. The algorithm for the conversion is,
complex_coefficient[0].real
= halfcomplex_coefficient[0];
complex_coefficient[0].imag
= 0.0;
for (i = 1; i < n - i; i++)
{
double hc_real
= halfcomplex_coefficient[(2 * i - 1)*stride];
double hc_imag
= halfcomplex_coefficient[(2 * i)*stride];
complex_coefficient[i*stride].real = hc_real;
complex_coefficient[i*stride].imag = hc_imag;
complex_coefficient[(n - i)*stride].real = hc_real;
complex_coefficient[(n - i)*stride].imag = -hc_imag;
}
if (i == n - i)
{
complex_coefficient[i*stride].real
= halfcomplex_coefficient[(n - 1)*stride];
complex_coefficient[i*stride].imag
= 0.0;
}
Here is an example program using 'gsl_fft_real_transform' and
'gsl_fft_halfcomplex_inverse'. It generates a real signal in the shape
of a square pulse. The pulse is Fourier transformed to frequency space,
and all but the lowest ten frequency components are removed from the
array of Fourier coefficients returned by 'gsl_fft_real_transform'.
The remaining Fourier coefficients are transformed back to the
time-domain, to give a filtered version of the square pulse. Since
Fourier coefficients are stored using the half-complex symmetry both
positive and negative frequencies are removed and the final filtered
signal is also real.
#include
#include
#include
#include
#include
int
main (void)
{
int i, n = 100;
double data[n];
gsl_fft_real_wavetable * real;
gsl_fft_halfcomplex_wavetable * hc;
gsl_fft_real_workspace * work;
for (i = 0; i < n; i++)
{
data[i] = 0.0;
}
for (i = n / 3; i < 2 * n / 3; i++)
{
data[i] = 1.0;
}
for (i = 0; i < n; i++)
{
printf ("%d: %e\n", i, data[i]);
}
printf ("\n");
work = gsl_fft_real_workspace_alloc (n);
real = gsl_fft_real_wavetable_alloc (n);
gsl_fft_real_transform (data, 1, n,
real, work);
gsl_fft_real_wavetable_free (real);
for (i = 11; i < n; i++)
{
data[i] = 0;
}
hc = gsl_fft_halfcomplex_wavetable_alloc (n);
gsl_fft_halfcomplex_inverse (data, 1, n,
hc, work);
gsl_fft_halfcomplex_wavetable_free (hc);
for (i = 0; i < n; i++)
{
printf ("%d: %e\n", i, data[i]);
}
gsl_fft_real_workspace_free (work);
return 0;
}
16.8 References and Further Reading
===================================
A good starting point for learning more about the FFT is the review
article 'Fast Fourier Transforms: A Tutorial Review and A State of the
Art' by Duhamel and Vetterli,
P. Duhamel and M. Vetterli. Fast Fourier transforms: A tutorial
review and a state of the art. 'Signal Processing', 19:259-299,
1990.
To find out about the algorithms used in the GSL routines you may want
to consult the document 'GSL FFT Algorithms' (it is included in GSL, as
'doc/fftalgorithms.tex'). This has general information on FFTs and
explicit derivations of the implementation for each routine. There are
also references to the relevant literature. For convenience some of the
more important references are reproduced below.
There are several introductory books on the FFT with example programs,
such as 'The Fast Fourier Transform' by Brigham and 'DFT/FFT and
Convolution Algorithms' by Burrus and Parks,
E. Oran Brigham. 'The Fast Fourier Transform'. Prentice Hall,
1974.
C. S. Burrus and T. W. Parks. 'DFT/FFT and Convolution
Algorithms'. Wiley, 1984.
Both these introductory books cover the radix-2 FFT in some detail. The
mixed-radix algorithm at the heart of the FFTPACK routines is reviewed
in Clive Temperton's paper,
Clive Temperton. Self-sorting mixed-radix fast Fourier transforms.
'Journal of Computational Physics', 52(1):1-23, 1983.
The derivation of FFTs for real-valued data is explained in the
following two articles,
Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C.
Sidney Burrus. Real-valued fast Fourier transform algorithms.
'IEEE Transactions on Acoustics, Speech, and Signal Processing',
ASSP-35(6):849-863, 1987.
Clive Temperton. Fast mixed-radix real Fourier transforms.
'Journal of Computational Physics', 52:340-350, 1983.
In 1979 the IEEE published a compendium of carefully-reviewed Fortran
FFT programs in 'Programs for Digital Signal Processing'. It is a
useful reference for implementations of many different FFT algorithms,
Digital Signal Processing Committee and IEEE Acoustics, Speech, and
Signal Processing Committee, editors. 'Programs for Digital Signal
Processing'. IEEE Press, 1979.
For large-scale FFT work we recommend the use of the dedicated FFTW
library by Frigo and Johnson. The FFTW library is self-optimizing--it
automatically tunes itself for each hardware platform in order to
achieve maximum performance. It is available under the GNU GPL.
FFTW Website,
The source code for FFTPACK is available from Netlib,
FFTPACK,
17 Numerical Integration
************************
This chapter describes routines for performing numerical integration
(quadrature) of a function in one dimension. There are routines for
adaptive and non-adaptive integration of general functions, with
specialised routines for specific cases. These include integration over
infinite and semi-infinite ranges, singular integrals, including
logarithmic singularities, computation of Cauchy principal values and
oscillatory integrals. The library reimplements the algorithms used in
QUADPACK, a numerical integration package written by Piessens, de
Doncker-Kapenga, Ueberhuber and Kahaner. Fortran code for QUADPACK is
available on Netlib. Also included are non-adaptive, fixed-order
Gauss-Legendre integration routines with high precision coefficients by
Pavel Holoborodko.
The functions described in this chapter are declared in the header
file 'gsl_integration.h'.
17.1 Introduction
=================
Each algorithm computes an approximation to a definite integral of the
form,
I = \int_a^b f(x) w(x) dx
where w(x) is a weight function (for general integrands w(x)=1). The
user provides absolute and relative error bounds (epsabs, epsrel) which
specify the following accuracy requirement,
|RESULT - I| <= max(epsabs, epsrel |I|)
where RESULT is the numerical approximation obtained by the algorithm.
The algorithms attempt to estimate the absolute error ABSERR = |RESULT -
I| in such a way that the following inequality holds,
|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
In short, the routines return the first approximation which has an
absolute error smaller than epsabs or a relative error smaller than
epsrel.
Note that this is an either-or constraint, not simultaneous. To
compute to a specified absolute error, set epsrel to zero. To compute
to a specified relative error, set epsabs to zero. The routines will
fail to converge if the error bounds are too stringent, but always
return the best approximation obtained up to that stage.
The algorithms in QUADPACK use a naming convention based on the
following letters,
'Q' - quadrature routine
'N' - non-adaptive integrator
'A' - adaptive integrator
'G' - general integrand (user-defined)
'W' - weight function with integrand
'S' - singularities can be more readily integrated
'P' - points of special difficulty can be supplied
'I' - infinite range of integration
'O' - oscillatory weight function, cos or sin
'F' - Fourier integral
'C' - Cauchy principal value
The algorithms are built on pairs of quadrature rules, a higher order
rule and a lower order rule. The higher order rule is used to compute
the best approximation to an integral over a small range. The
difference between the results of the higher order rule and the lower
order rule gives an estimate of the error in the approximation.
17.1.1 Integrands without weight functions
------------------------------------------
The algorithms for general functions (without a weight function) are
based on Gauss-Kronrod rules.
A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule
of order m. This is extended with additional points between each of the
abscissae to give a higher order Kronrod rule of order 2m+1. The
Kronrod rule is efficient because it reuses existing function
evaluations from the Gaussian rule.
The higher order Kronrod rule is used as the best approximation to
the integral, and the difference between the two rules is used as an
estimate of the error in the approximation.
17.1.2 Integrands with weight functions
---------------------------------------
For integrands with weight functions the algorithms use Clenshaw-Curtis
quadrature rules.
A Clenshaw-Curtis rule begins with an n-th order Chebyshev polynomial
approximation to the integrand. This polynomial can be integrated
exactly to give an approximation to the integral of the original
function. The Chebyshev expansion can be extended to higher orders to
improve the approximation and provide an estimate of the error.
17.1.3 Integrands with singular weight functions
------------------------------------------------
The presence of singularities (or other behavior) in the integrand can
cause slow convergence in the Chebyshev approximation. The modified
Clenshaw-Curtis rules used in QUADPACK separate out several common
weight functions which cause slow convergence.
These weight functions are integrated analytically against the
Chebyshev polynomials to precompute "modified Chebyshev moments".
Combining the moments with the Chebyshev approximation to the function
gives the desired integral. The use of analytic integration for the
singular part of the function allows exact cancellations and
substantially improves the overall convergence behavior of the
integration.
17.2 QNG non-adaptive Gauss-Kronrod integration
===============================================
The QNG algorithm is a non-adaptive procedure which uses fixed
Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum
of 87 points. It is provided for fast integration of smooth functions.
-- Function: int gsl_integration_qng (const gsl_function * F, double A,
double B, double EPSABS, double EPSREL, double * RESULT,
double * ABSERR, size_t * NEVAL)
This function applies the Gauss-Kronrod 10-point, 21-point,
43-point and 87-point integration rules in succession until an
estimate of the integral of f over (a,b) is achieved within the
desired absolute and relative error limits, EPSABS and EPSREL. The
function returns the final approximation, RESULT, an estimate of
the absolute error, ABSERR and the number of function evaluations
used, NEVAL. The Gauss-Kronrod rules are designed in such a way
that each rule uses all the results of its predecessors, in order
to minimize the total number of function evaluations.
17.3 QAG adaptive integration
=============================
The QAG algorithm is a simple adaptive integration procedure. The
integration region is divided into subintervals, and on each iteration
the subinterval with the largest estimated error is bisected. This
reduces the overall error rapidly, as the subintervals become
concentrated around local difficulties in the integrand. These
subintervals are managed by a 'gsl_integration_workspace' struct, which
handles the memory for the subinterval ranges, results and error
estimates.
-- Function: gsl_integration_workspace *
gsl_integration_workspace_alloc (size_t N)
This function allocates a workspace sufficient to hold N double
precision intervals, their integration results and error estimates.
One workspace may be used multiple times as all necessary
reinitialization is performed automatically by the integration
routines.
-- Function: void gsl_integration_workspace_free
(gsl_integration_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_integration_qag (const gsl_function * F, double A,
double B, double EPSABS, double EPSREL, size_t LIMIT, int KEY,
gsl_integration_workspace * WORKSPACE, double * RESULT, double
* ABSERR)
This function applies an integration rule adaptively until an
estimate of the integral of f over (a,b) is achieved within the
desired absolute and relative error limits, EPSABS and EPSREL. The
function returns the final approximation, RESULT, and an estimate
of the absolute error, ABSERR. The integration rule is determined
by the value of KEY, which should be chosen from the following
symbolic names,
GSL_INTEG_GAUSS15 (key = 1)
GSL_INTEG_GAUSS21 (key = 2)
GSL_INTEG_GAUSS31 (key = 3)
GSL_INTEG_GAUSS41 (key = 4)
GSL_INTEG_GAUSS51 (key = 5)
GSL_INTEG_GAUSS61 (key = 6)
corresponding to the 15, 21, 31, 41, 51 and 61 point Gauss-Kronrod
rules. The higher-order rules give better accuracy for smooth
functions, while lower-order rules save time when the function
contains local difficulties, such as discontinuities.
On each iteration the adaptive integration strategy bisects the
interval with the largest error estimate. The subintervals and
their results are stored in the memory provided by WORKSPACE. The
maximum number of subintervals is given by LIMIT, which may not
exceed the allocated size of the workspace.
17.4 QAGS adaptive integration with singularities
=================================================
The presence of an integrable singularity in the integration region
causes an adaptive routine to concentrate new subintervals around the
singularity. As the subintervals decrease in size the successive
approximations to the integral converge in a limiting fashion. This
approach to the limit can be accelerated using an extrapolation
procedure. The QAGS algorithm combines adaptive bisection with the Wynn
epsilon-algorithm to speed up the integration of many types of
integrable singularities.
-- Function: int gsl_integration_qags (const gsl_function * F, double
A, double B, double EPSABS, double EPSREL, size_t LIMIT,
gsl_integration_workspace * WORKSPACE, double * RESULT, double
* ABSERR)
This function applies the Gauss-Kronrod 21-point integration rule
adaptively until an estimate of the integral of f over (a,b) is
achieved within the desired absolute and relative error limits,
EPSABS and EPSREL. The results are extrapolated using the
epsilon-algorithm, which accelerates the convergence of the
integral in the presence of discontinuities and integrable
singularities. The function returns the final approximation from
the extrapolation, RESULT, and an estimate of the absolute error,
ABSERR. The subintervals and their results are stored in the
memory provided by WORKSPACE. The maximum number of subintervals
is given by LIMIT, which may not exceed the allocated size of the
workspace.
17.5 QAGP adaptive integration with known singular points
=========================================================
-- Function: int gsl_integration_qagp (const gsl_function * F, double *
PTS, size_t NPTS, double EPSABS, double EPSREL, size_t LIMIT,
gsl_integration_workspace * WORKSPACE, double * RESULT, double
* ABSERR)
This function applies the adaptive integration algorithm QAGS
taking account of the user-supplied locations of singular points.
The array PTS of length NPTS should contain the endpoints of the
integration ranges defined by the integration region and locations
of the singularities. For example, to integrate over the region
(a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 < x_3
< b) the following PTS array should be used
pts[0] = a
pts[1] = x_1
pts[2] = x_2
pts[3] = x_3
pts[4] = b
with NPTS = 5.
If you know the locations of the singular points in the integration
region then this routine will be faster than 'QAGS'.
17.6 QAGI adaptive integration on infinite intervals
====================================================
-- Function: int gsl_integration_qagi (gsl_function * F, double EPSABS,
double EPSREL, size_t LIMIT, gsl_integration_workspace *
WORKSPACE, double * RESULT, double * ABSERR)
This function computes the integral of the function F over the
infinite interval (-\infty,+\infty). The integral is mapped onto
the semi-open interval (0,1] using the transformation x = (1-t)/t,
\int_{-\infty}^{+\infty} dx f(x) =
\int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.
It is then integrated using the QAGS algorithm. The normal
21-point Gauss-Kronrod rule of QAGS is replaced by a 15-point rule,
because the transformation can generate an integrable singularity
at the origin. In this case a lower-order rule is more efficient.
-- Function: int gsl_integration_qagiu (gsl_function * F, double A,
double EPSABS, double EPSREL, size_t LIMIT,
gsl_integration_workspace * WORKSPACE, double * RESULT, double
* ABSERR)
This function computes the integral of the function F over the
semi-infinite interval (a,+\infty). The integral is mapped onto
the semi-open interval (0,1] using the transformation x = a +
(1-t)/t,
\int_{a}^{+\infty} dx f(x) =
\int_0^1 dt f(a + (1-t)/t)/t^2
and then integrated using the QAGS algorithm.
-- Function: int gsl_integration_qagil (gsl_function * F, double B,
double EPSABS, double EPSREL, size_t LIMIT,
gsl_integration_workspace * WORKSPACE, double * RESULT, double
* ABSERR)
This function computes the integral of the function F over the
semi-infinite interval (-\infty,b). The integral is mapped onto
the semi-open interval (0,1] using the transformation x = b -
(1-t)/t,
\int_{-\infty}^{b} dx f(x) =
\int_0^1 dt f(b - (1-t)/t)/t^2
and then integrated using the QAGS algorithm.
17.7 QAWC adaptive integration for Cauchy principal values
==========================================================
-- Function: int gsl_integration_qawc (gsl_function * F, double A,
double B, double C, double EPSABS, double EPSREL, size_t
LIMIT, gsl_integration_workspace * WORKSPACE, double * RESULT,
double * ABSERR)
This function computes the Cauchy principal value of the integral
of f over (a,b), with a singularity at C,
I = \int_a^b dx f(x) / (x - c)
The adaptive bisection algorithm of QAG is used, with modifications
to ensure that subdivisions do not occur at the singular point x =
c. When a subinterval contains the point x = c or is close to it
then a special 25-point modified Clenshaw-Curtis rule is used to
control the singularity. Further away from the singularity the
algorithm uses an ordinary 15-point Gauss-Kronrod integration rule.
17.8 QAWS adaptive integration for singular functions
=====================================================
The QAWS algorithm is designed for integrands with algebraic-logarithmic
singularities at the end-points of an integration region. In order to
work efficiently the algorithm requires a precomputed table of Chebyshev
moments.
-- Function: gsl_integration_qaws_table *
gsl_integration_qaws_table_alloc (double ALPHA, double BETA,
int MU, int NU)
This function allocates space for a 'gsl_integration_qaws_table'
struct describing a singular weight function W(x) with the
parameters (\alpha, \beta, \mu, \nu),
W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1. The
weight function can take four different forms depending on the
values of \mu and \nu,
W(x) = (x-a)^alpha (b-x)^beta (mu = 0, nu = 0)
W(x) = (x-a)^alpha (b-x)^beta log(x-a) (mu = 1, nu = 0)
W(x) = (x-a)^alpha (b-x)^beta log(b-x) (mu = 0, nu = 1)
W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)
The singular points (a,b) do not have to be specified until the
integral is computed, where they are the endpoints of the
integration range.
The function returns a pointer to the newly allocated table
'gsl_integration_qaws_table' if no errors were detected, and 0 in
the case of error.
-- Function: int gsl_integration_qaws_table_set
(gsl_integration_qaws_table * T, double ALPHA, double BETA,
int MU, int NU)
This function modifies the parameters (\alpha, \beta, \mu, \nu) of
an existing 'gsl_integration_qaws_table' struct T.
-- Function: void gsl_integration_qaws_table_free
(gsl_integration_qaws_table * T)
This function frees all the memory associated with the
'gsl_integration_qaws_table' struct T.
-- Function: int gsl_integration_qaws (gsl_function * F, const double
A, const double B, gsl_integration_qaws_table * T, const
double EPSABS, const double EPSREL, const size_t LIMIT,
gsl_integration_workspace * WORKSPACE, double * RESULT, double
* ABSERR)
This function computes the integral of the function f(x) over the
interval (a,b) with the singular weight function (x-a)^\alpha
(b-x)^\beta \log^\mu (x-a) \log^\nu (b-x). The parameters of the
weight function (\alpha, \beta, \mu, \nu) are taken from the table
T. The integral is,
I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).
The adaptive bisection algorithm of QAG is used. When a
subinterval contains one of the endpoints then a special 25-point
modified Clenshaw-Curtis rule is used to control the singularities.
For subintervals which do not include the endpoints an ordinary
15-point Gauss-Kronrod integration rule is used.
17.9 QAWO adaptive integration for oscillatory functions
========================================================
The QAWO algorithm is designed for integrands with an oscillatory
factor, \sin(\omega x) or \cos(\omega x). In order to work efficiently
the algorithm requires a table of Chebyshev moments which must be
pre-computed with calls to the functions below.
-- Function: gsl_integration_qawo_table *
gsl_integration_qawo_table_alloc (double OMEGA, double L, enum
gsl_integration_qawo_enum SINE, size_t N)
This function allocates space for a 'gsl_integration_qawo_table'
struct and its associated workspace describing a sine or cosine
weight function W(x) with the parameters (\omega, L),
W(x) = sin(omega x)
W(x) = cos(omega x)
The parameter L must be the length of the interval over which the
function will be integrated L = b - a. The choice of sine or
cosine is made with the parameter SINE which should be chosen from
one of the two following symbolic values:
GSL_INTEG_COSINE
GSL_INTEG_SINE
The 'gsl_integration_qawo_table' is a table of the trigonometric
coefficients required in the integration process. The parameter N
determines the number of levels of coefficients that are computed.
Each level corresponds to one bisection of the interval L, so that
N levels are sufficient for subintervals down to the length L/2^n.
The integration routine 'gsl_integration_qawo' returns the error
'GSL_ETABLE' if the number of levels is insufficient for the
requested accuracy.
-- Function: int gsl_integration_qawo_table_set
(gsl_integration_qawo_table * T, double OMEGA, double L, enum
gsl_integration_qawo_enum SINE)
This function changes the parameters OMEGA, L and SINE of the
existing workspace T.
-- Function: int gsl_integration_qawo_table_set_length
(gsl_integration_qawo_table * T, double L)
This function allows the length parameter L of the workspace T to
be changed.
-- Function: void gsl_integration_qawo_table_free
(gsl_integration_qawo_table * T)
This function frees all the memory associated with the workspace T.
-- Function: int gsl_integration_qawo (gsl_function * F, const double
A, const double EPSABS, const double EPSREL, const size_t
LIMIT, gsl_integration_workspace * WORKSPACE,
gsl_integration_qawo_table * WF, double * RESULT, double *
ABSERR)
This function uses an adaptive algorithm to compute the integral of
f over (a,b) with the weight function \sin(\omega x) or \cos(\omega
x) defined by the table WF,
I = \int_a^b dx f(x) sin(omega x)
I = \int_a^b dx f(x) cos(omega x)
The results are extrapolated using the epsilon-algorithm to
accelerate the convergence of the integral. The function returns
the final approximation from the extrapolation, RESULT, and an
estimate of the absolute error, ABSERR. The subintervals and their
results are stored in the memory provided by WORKSPACE. The
maximum number of subintervals is given by LIMIT, which may not
exceed the allocated size of the workspace.
Those subintervals with "large" widths d where d\omega > 4 are
computed using a 25-point Clenshaw-Curtis integration rule, which
handles the oscillatory behavior. Subintervals with a "small"
widths where d\omega < 4 are computed using a 15-point
Gauss-Kronrod integration.
17.10 QAWF adaptive integration for Fourier integrals
=====================================================
-- Function: int gsl_integration_qawf (gsl_function * F, const double
A, const double EPSABS, const size_t LIMIT,
gsl_integration_workspace * WORKSPACE,
gsl_integration_workspace * CYCLE_WORKSPACE,
gsl_integration_qawo_table * WF, double * RESULT, double *
ABSERR)
This function attempts to compute a Fourier integral of the
function F over the semi-infinite interval [a,+\infty).
I = \int_a^{+\infty} dx f(x) sin(omega x)
I = \int_a^{+\infty} dx f(x) cos(omega x)
The parameter \omega and choice of \sin or \cos is taken from the
table WF (the length L can take any value, since it is overridden
by this function to a value appropriate for the Fourier
integration). The integral is computed using the QAWO algorithm
over each of the subintervals,
C_1 = [a, a + c]
C_2 = [a + c, a + 2 c]
... = ...
C_k = [a + (k-1) c, a + k c]
where c = (2 floor(|\omega|) + 1) \pi/|\omega|. The width c is
chosen to cover an odd number of periods so that the contributions
from the intervals alternate in sign and are monotonically
decreasing when F is positive and monotonically decreasing. The
sum of this sequence of contributions is accelerated using the
epsilon-algorithm.
This function works to an overall absolute tolerance of ABSERR.
The following strategy is used: on each interval C_k the algorithm
tries to achieve the tolerance
TOL_k = u_k abserr
where u_k = (1 - p)p^{k-1} and p = 9/10. The sum of the geometric
series of contributions from each interval gives an overall
tolerance of ABSERR.
If the integration of a subinterval leads to difficulties then the
accuracy requirement for subsequent intervals is relaxed,
TOL_k = u_k max(abserr, max_{in - 1], this function obtains the I-th
Gauss-Legendre point XI and weight WI on the interval [A,B]. The
points and weights are ordered by increasing point value. A
function f may be integrated on [A,B] by summing wi * f(xi) over I.
-- Function: void gsl_integration_glfixed_table_free
(gsl_integration_glfixed_table * T)
This function frees the memory associated with the table T.
17.13 Error codes
=================
In addition to the standard error codes for invalid arguments the
functions can return the following values,
'GSL_EMAXITER'
the maximum number of subdivisions was exceeded.
'GSL_EROUND'
cannot reach tolerance because of roundoff error, or roundoff error
was detected in the extrapolation table.
'GSL_ESING'
a non-integrable singularity or other bad integrand behavior was
found in the integration interval.
'GSL_EDIVERGE'
the integral is divergent, or too slowly convergent to be
integrated numerically.
17.14 Examples
==============
The integrator 'QAGS' will handle a large class of definite integrals.
For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,
\int_0^1 x^{-1/2} log(x) dx = -4
The program below computes this integral to a relative accuracy bound of
'1e-7'.
#include
#include
#include
double f (double x, void * params) {
double alpha = *(double *) params;
double f = log(alpha*x) / sqrt(x);
return f;
}
int
main (void)
{
gsl_integration_workspace * w
= gsl_integration_workspace_alloc (1000);
double result, error;
double expected = -4.0;
double alpha = 1.0;
gsl_function F;
F.function = &f;
F.params = α
gsl_integration_qags (&F, 0, 1, 0, 1e-7, 1000,
w, &result, &error);
printf ("result = % .18f\n", result);
printf ("exact result = % .18f\n", expected);
printf ("estimated error = % .18f\n", error);
printf ("actual error = % .18f\n", result - expected);
printf ("intervals = %zu\n", w->size);
gsl_integration_workspace_free (w);
return 0;
}
The results below show that the desired accuracy is achieved after 8
subdivisions.
$ ./a.out
result = -4.000000000000085265
exact result = -4.000000000000000000
estimated error = 0.000000000000135447
actual error = -0.000000000000085265
intervals = 8
In fact, the extrapolation procedure used by 'QAGS' produces an accuracy
of almost twice as many digits. The error estimate returned by the
extrapolation procedure is larger than the actual error, giving a margin
of safety of one order of magnitude.
17.15 References and Further Reading
====================================
The following book is the definitive reference for QUADPACK, and was
written by the original authors. It provides descriptions of the
algorithms, program listings, test programs and examples. It also
includes useful advice on numerical integration and many references to
the numerical integration literature used in developing QUADPACK.
R. Piessens, E. de Doncker-Kapenga, C.W. Ueberhuber, D.K. Kahaner.
'QUADPACK A subroutine package for automatic integration' Springer
Verlag, 1983.
The CQUAD integration algorithm is described in the following paper:
P. Gonnet, "Increasing the Reliability of Adaptive Quadrature Using
Explicit Interpolants", 'ACM Transactions on Mathematical
Software', Volume 37 (2010), Issue 3, Article 26.
18 Random Number Generation
***************************
The library provides a large collection of random number generators
which can be accessed through a uniform interface. Environment
variables allow you to select different generators and seeds at runtime,
so that you can easily switch between generators without needing to
recompile your program. Each instance of a generator keeps track of its
own state, allowing the generators to be used in multi-threaded
programs. Additional functions are available for transforming uniform
random numbers into samples from continuous or discrete probability
distributions such as the Gaussian, log-normal or Poisson distributions.
These functions are declared in the header file 'gsl_rng.h'.
18.1 General comments on random numbers
=======================================
In 1988, Park and Miller wrote a paper entitled "Random number
generators: good ones are hard to find." [Commun. ACM, 31, 1192-1201].
Fortunately, some excellent random number generators are available,
though poor ones are still in common use. You may be happy with the
system-supplied random number generator on your computer, but you should
be aware that as computers get faster, requirements on random number
generators increase. Nowadays, a simulation that calls a random number
generator millions of times can often finish before you can make it down
the hall to the coffee machine and back.
A very nice review of random number generators was written by Pierre
L'Ecuyer, as Chapter 4 of the book: Handbook on Simulation, Jerry Banks,
ed. (Wiley, 1997). The chapter is available in postscript from
L'Ecuyer's ftp site (see references). Knuth's volume on Seminumerical
Algorithms (originally published in 1968) devotes 170 pages to random
number generators, and has recently been updated in its 3rd edition
(1997). It is brilliant, a classic. If you don't own it, you should
stop reading right now, run to the nearest bookstore, and buy it.
A good random number generator will satisfy both theoretical and
statistical properties. Theoretical properties are often hard to obtain
(they require real math!), but one prefers a random number generator
with a long period, low serial correlation, and a tendency _not_ to
"fall mainly on the planes." Statistical tests are performed with
numerical simulations. Generally, a random number generator is used to
estimate some quantity for which the theory of probability provides an
exact answer. Comparison to this exact answer provides a measure of
"randomness".
18.2 The Random Number Generator Interface
==========================================
It is important to remember that a random number generator is not a
"real" function like sine or cosine. Unlike real functions, successive
calls to a random number generator yield different return values. Of
course that is just what you want for a random number generator, but to
achieve this effect, the generator must keep track of some kind of
"state" variable. Sometimes this state is just an integer (sometimes
just the value of the previously generated random number), but often it
is more complicated than that and may involve a whole array of numbers,
possibly with some indices thrown in. To use the random number
generators, you do not need to know the details of what comprises the
state, and besides that varies from algorithm to algorithm.
The random number generator library uses two special structs,
'gsl_rng_type' which holds static information about each type of
generator and 'gsl_rng' which describes an instance of a generator
created from a given 'gsl_rng_type'.
The functions described in this section are declared in the header
file 'gsl_rng.h'.
18.3 Random number generator initialization
===========================================
-- Function: gsl_rng * gsl_rng_alloc (const gsl_rng_type * T)
This function returns a pointer to a newly-created instance of a
random number generator of type T. For example, the following code
creates an instance of the Tausworthe generator,
gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
If there is insufficient memory to create the generator then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
The generator is automatically initialized with the default seed,
'gsl_rng_default_seed'. This is zero by default but can be changed
either directly or by using the environment variable 'GSL_RNG_SEED'
(*note Random number environment variables::).
The details of the available generator types are described later in
this chapter.
-- Function: void gsl_rng_set (const gsl_rng * R, unsigned long int S)
This function initializes (or 'seeds') the random number generator.
If the generator is seeded with the same value of S on two
different runs, the same stream of random numbers will be generated
by successive calls to the routines below. If different values of
S >= 1 are supplied, then the generated streams of random numbers
should be completely different. If the seed S is zero then the
standard seed from the original implementation is used instead.
For example, the original Fortran source code for the 'ranlux'
generator used a seed of 314159265, and so choosing S equal to zero
reproduces this when using 'gsl_rng_ranlux'.
When using multiple seeds with the same generator, choose seed
values greater than zero to avoid collisions with the default
setting.
Note that the most generators only accept 32-bit seeds, with higher
values being reduced modulo 2^32. For generators with smaller
ranges the maximum seed value will typically be lower.
-- Function: void gsl_rng_free (gsl_rng * R)
This function frees all the memory associated with the generator R.
18.4 Sampling from a random number generator
============================================
The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers. Inline
versions of these functions are used when 'HAVE_INLINE' is defined. To
obtain non-uniform distributions *note Random Number Distributions::.
-- Function: unsigned long int gsl_rng_get (const gsl_rng * R)
This function returns a random integer from the generator R. The
minimum and maximum values depend on the algorithm used, but all
integers in the range [MIN,MAX] are equally likely. The values of
MIN and MAX can be determined using the auxiliary functions
'gsl_rng_max (r)' and 'gsl_rng_min (r)'.
-- Function: double gsl_rng_uniform (const gsl_rng * R)
This function returns a double precision floating point number
uniformly distributed in the range [0,1). The range includes 0.0
but excludes 1.0. The value is typically obtained by dividing the
result of 'gsl_rng_get(r)' by 'gsl_rng_max(r) + 1.0' in double
precision. Some generators compute this ratio internally so that
they can provide floating point numbers with more than 32 bits of
randomness (the maximum number of bits that can be portably
represented in a single 'unsigned long int').
-- Function: double gsl_rng_uniform_pos (const gsl_rng * R)
This function returns a positive double precision floating point
number uniformly distributed in the range (0,1), excluding both 0.0
and 1.0. The number is obtained by sampling the generator with the
algorithm of 'gsl_rng_uniform' until a non-zero value is obtained.
You can use this function if you need to avoid a singularity at
0.0.
-- Function: unsigned long int gsl_rng_uniform_int (const gsl_rng * R,
unsigned long int N)
This function returns a random integer from 0 to n-1 inclusive by
scaling down and/or discarding samples from the generator R. All
integers in the range [0,n-1] are produced with equal probability.
For generators with a non-zero minimum value an offset is applied
so that zero is returned with the correct probability.
Note that this function is designed for sampling from ranges
smaller than the range of the underlying generator. The parameter
N must be less than or equal to the range of the generator R. If N
is larger than the range of the generator then the function calls
the error handler with an error code of 'GSL_EINVAL' and returns
zero.
In particular, this function is not intended for generating the
full range of unsigned integer values [0,2^32-1]. Instead choose a
generator with the maximal integer range and zero minimum value,
such as 'gsl_rng_ranlxd1', 'gsl_rng_mt19937' or 'gsl_rng_taus', and
sample it directly using 'gsl_rng_get'. The range of each
generator can be found using the auxiliary functions described in
the next section.
18.5 Auxiliary random number generator functions
================================================
The following functions provide information about an existing generator.
You should use them in preference to hard-coding the generator
parameters into your own code.
-- Function: const char * gsl_rng_name (const gsl_rng * R)
This function returns a pointer to the name of the generator. For
example,
printf ("r is a '%s' generator\n",
gsl_rng_name (r));
would print something like 'r is a 'taus' generator'.
-- Function: unsigned long int gsl_rng_max (const gsl_rng * R)
'gsl_rng_max' returns the largest value that 'gsl_rng_get' can
return.
-- Function: unsigned long int gsl_rng_min (const gsl_rng * R)
'gsl_rng_min' returns the smallest value that 'gsl_rng_get' can
return. Usually this value is zero. There are some generators
with algorithms that cannot return zero, and for these generators
the minimum value is 1.
-- Function: void * gsl_rng_state (const gsl_rng * R)
-- Function: size_t gsl_rng_size (const gsl_rng * R)
These functions return a pointer to the state of generator R and
its size. You can use this information to access the state
directly. For example, the following code will write the state of
a generator to a stream,
void * state = gsl_rng_state (r);
size_t n = gsl_rng_size (r);
fwrite (state, n, 1, stream);
-- Function: const gsl_rng_type ** gsl_rng_types_setup (void)
This function returns a pointer to an array of all the available
generator types, terminated by a null pointer. The function should
be called once at the start of the program, if needed. The
following code fragment shows how to iterate over the array of
generator types to print the names of the available algorithms,
const gsl_rng_type **t, **t0;
t0 = gsl_rng_types_setup ();
printf ("Available generators:\n");
for (t = t0; *t != 0; t++)
{
printf ("%s\n", (*t)->name);
}
18.6 Random number environment variables
========================================
The library allows you to choose a default generator and seed from the
environment variables 'GSL_RNG_TYPE' and 'GSL_RNG_SEED' and the function
'gsl_rng_env_setup'. This makes it easy try out different generators
and seeds without having to recompile your program.
-- Function: const gsl_rng_type * gsl_rng_env_setup (void)
This function reads the environment variables 'GSL_RNG_TYPE' and
'GSL_RNG_SEED' and uses their values to set the corresponding
library variables 'gsl_rng_default' and 'gsl_rng_default_seed'.
These global variables are defined as follows,
extern const gsl_rng_type *gsl_rng_default
extern unsigned long int gsl_rng_default_seed
The environment variable 'GSL_RNG_TYPE' should be the name of a
generator, such as 'taus' or 'mt19937'. The environment variable
'GSL_RNG_SEED' should contain the desired seed value. It is
converted to an 'unsigned long int' using the C library function
'strtoul'.
If you don't specify a generator for 'GSL_RNG_TYPE' then
'gsl_rng_mt19937' is used as the default. The initial value of
'gsl_rng_default_seed' is zero.
Here is a short program which shows how to create a global generator
using the environment variables 'GSL_RNG_TYPE' and 'GSL_RNG_SEED',
#include
#include
gsl_rng * r; /* global generator */
int
main (void)
{
const gsl_rng_type * T;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf ("generator type: %s\n", gsl_rng_name (r));
printf ("seed = %lu\n", gsl_rng_default_seed);
printf ("first value = %lu\n", gsl_rng_get (r));
gsl_rng_free (r);
return 0;
}
Running the program without any environment variables uses the initial
defaults, an 'mt19937' generator with a seed of 0,
$ ./a.out
generator type: mt19937
seed = 0
first value = 4293858116
By setting the two variables on the command line we can change the
default generator and the seed,
$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out
GSL_RNG_TYPE=taus
GSL_RNG_SEED=123
generator type: taus
seed = 123
first value = 2720986350
18.7 Copying random number generator state
==========================================
The above methods do not expose the random number 'state' which changes
from call to call. It is often useful to be able to save and restore
the state. To permit these practices, a few somewhat more advanced
functions are supplied. These include:
-- Function: int gsl_rng_memcpy (gsl_rng * DEST, const gsl_rng * SRC)
This function copies the random number generator SRC into the
pre-existing generator DEST, making DEST into an exact copy of SRC.
The two generators must be of the same type.
-- Function: gsl_rng * gsl_rng_clone (const gsl_rng * R)
This function returns a pointer to a newly created generator which
is an exact copy of the generator R.
18.8 Reading and writing random number generator state
======================================================
The library provides functions for reading and writing the random number
state to a file as binary data.
-- Function: int gsl_rng_fwrite (FILE * STREAM, const gsl_rng * R)
This function writes the random number state of the random number
generator R to the stream STREAM in binary format. The return
value is 0 for success and 'GSL_EFAILED' if there was a problem
writing to the file. Since the data is written in the native
binary format it may not be portable between different
architectures.
-- Function: int gsl_rng_fread (FILE * STREAM, gsl_rng * R)
This function reads the random number state into the random number
generator R from the open stream STREAM in binary format. The
random number generator R must be preinitialized with the correct
random number generator type since type information is not saved.
The return value is 0 for success and 'GSL_EFAILED' if there was a
problem reading from the file. The data is assumed to have been
written in the native binary format on the same architecture.
18.9 Random number generator algorithms
=======================================
The functions described above make no reference to the actual algorithm
used. This is deliberate so that you can switch algorithms without
having to change any of your application source code. The library
provides a large number of generators of different types, including
simulation quality generators, generators provided for compatibility
with other libraries and historical generators from the past.
The following generators are recommended for use in simulation. They
have extremely long periods, low correlation and pass most statistical
tests. For the most reliable source of uncorrelated numbers, the
second-generation RANLUX generators have the strongest proof of
randomness.
-- Generator: gsl_rng_mt19937
The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
variant of the twisted generalized feedback shift-register
algorithm, and is known as the "Mersenne Twister" generator. It
has a Mersenne prime period of 2^19937 - 1 (about 10^6000) and is
equi-distributed in 623 dimensions. It has passed the DIEHARD
statistical tests. It uses 624 words of state per generator and is
comparable in speed to the other generators. The original
generator used a default seed of 4357 and choosing S equal to zero
in 'gsl_rng_set' reproduces this. Later versions switched to 5489
as the default seed, you can choose this explicitly via
'gsl_rng_set' instead if you require it.
For more information see,
Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
623-dimensionally equidistributed uniform pseudorandom number
generator". 'ACM Transactions on Modeling and Computer
Simulation', Vol. 8, No. 1 (Jan. 1998), Pages 3-30
The generator 'gsl_rng_mt19937' uses the second revision of the
seeding procedure published by the two authors above in 2002. The
original seeding procedures could cause spurious artifacts for some
seed values. They are still available through the alternative
generators 'gsl_rng_mt19937_1999' and 'gsl_rng_mt19937_1998'.
-- Generator: gsl_rng_ranlxs0
-- Generator: gsl_rng_ranlxs1
-- Generator: gsl_rng_ranlxs2
The generator 'ranlxs0' is a second-generation version of the
RANLUX algorithm of Lu"scher, which produces "luxury random
numbers". This generator provides single precision output (24
bits) at three luxury levels 'ranlxs0', 'ranlxs1' and 'ranlxs2', in
increasing order of strength. It uses double-precision floating
point arithmetic internally and can be significantly faster than
the integer version of 'ranlux', particularly on 64-bit
architectures. The period of the generator is about 10^171. The
algorithm has mathematically proven properties and can provide
truly decorrelated numbers at a known level of randomness. The
higher luxury levels provide increased decorrelation between
samples as an additional safety margin.
Note that the range of allowed seeds for this generator is
[0,2^31-1]. Higher seed values are wrapped modulo 2^31.
-- Generator: gsl_rng_ranlxd1
-- Generator: gsl_rng_ranlxd2
These generators produce double precision output (48 bits) from the
RANLXS generator. The library provides two luxury levels 'ranlxd1'
and 'ranlxd2', in increasing order of strength.
-- Generator: gsl_rng_ranlux
-- Generator: gsl_rng_ranlux389
The 'ranlux' generator is an implementation of the original
algorithm developed by Lu"scher. It uses a
lagged-fibonacci-with-skipping algorithm to produce "luxury random
numbers". It is a 24-bit generator, originally designed for
single-precision IEEE floating point numbers. This implementation
is based on integer arithmetic, while the second-generation
versions RANLXS and RANLXD described above provide floating-point
implementations which will be faster on many platforms. The period
of the generator is about 10^171. The algorithm has mathematically
proven properties and it can provide truly decorrelated numbers at
a known level of randomness. The default level of decorrelation
recommended by Lu"scher is provided by 'gsl_rng_ranlux', while
'gsl_rng_ranlux389' gives the highest level of randomness, with all
24 bits decorrelated. Both types of generator use 24 words of
state per generator.
For more information see,
M. Lu"scher, "A portable high-quality random number generator
for lattice field theory calculations", 'Computer Physics
Communications', 79 (1994) 100-110.
F. James, "RANLUX: A Fortran implementation of the
high-quality pseudo-random number generator of Lu"scher",
'Computer Physics Communications', 79 (1994) 111-114
-- Generator: gsl_rng_cmrg
This is a combined multiple recursive generator by L'Ecuyer. Its
sequence is,
z_n = (x_n - y_n) mod m_1
where the two underlying generators x_n and y_n are,
x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1
y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2
with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098,
b_2 = 0, b_3 = -539608, and moduli m_1 = 2^31 - 1 = 2147483647 and
m_2 = 2145483479.
The period of this generator is lcm(m_1^3-1, m_2^3-1), which is
approximately 2^185 (about 10^56). It uses 6 words of state per
generator. For more information see,
P. L'Ecuyer, "Combined Multiple Recursive Random Number
Generators", 'Operations Research', 44, 5 (1996), 816-822.
-- Generator: gsl_rng_mrg
This is a fifth-order multiple recursive generator by L'Ecuyer,
Blouin and Coutre. Its sequence is,
x_n = (a_1 x_{n-1} + a_5 x_{n-5}) mod m
with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m =
2^31 - 1.
The period of this generator is about 10^46. It uses 5 words of
state per generator. More information can be found in the
following paper,
P. L'Ecuyer, F. Blouin, and R. Coutre, "A search for good
multiple recursive random number generators", 'ACM
Transactions on Modeling and Computer Simulation' 3, 87-98
(1993).
-- Generator: gsl_rng_taus
-- Generator: gsl_rng_taus2
This is a maximally equidistributed combined Tausworthe generator
by L'Ecuyer. The sequence is,
x_n = (s1_n ^^ s2_n ^^ s3_n)
where,
s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19))
s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25))
s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))
computed modulo 2^32. In the formulas above ^^ denotes
"exclusive-or". Note that the algorithm relies on the properties
of 32-bit unsigned integers and has been implemented using a
bitmask of '0xFFFFFFFF' to make it work on 64 bit machines.
The period of this generator is 2^88 (about 10^26). It uses 3
words of state per generator. For more information see,
P. L'Ecuyer, "Maximally Equidistributed Combined Tausworthe
Generators", 'Mathematics of Computation', 65, 213 (1996),
203-213.
The generator 'gsl_rng_taus2' uses the same algorithm as
'gsl_rng_taus' but with an improved seeding procedure described in
the paper,
P. L'Ecuyer, "Tables of Maximally Equidistributed Combined
LFSR Generators", 'Mathematics of Computation', 68, 225
(1999), 261-269
The generator 'gsl_rng_taus2' should now be used in preference to
'gsl_rng_taus'.
-- Generator: gsl_rng_gfsr4
The 'gfsr4' generator is like a lagged-fibonacci generator, and
produces each number as an 'xor''d sum of four previous values.
r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}
Ziff (ref below) notes that "it is now widely known" that two-tap
registers (such as R250, which is described below) have serious
flaws, the most obvious one being the three-point correlation that
comes from the definition of the generator. Nice mathematical
properties can be derived for GFSR's, and numerics bears out the
claim that 4-tap GFSR's with appropriately chosen offsets are as
random as can be measured, using the author's test.
This implementation uses the values suggested the example on p392
of Ziff's article: A=471, B=1586, C=6988, D=9689.
If the offsets are appropriately chosen (such as the one ones in
this implementation), then the sequence is said to be maximal; that
means that the period is 2^D - 1, where D is the longest lag. (It
is one less than 2^D because it is not permitted to have all zeros
in the 'ra[]' array.) For this implementation with D=9689 that
works out to about 10^2917.
Note that the implementation of this generator using a 32-bit
integer amounts to 32 parallel implementations of one-bit
generators. One consequence of this is that the period of this
32-bit generator is the same as for the one-bit generator.
Moreover, this independence means that all 32-bit patterns are
equally likely, and in particular that 0 is an allowed random
value. (We are grateful to Heiko Bauke for clarifying for us these
properties of GFSR random number generators.)
For more information see,
Robert M. Ziff, "Four-tap shift-register-sequence
random-number generators", 'Computers in Physics', 12(4),
Jul/Aug 1998, pp 385-392.
18.10 Unix random number generators
===================================
The standard Unix random number generators 'rand', 'random' and 'rand48'
are provided as part of GSL. Although these generators are widely
available individually often they aren't all available on the same
platform. This makes it difficult to write portable code using them and
so we have included the complete set of Unix generators in GSL for
convenience. Note that these generators don't produce high-quality
randomness and aren't suitable for work requiring accurate statistics.
However, if you won't be measuring statistical quantities and just want
to introduce some variation into your program then these generators are
quite acceptable.
-- Generator: gsl_rng_rand
This is the BSD 'rand' generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
with a = 1103515245, c = 12345 and m = 2^31. The seed specifies
the initial value, x_1. The period of this generator is 2^31, and
it uses 1 word of storage per generator.
-- Generator: gsl_rng_random_bsd
-- Generator: gsl_rng_random_libc5
-- Generator: gsl_rng_random_glibc2
These generators implement the 'random' family of functions, a set
of linear feedback shift register generators originally used in BSD
Unix. There are several versions of 'random' in use today: the
original BSD version (e.g. on SunOS4), a libc5 version (found on
older GNU/Linux systems) and a glibc2 version. Each version uses a
different seeding procedure, and thus produces different sequences.
The original BSD routines accepted a variable length buffer for the
generator state, with longer buffers providing higher-quality
randomness. The 'random' function implemented algorithms for
buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm
with the largest length that would fit into the user-supplied
buffer was used. To support these algorithms additional generators
are available with the following names,
gsl_rng_random8_bsd
gsl_rng_random32_bsd
gsl_rng_random64_bsd
gsl_rng_random128_bsd
gsl_rng_random256_bsd
where the numeric suffix indicates the buffer length. The original
BSD 'random' function used a 128-byte default buffer and so
'gsl_rng_random_bsd' has been made equivalent to
'gsl_rng_random128_bsd'. Corresponding versions of the 'libc5' and
'glibc2' generators are also available, with the names
'gsl_rng_random8_libc5', 'gsl_rng_random8_glibc2', etc.
-- Generator: gsl_rng_rand48
This is the Unix 'rand48' generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
defined on 48-bit unsigned integers with a = 25214903917, c = 11
and m = 2^48. The seed specifies the upper 32 bits of the initial
value, x_1, with the lower 16 bits set to '0x330E'. The function
'gsl_rng_get' returns the upper 32 bits from each term of the
sequence. This does not have a direct parallel in the original
'rand48' functions, but forcing the result to type 'long int'
reproduces the output of 'mrand48'. The function 'gsl_rng_uniform'
uses the full 48 bits of internal state to return the double
precision number x_n/m, which is equivalent to the function
'drand48'. Note that some versions of the GNU C Library contained
a bug in 'mrand48' function which caused it to produce different
results (only the lower 16-bits of the return value were set).
18.11 Other random number generators
====================================
The generators in this section are provided for compatibility with
existing libraries. If you are converting an existing program to use
GSL then you can select these generators to check your new
implementation against the original one, using the same random number
generator. After verifying that your new program reproduces the
original results you can then switch to a higher-quality generator.
Note that most of the generators in this section are based on single
linear congruence relations, which are the least sophisticated type of
generator. In particular, linear congruences have poor properties when
used with a non-prime modulus, as several of these routines do (e.g.
with a power of two modulus, 2^31 or 2^32). This leads to periodicity
in the least significant bits of each number, with only the higher bits
having any randomness. Thus if you want to produce a random bitstream
it is best to avoid using the least significant bits.
-- Generator: gsl_rng_ranf
This is the CRAY random number generator 'RANF'. Its sequence is
x_{n+1} = (a x_n) mod m
defined on 48-bit unsigned integers with a = 44485709377909 and m =
2^48. The seed specifies the lower 32 bits of the initial value,
x_1, with the lowest bit set to prevent the seed taking an even
value. The upper 16 bits of x_1 are set to 0. A consequence of
this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc.
produce the same sequences.
The generator compatible with the CRAY MATHLIB routine RANF. It
produces double precision floating point numbers which should be
identical to those from the original RANF.
There is a subtlety in the implementation of the seeding. The
initial state is reversed through one step, by multiplying by the
modular inverse of a mod m. This is done for compatibility with
the original CRAY implementation.
Note that you can only seed the generator with integers up to 2^32,
while the original CRAY implementation uses non-portable wide
integers which can cover all 2^48 states of the generator.
The function 'gsl_rng_get' returns the upper 32 bits from each term
of the sequence. The function 'gsl_rng_uniform' uses the full 48
bits to return the double precision number x_n/m.
The period of this generator is 2^46.
-- Generator: gsl_rng_ranmar
This is the RANMAR lagged-fibonacci generator of Marsaglia, Zaman
and Tsang. It is a 24-bit generator, originally designed for
single-precision IEEE floating point numbers. It was included in
the CERNLIB high-energy physics library.
-- Generator: gsl_rng_r250
This is the shift-register generator of Kirkpatrick and Stoll. The
sequence is based on the recurrence
x_n = x_{n-103} ^^ x_{n-250}
where ^^ denotes "exclusive-or", defined on 32-bit words. The
period of this generator is about 2^250 and it uses 250 words of
state per generator.
For more information see,
S. Kirkpatrick and E. Stoll, "A very fast shift-register
sequence random number generator", 'Journal of Computational
Physics', 40, 517-526 (1981)
-- Generator: gsl_rng_tt800
This is an earlier version of the twisted generalized feedback
shift-register generator, and has been superseded by the
development of MT19937. However, it is still an acceptable
generator in its own right. It has a period of 2^800 and uses 33
words of storage per generator.
For more information see,
Makoto Matsumoto and Yoshiharu Kurita, "Twisted GFSR
Generators II", 'ACM Transactions on Modelling and Computer
Simulation', Vol. 4, No. 3, 1994, pages 254-266.
-- Generator: gsl_rng_vax
This is the VAX generator 'MTH$RANDOM'. Its sequence is,
x_{n+1} = (a x_n + c) mod m
with a = 69069, c = 1 and m = 2^32. The seed specifies the initial
value, x_1. The period of this generator is 2^32 and it uses 1
word of storage per generator.
-- Generator: gsl_rng_transputer
This is the random number generator from the INMOS Transputer
Development system. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 1664525 and m = 2^32. The seed specifies the initial
value, x_1.
-- Generator: gsl_rng_randu
This is the IBM 'RANDU' generator. Its sequence is
x_{n+1} = (a x_n) mod m
with a = 65539 and m = 2^31. The seed specifies the initial value,
x_1. The period of this generator was only 2^29. It has become a
textbook example of a poor generator.
-- Generator: gsl_rng_minstd
This is Park and Miller's "minimal standard" MINSTD generator, a
simple linear congruence which takes care to avoid the major
pitfalls of such algorithms. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 16807 and m = 2^31 - 1 = 2147483647. The seed specifies
the initial value, x_1. The period of this generator is about
2^31.
This generator was used in the IMSL Library (subroutine RNUN) and
in MATLAB (the RAND function) in the past. It is also sometimes
known by the acronym "GGL" (I'm not sure what that stands for).
For more information see,
Park and Miller, "Random Number Generators: Good ones are hard
to find", 'Communications of the ACM', October 1988, Volume
31, No 10, pages 1192-1201.
-- Generator: gsl_rng_uni
-- Generator: gsl_rng_uni32
This is a reimplementation of the 16-bit SLATEC random number
generator RUNIF. A generalization of the generator to 32 bits is
provided by 'gsl_rng_uni32'. The original source code is available
from NETLIB.
-- Generator: gsl_rng_slatec
This is the SLATEC random number generator RAND. It is ancient.
The original source code is available from NETLIB.
-- Generator: gsl_rng_zuf
This is the ZUFALL lagged Fibonacci series generator of Peterson.
Its sequence is,
t = u_{n-273} + u_{n-607}
u_n = t - floor(t)
The original source code is available from NETLIB. For more
information see,
W. Petersen, "Lagged Fibonacci Random Number Generators for
the NEC SX-3", 'International Journal of High Speed Computing'
(1994).
-- Generator: gsl_rng_knuthran2
This is a second-order multiple recursive generator described by
Knuth in 'Seminumerical Algorithms', 3rd Ed., page 108. Its
sequence is,
x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m
with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.
-- Generator: gsl_rng_knuthran2002
-- Generator: gsl_rng_knuthran
This is a second-order multiple recursive generator described by
Knuth in 'Seminumerical Algorithms', 3rd Ed., Section 3.6. Knuth
provides its C code. The updated routine 'gsl_rng_knuthran2002' is
from the revised 9th printing and corrects some weaknesses in the
earlier version, which is implemented as 'gsl_rng_knuthran'.
-- Generator: gsl_rng_borosh13
-- Generator: gsl_rng_fishman18
-- Generator: gsl_rng_fishman20
-- Generator: gsl_rng_lecuyer21
-- Generator: gsl_rng_waterman14
These multiplicative generators are taken from Knuth's
'Seminumerical Algorithms', 3rd Ed., pages 106-108. Their sequence
is,
x_{n+1} = (a x_n) mod m
where the seed specifies the initial value, x_1. The parameters a
and m are as follows, Borosh-Niederreiter: a = 1812433253, m =
2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271,
m = 2^31 - 1, L'Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a =
1566083941, m = 2^32.
-- Generator: gsl_rng_fishman2x
This is the L'Ecuyer-Fishman random number generator. It is taken
from Knuth's 'Seminumerical Algorithms', 3rd Ed., page 108. Its
sequence is,
z_{n+1} = (x_n - y_n) mod m
with m = 2^31 - 1. x_n and y_n are given by the 'fishman20' and
'lecuyer21' algorithms. The seed specifies the initial value, x_1.
-- Generator: gsl_rng_coveyou
This is the Coveyou random number generator. It is taken from
Knuth's 'Seminumerical Algorithms', 3rd Ed., Section 3.2.2. Its
sequence is,
x_{n+1} = (x_n (x_n + 1)) mod m
with m = 2^32. The seed specifies the initial value, x_1.
18.12 Performance
=================
The following table shows the relative performance of a selection the
available random number generators. The fastest simulation quality
generators are 'taus', 'gfsr4' and 'mt19937'. The generators which
offer the best mathematically-proven quality are those based on the
RANLUX algorithm.
1754 k ints/sec, 870 k doubles/sec, taus
1613 k ints/sec, 855 k doubles/sec, gfsr4
1370 k ints/sec, 769 k doubles/sec, mt19937
565 k ints/sec, 571 k doubles/sec, ranlxs0
400 k ints/sec, 405 k doubles/sec, ranlxs1
490 k ints/sec, 389 k doubles/sec, mrg
407 k ints/sec, 297 k doubles/sec, ranlux
243 k ints/sec, 254 k doubles/sec, ranlxd1
251 k ints/sec, 253 k doubles/sec, ranlxs2
238 k ints/sec, 215 k doubles/sec, cmrg
247 k ints/sec, 198 k doubles/sec, ranlux389
141 k ints/sec, 140 k doubles/sec, ranlxd2
18.13 Examples
==============
The following program demonstrates the use of a random number generator
to produce uniform random numbers in the range [0.0, 1.0),
#include
#include
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < n; i++)
{
double u = gsl_rng_uniform (r);
printf ("%.5f\n", u);
}
gsl_rng_free (r);
return 0;
}
Here is the output of the program,
$ ./a.out
0.99974
0.16291
0.28262
0.94720
0.23166
0.48497
0.95748
0.74431
0.54004
0.73995
The numbers depend on the seed used by the generator. The default seed
can be changed with the 'GSL_RNG_SEED' environment variable to produce a
different stream of numbers. The generator itself can be changed using
the environment variable 'GSL_RNG_TYPE'. Here is the output of the
program using a seed value of 123 and the multiple-recursive generator
'mrg',
$ GSL_RNG_SEED=123 GSL_RNG_TYPE=mrg ./a.out
0.33050
0.86631
0.32982
0.67620
0.53391
0.06457
0.16847
0.70229
0.04371
0.86374
18.14 References and Further Reading
====================================
The subject of random number generation and testing is reviewed
extensively in Knuth's 'Seminumerical Algorithms'.
Donald E. Knuth, 'The Art of Computer Programming: Seminumerical
Algorithms' (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
Further information is available in the review paper written by Pierre
L'Ecuyer,
P. L'Ecuyer, "Random Number Generation", Chapter 4 of the Handbook
on Simulation, Jerry Banks Ed., Wiley, 1998, 93-137.
in the file
'handsim.ps'.
The source code for the DIEHARD random number generator tests is also
available online,
'DIEHARD source code' G. Marsaglia,
A comprehensive set of random number generator tests is available from
NIST,
NIST Special Publication 800-22, "A Statistical Test Suite for the
Validation of Random Number Generators and Pseudo Random Number
Generators for Cryptographic Applications".
18.15 Acknowledgements
======================
Thanks to Makoto Matsumoto, Takuji Nishimura and Yoshiharu Kurita for
making the source code to their generators (MT19937, MM&TN; TT800,
MM&YK) available under the GNU General Public License. Thanks to Martin
Lu"scher for providing notes and source code for the RANLXS and RANLXD
generators.
19 Quasi-Random Sequences
*************************
This chapter describes functions for generating quasi-random sequences
in arbitrary dimensions. A quasi-random sequence progressively covers a
d-dimensional space with a set of points that are uniformly distributed.
Quasi-random sequences are also known as low-discrepancy sequences. The
quasi-random sequence generators use an interface that is similar to the
interface for random number generators, except that seeding is not
required--each generator produces a single sequence.
The functions described in this section are declared in the header
file 'gsl_qrng.h'.
19.1 Quasi-random number generator initialization
=================================================
-- Function: gsl_qrng * gsl_qrng_alloc (const gsl_qrng_type * T,
unsigned int D)
This function returns a pointer to a newly-created instance of a
quasi-random sequence generator of type T and dimension D. If
there is insufficient memory to create the generator then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
-- Function: void gsl_qrng_free (gsl_qrng * Q)
This function frees all the memory associated with the generator Q.
-- Function: void gsl_qrng_init (gsl_qrng * Q)
This function reinitializes the generator Q to its starting point.
Note that quasi-random sequences do not use a seed and always
produce the same set of values.
19.2 Sampling from a quasi-random number generator
==================================================
-- Function: int gsl_qrng_get (const gsl_qrng * Q, double X[])
This function stores the next point from the sequence generator Q
in the array X. The space available for X must match the dimension
of the generator. The point X will lie in the range 0 < x_i < 1
for each x_i. An inline version of this function is used when
'HAVE_INLINE' is defined.
19.3 Auxiliary quasi-random number generator functions
======================================================
-- Function: const char * gsl_qrng_name (const gsl_qrng * Q)
This function returns a pointer to the name of the generator.
-- Function: size_t gsl_qrng_size (const gsl_qrng * Q)
-- Function: void * gsl_qrng_state (const gsl_qrng * Q)
These functions return a pointer to the state of generator R and
its size. You can use this information to access the state
directly. For example, the following code will write the state of
a generator to a stream,
void * state = gsl_qrng_state (q);
size_t n = gsl_qrng_size (q);
fwrite (state, n, 1, stream);
19.4 Saving and restoring quasi-random number generator state
=============================================================
-- Function: int gsl_qrng_memcpy (gsl_qrng * DEST, const gsl_qrng *
SRC)
This function copies the quasi-random sequence generator SRC into
the pre-existing generator DEST, making DEST into an exact copy of
SRC. The two generators must be of the same type.
-- Function: gsl_qrng * gsl_qrng_clone (const gsl_qrng * Q)
This function returns a pointer to a newly created generator which
is an exact copy of the generator Q.
19.5 Quasi-random number generator algorithms
=============================================
The following quasi-random sequence algorithms are available,
-- Generator: gsl_qrng_niederreiter_2
This generator uses the algorithm described in Bratley, Fox,
Niederreiter, 'ACM Trans. Model. Comp. Sim.' 2, 195 (1992). It is
valid up to 12 dimensions.
-- Generator: gsl_qrng_sobol
This generator uses the Sobol sequence described in Antonov,
Saleev, 'USSR Comput. Maths. Math. Phys.' 19, 252 (1980). It is
valid up to 40 dimensions.
-- Generator: gsl_qrng_halton
-- Generator: gsl_qrng_reversehalton
These generators use the Halton and reverse Halton sequences
described in J.H. Halton, 'Numerische Mathematik' 2, 84-90 (1960)
and B. Vandewoestyne and R. Cools 'Computational and Applied
Mathematics' 189, 1&2, 341-361 (2006). They are valid up to 1229
dimensions.
19.6 Examples
=============
The following program prints the first 1024 points of the 2-dimensional
Sobol sequence.
#include
#include
int
main (void)
{
int i;
gsl_qrng * q = gsl_qrng_alloc (gsl_qrng_sobol, 2);
for (i = 0; i < 1024; i++)
{
double v[2];
gsl_qrng_get (q, v);
printf ("%.5f %.5f\n", v[0], v[1]);
}
gsl_qrng_free (q);
return 0;
}
Here is the output from the program,
$ ./a.out
0.50000 0.50000
0.75000 0.25000
0.25000 0.75000
0.37500 0.37500
0.87500 0.87500
0.62500 0.12500
0.12500 0.62500
....
It can be seen that successive points progressively fill-in the spaces
between previous points.
19.7 References
===============
The implementations of the quasi-random sequence routines are based on
the algorithms described in the following paper,
P. Bratley and B.L. Fox and H. Niederreiter, "Algorithm 738:
Programs to Generate Niederreiter's Low-discrepancy Sequences",
'ACM Transactions on Mathematical Software', Vol. 20, No. 4,
December, 1994, p. 494-495.
20 Random Number Distributions
******************************
This chapter describes functions for generating random variates and
computing their probability distributions. Samples from the
distributions described in this chapter can be obtained using any of the
random number generators in the library as an underlying source of
randomness.
In the simplest cases a non-uniform distribution can be obtained
analytically from the uniform distribution of a random number generator
by applying an appropriate transformation. This method uses one call to
the random number generator. More complicated distributions are created
by the "acceptance-rejection" method, which compares the desired
distribution against a distribution which is similar and known
analytically. This usually requires several samples from the generator.
The library also provides cumulative distribution functions and
inverse cumulative distribution functions, sometimes referred to as
quantile functions. The cumulative distribution functions and their
inverses are computed separately for the upper and lower tails of the
distribution, allowing full accuracy to be retained for small results.
The functions for random variates and probability density functions
described in this section are declared in 'gsl_randist.h'. The
corresponding cumulative distribution functions are declared in
'gsl_cdf.h'.
Note that the discrete random variate functions always return a value
of type 'unsigned int', and on most platforms this has a maximum value
of 2^32-1 ~=~ 4.29e9. They should only be called with a safe range of
parameters (where there is a negligible probability of a variate
exceeding this limit) to prevent incorrect results due to overflow.
20.1 Introduction
=================
Continuous random number distributions are defined by a probability
density function, p(x), such that the probability of x occurring in the
infinitesimal range x to x+dx is p dx.
The cumulative distribution function for the lower tail P(x) is
defined by the integral,
P(x) = \int_{-\infty}^{x} dx' p(x')
and gives the probability of a variate taking a value less than x.
The cumulative distribution function for the upper tail Q(x) is
defined by the integral,
Q(x) = \int_{x}^{+\infty} dx' p(x')
and gives the probability of a variate taking a value greater than x.
The upper and lower cumulative distribution functions are related by
P(x) + Q(x) = 1 and satisfy 0 <= P(x) <= 1, 0 <= Q(x) <= 1.
The inverse cumulative distributions, x=P^{-1}(P) and x=Q^{-1}(Q)
give the values of x which correspond to a specific value of P or Q.
They can be used to find confidence limits from probability values.
For discrete distributions the probability of sampling the integer
value k is given by p(k), where \sum_k p(k) = 1. The cumulative
distribution for the lower tail P(k) of a discrete distribution is
defined as,
P(k) = \sum_{i <= k} p(i)
where the sum is over the allowed range of the distribution less than or
equal to k.
The cumulative distribution for the upper tail of a discrete
distribution Q(k) is defined as
Q(k) = \sum_{i > k} p(i)
giving the sum of probabilities for all values greater than k. These
two definitions satisfy the identity P(k)+Q(k)=1.
If the range of the distribution is 1 to n inclusive then P(n)=1,
Q(n)=0 while P(1) = p(1), Q(1)=1-p(1).
20.2 The Gaussian Distribution
==============================
-- Function: double gsl_ran_gaussian (const gsl_rng * R, double SIGMA)
This function returns a Gaussian random variate, with mean zero and
standard deviation SIGMA. The probability distribution for
Gaussian random variates is,
p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
for x in the range -\infty to +\infty. Use the transformation z =
\mu + x on the numbers returned by 'gsl_ran_gaussian' to obtain a
Gaussian distribution with mean \mu. This function uses the
Box-Muller algorithm which requires two calls to the random number
generator R.
-- Function: double gsl_ran_gaussian_pdf (double X, double SIGMA)
This function computes the probability density p(x) at X for a
Gaussian distribution with standard deviation SIGMA, using the
formula given above.
-- Function: double gsl_ran_gaussian_ziggurat (const gsl_rng * R,
double SIGMA)
-- Function: double gsl_ran_gaussian_ratio_method (const gsl_rng * R,
double SIGMA)
This function computes a Gaussian random variate using the
alternative Marsaglia-Tsang ziggurat and Kinderman-Monahan-Leva
ratio methods. The Ziggurat algorithm is the fastest available
algorithm in most cases.
-- Function: double gsl_ran_ugaussian (const gsl_rng * R)
-- Function: double gsl_ran_ugaussian_pdf (double X)
-- Function: double gsl_ran_ugaussian_ratio_method (const gsl_rng * R)
These functions compute results for the unit Gaussian distribution.
They are equivalent to the functions above with a standard
deviation of one, SIGMA = 1.
-- Function: double gsl_cdf_gaussian_P (double X, double SIGMA)
-- Function: double gsl_cdf_gaussian_Q (double X, double SIGMA)
-- Function: double gsl_cdf_gaussian_Pinv (double P, double SIGMA)
-- Function: double gsl_cdf_gaussian_Qinv (double Q, double SIGMA)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Gaussian distribution with standard
deviation SIGMA.
-- Function: double gsl_cdf_ugaussian_P (double X)
-- Function: double gsl_cdf_ugaussian_Q (double X)
-- Function: double gsl_cdf_ugaussian_Pinv (double P)
-- Function: double gsl_cdf_ugaussian_Qinv (double Q)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the unit Gaussian distribution.
20.3 The Gaussian Tail Distribution
===================================
-- Function: double gsl_ran_gaussian_tail (const gsl_rng * R, double A,
double SIGMA)
This function provides random variates from the upper tail of a
Gaussian distribution with standard deviation SIGMA. The values
returned are larger than the lower limit A, which must be positive.
The method is based on Marsaglia's famous rectangle-wedge-tail
algorithm (Ann. Math. Stat. 32, 894-899 (1961)), with this
aspect explained in Knuth, v2, 3rd ed, p139,586 (exercise 11).
The probability distribution for Gaussian tail random variates is,
p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx
for x > a where N(a;\sigma) is the normalization constant,
N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
-- Function: double gsl_ran_gaussian_tail_pdf (double X, double A,
double SIGMA)
This function computes the probability density p(x) at X for a
Gaussian tail distribution with standard deviation SIGMA and lower
limit A, using the formula given above.
-- Function: double gsl_ran_ugaussian_tail (const gsl_rng * R, double
A)
-- Function: double gsl_ran_ugaussian_tail_pdf (double X, double A)
These functions compute results for the tail of a unit Gaussian
distribution. They are equivalent to the functions above with a
standard deviation of one, SIGMA = 1.
20.4 The Bivariate Gaussian Distribution
========================================
-- Function: void gsl_ran_bivariate_gaussian (const gsl_rng * R, double
SIGMA_X, double SIGMA_Y, double RHO, double * X, double * Y)
This function generates a pair of correlated Gaussian variates,
with mean zero, correlation coefficient RHO and standard deviations
SIGMA_X and SIGMA_Y in the x and y directions. The probability
distribution for bivariate Gaussian random variates is,
p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy
for x,y in the range -\infty to +\infty. The correlation
coefficient RHO should lie between 1 and -1.
-- Function: double gsl_ran_bivariate_gaussian_pdf (double X, double Y,
double SIGMA_X, double SIGMA_Y, double RHO)
This function computes the probability density p(x,y) at (X,Y) for
a bivariate Gaussian distribution with standard deviations SIGMA_X,
SIGMA_Y and correlation coefficient RHO, using the formula given
above.
20.5 The Multivariate Gaussian Distribution
===========================================
-- Function: int gsl_ran_multivariate_gaussian (const gsl_rng * R,
const gsl_vector * MU, const gsl_matrix * L, gsl_vector *
RESULT)
This function generates a random vector satisfying the
k-dimensional multivariate Gaussian distribution with mean \mu and
variance-covariance matrix \Sigma. On input, the k-vector \mu is
given in MU, and the Cholesky factor of the k-by-k matrix \Sigma =
L L^T is given in the lower triangle of L, as output from
'gsl_linalg_cholesky_decomp'. The random vector is stored in
RESULT on output. The probability distribution for multivariate
Gaussian random variates is
p(x_1,...,x_k) dx_1 ... dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k
-- Function: int gsl_ran_multivariate_gaussian_pdf (const gsl_vector *
X, const gsl_vector * MU, const gsl_matrix * L, double *
RESULT, gsl_vector * WORK)
-- Function: int gsl_ran_multivariate_gaussian_log_pdf (const
gsl_vector * X, const gsl_vector * MU, const gsl_matrix * L,
double * RESULT, gsl_vector * WORK)
These functions compute p(x) or \log{p(x)} at the point X, using
mean vector MU and variance-covariance matrix specified by its
Cholesky factor L using the formula above. Additional workspace of
length k is required in WORK.
-- Function: int gsl_ran_multivariate_gaussian_mean (const gsl_matrix *
X, gsl_vector * MU_HAT)
Given a set of n samples X_j from a k-dimensional multivariate
Gaussian distribution, this function computes the maximum
likelihood estimate of the mean of the distribution, given by
\Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j
The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix X, and
the maximum likelihood estimate of the mean is stored in MU_HAT on
output.
-- Function: int gsl_ran_multivariate_gaussian_vcov (const gsl_matrix *
X, gsl_matrix * SIGMA_HAT)
Given a set of n samples X_j from a k-dimensional multivariate
Gaussian distribution, this function computes the maximum
likelihood estimate of the variance-covariance matrix of the
distribution, given by
\Hat{\Sigma} = {1 \over n} \sum_{j=1}^n \left( X_j - \Hat{\mu} \right) \left( X_j - \Hat{\mu} \right)^T
The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix X and
the maximum likelihood estimate of the variance-covariance matrix
is stored in SIGMA_HAT on output.
20.6 The Exponential Distribution
=================================
-- Function: double gsl_ran_exponential (const gsl_rng * R, double MU)
This function returns a random variate from the exponential
distribution with mean MU. The distribution is,
p(x) dx = {1 \over \mu} \exp(-x/\mu) dx
for x >= 0.
-- Function: double gsl_ran_exponential_pdf (double X, double MU)
This function computes the probability density p(x) at X for an
exponential distribution with mean MU, using the formula given
above.
-- Function: double gsl_cdf_exponential_P (double X, double MU)
-- Function: double gsl_cdf_exponential_Q (double X, double MU)
-- Function: double gsl_cdf_exponential_Pinv (double P, double MU)
-- Function: double gsl_cdf_exponential_Qinv (double Q, double MU)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the exponential distribution with mean
MU.
20.7 The Laplace Distribution
=============================
-- Function: double gsl_ran_laplace (const gsl_rng * R, double A)
This function returns a random variate from the Laplace
distribution with width A. The distribution is,
p(x) dx = {1 \over 2 a} \exp(-|x/a|) dx
for -\infty < x < \infty.
-- Function: double gsl_ran_laplace_pdf (double X, double A)
This function computes the probability density p(x) at X for a
Laplace distribution with width A, using the formula given above.
-- Function: double gsl_cdf_laplace_P (double X, double A)
-- Function: double gsl_cdf_laplace_Q (double X, double A)
-- Function: double gsl_cdf_laplace_Pinv (double P, double A)
-- Function: double gsl_cdf_laplace_Qinv (double Q, double A)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Laplace distribution with width A.
20.8 The Exponential Power Distribution
=======================================
-- Function: double gsl_ran_exppow (const gsl_rng * R, double A, double
B)
This function returns a random variate from the exponential power
distribution with scale parameter A and exponent B. The
distribution is,
p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx
for x >= 0. For b = 1 this reduces to the Laplace distribution.
For b = 2 it has the same form as a Gaussian distribution, but with
a = \sqrt{2} \sigma.
-- Function: double gsl_ran_exppow_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for an
exponential power distribution with scale parameter A and exponent
B, using the formula given above.
-- Function: double gsl_cdf_exppow_P (double X, double A, double B)
-- Function: double gsl_cdf_exppow_Q (double X, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) for the exponential power distribution with parameters A and
B.
20.9 The Cauchy Distribution
============================
-- Function: double gsl_ran_cauchy (const gsl_rng * R, double A)
This function returns a random variate from the Cauchy distribution
with scale parameter A. The probability distribution for Cauchy
random variates is,
p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx
for x in the range -\infty to +\infty. The Cauchy distribution is
also known as the Lorentz distribution.
-- Function: double gsl_ran_cauchy_pdf (double X, double A)
This function computes the probability density p(x) at X for a
Cauchy distribution with scale parameter A, using the formula given
above.
-- Function: double gsl_cdf_cauchy_P (double X, double A)
-- Function: double gsl_cdf_cauchy_Q (double X, double A)
-- Function: double gsl_cdf_cauchy_Pinv (double P, double A)
-- Function: double gsl_cdf_cauchy_Qinv (double Q, double A)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Cauchy distribution with scale
parameter A.
20.10 The Rayleigh Distribution
===============================
-- Function: double gsl_ran_rayleigh (const gsl_rng * R, double SIGMA)
This function returns a random variate from the Rayleigh
distribution with scale parameter SIGMA. The distribution is,
p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx
for x > 0.
-- Function: double gsl_ran_rayleigh_pdf (double X, double SIGMA)
This function computes the probability density p(x) at X for a
Rayleigh distribution with scale parameter SIGMA, using the formula
given above.
-- Function: double gsl_cdf_rayleigh_P (double X, double SIGMA)
-- Function: double gsl_cdf_rayleigh_Q (double X, double SIGMA)
-- Function: double gsl_cdf_rayleigh_Pinv (double P, double SIGMA)
-- Function: double gsl_cdf_rayleigh_Qinv (double Q, double SIGMA)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Rayleigh distribution with scale
parameter SIGMA.
20.11 The Rayleigh Tail Distribution
====================================
-- Function: double gsl_ran_rayleigh_tail (const gsl_rng * R, double A,
double SIGMA)
This function returns a random variate from the tail of the
Rayleigh distribution with scale parameter SIGMA and a lower limit
of A. The distribution is,
p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
for x > a.
-- Function: double gsl_ran_rayleigh_tail_pdf (double X, double A,
double SIGMA)
This function computes the probability density p(x) at X for a
Rayleigh tail distribution with scale parameter SIGMA and lower
limit A, using the formula given above.
20.12 The Landau Distribution
=============================
-- Function: double gsl_ran_landau (const gsl_rng * R)
This function returns a random variate from the Landau
distribution. The probability distribution for Landau random
variates is defined analytically by the complex integral,
p(x) = (1/(2 \pi i)) \int_{c-i\infty}^{c+i\infty} ds exp(s log(s) + x s)
For numerical purposes it is more convenient to use the following
equivalent form of the integral,
p(x) = (1/\pi) \int_0^\infty dt \exp(-t \log(t) - x t) \sin(\pi t).
-- Function: double gsl_ran_landau_pdf (double X)
This function computes the probability density p(x) at X for the
Landau distribution using an approximation to the formula given
above.
20.13 The Levy alpha-Stable Distributions
=========================================
-- Function: double gsl_ran_levy (const gsl_rng * R, double C, double
ALPHA)
This function returns a random variate from the Levy symmetric
stable distribution with scale C and exponent ALPHA. The symmetric
stable probability distribution is defined by a Fourier transform,
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)
There is no explicit solution for the form of p(x) and the library
does not define a corresponding 'pdf' function. For \alpha = 1 the
distribution reduces to the Cauchy distribution. For \alpha = 2 it
is a Gaussian distribution with \sigma = \sqrt{2} c. For \alpha <
1 the tails of the distribution become extremely wide.
The algorithm only works for 0 < alpha <= 2.
20.14 The Levy skew alpha-Stable Distribution
=============================================
-- Function: double gsl_ran_levy_skew (const gsl_rng * R, double C,
double ALPHA, double BETA)
This function returns a random variate from the Levy skew stable
distribution with scale C, exponent ALPHA and skewness parameter
BETA. The skewness parameter must lie in the range [-1,1]. The
Levy skew stable probability distribution is defined by a Fourier
transform,
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
-(2/\pi)\log|t|. There is no explicit solution for the form of
p(x) and the library does not define a corresponding 'pdf'
function. For \alpha = 2 the distribution reduces to a Gaussian
distribution with \sigma = \sqrt{2} c and the skewness parameter
has no effect. For \alpha < 1 the tails of the distribution become
extremely wide. The symmetric distribution corresponds to \beta =
0.
The algorithm only works for 0 < alpha <= 2.
The Levy alpha-stable distributions have the property that if N
alpha-stable variates are drawn from the distribution p(c, \alpha,
\beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed
as an alpha-stable variate, p(N^(1/\alpha) c, \alpha, \beta).
20.15 The Gamma Distribution
============================
-- Function: double gsl_ran_gamma (const gsl_rng * R, double A, double
B)
This function returns a random variate from the gamma distribution.
The distribution function is,
p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
for x > 0.
The gamma distribution with an integer parameter A is known as the
Erlang distribution.
The variates are computed using the Marsaglia-Tsang fast gamma
method. This function for this method was previously called
'gsl_ran_gamma_mt' and can still be accessed using this name.
-- Function: double gsl_ran_gamma_knuth (const gsl_rng * R, double A,
double B)
This function returns a gamma variate using the algorithms from
Knuth (vol 2).
-- Function: double gsl_ran_gamma_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a
gamma distribution with parameters A and B, using the formula given
above.
-- Function: double gsl_cdf_gamma_P (double X, double A, double B)
-- Function: double gsl_cdf_gamma_Q (double X, double A, double B)
-- Function: double gsl_cdf_gamma_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_gamma_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the gamma distribution with parameters
A and B.
20.16 The Flat (Uniform) Distribution
=====================================
-- Function: double gsl_ran_flat (const gsl_rng * R, double A, double
B)
This function returns a random variate from the flat (uniform)
distribution from A to B. The distribution is,
p(x) dx = {1 \over (b-a)} dx
if a <= x < b and 0 otherwise.
-- Function: double gsl_ran_flat_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a
uniform distribution from A to B, using the formula given above.
-- Function: double gsl_cdf_flat_P (double X, double A, double B)
-- Function: double gsl_cdf_flat_Q (double X, double A, double B)
-- Function: double gsl_cdf_flat_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_flat_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for a uniform distribution from A to B.
20.17 The Lognormal Distribution
================================
-- Function: double gsl_ran_lognormal (const gsl_rng * R, double ZETA,
double SIGMA)
This function returns a random variate from the lognormal
distribution. The distribution function is,
p(x) dx = {1 \over x \sqrt{2 \pi \sigma^2} } \exp(-(\ln(x) - \zeta)^2/2 \sigma^2) dx
for x > 0.
-- Function: double gsl_ran_lognormal_pdf (double X, double ZETA,
double SIGMA)
This function computes the probability density p(x) at X for a
lognormal distribution with parameters ZETA and SIGMA, using the
formula given above.
-- Function: double gsl_cdf_lognormal_P (double X, double ZETA, double
SIGMA)
-- Function: double gsl_cdf_lognormal_Q (double X, double ZETA, double
SIGMA)
-- Function: double gsl_cdf_lognormal_Pinv (double P, double ZETA,
double SIGMA)
-- Function: double gsl_cdf_lognormal_Qinv (double Q, double ZETA,
double SIGMA)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the lognormal distribution with
parameters ZETA and SIGMA.
20.18 The Chi-squared Distribution
==================================
The chi-squared distribution arises in statistics. If Y_i are n
independent Gaussian random variates with unit variance then the
sum-of-squares,
X_i = \sum_i Y_i^2
has a chi-squared distribution with n degrees of freedom.
-- Function: double gsl_ran_chisq (const gsl_rng * R, double NU)
This function returns a random variate from the chi-squared
distribution with NU degrees of freedom. The distribution function
is,
p(x) dx = {1 \over 2 \Gamma(\nu/2) } (x/2)^{\nu/2 - 1} \exp(-x/2) dx
for x >= 0.
-- Function: double gsl_ran_chisq_pdf (double X, double NU)
This function computes the probability density p(x) at X for a
chi-squared distribution with NU degrees of freedom, using the
formula given above.
-- Function: double gsl_cdf_chisq_P (double X, double NU)
-- Function: double gsl_cdf_chisq_Q (double X, double NU)
-- Function: double gsl_cdf_chisq_Pinv (double P, double NU)
-- Function: double gsl_cdf_chisq_Qinv (double Q, double NU)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the chi-squared distribution with NU
degrees of freedom.
20.19 The F-distribution
========================
The F-distribution arises in statistics. If Y_1 and Y_2 are chi-squared
deviates with \nu_1 and \nu_2 degrees of freedom then the ratio,
X = { (Y_1 / \nu_1) \over (Y_2 / \nu_2) }
has an F-distribution F(x;\nu_1,\nu_2).
-- Function: double gsl_ran_fdist (const gsl_rng * R, double NU1,
double NU2)
This function returns a random variate from the F-distribution with
degrees of freedom NU1 and NU2. The distribution function is,
p(x) dx =
{ \Gamma((\nu_1 + \nu_2)/2)
\over \Gamma(\nu_1/2) \Gamma(\nu_2/2) }
\nu_1^{\nu_1/2} \nu_2^{\nu_2/2}
x^{\nu_1/2 - 1} (\nu_2 + \nu_1 x)^{-\nu_1/2 -\nu_2/2}
for x >= 0.
-- Function: double gsl_ran_fdist_pdf (double X, double NU1, double
NU2)
This function computes the probability density p(x) at X for an
F-distribution with NU1 and NU2 degrees of freedom, using the
formula given above.
-- Function: double gsl_cdf_fdist_P (double X, double NU1, double NU2)
-- Function: double gsl_cdf_fdist_Q (double X, double NU1, double NU2)
-- Function: double gsl_cdf_fdist_Pinv (double P, double NU1, double
NU2)
-- Function: double gsl_cdf_fdist_Qinv (double Q, double NU1, double
NU2)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the F-distribution with NU1 and NU2
degrees of freedom.
20.20 The t-distribution
========================
The t-distribution arises in statistics. If Y_1 has a normal
distribution and Y_2 has a chi-squared distribution with \nu degrees of
freedom then the ratio,
X = { Y_1 \over \sqrt{Y_2 / \nu} }
has a t-distribution t(x;\nu) with \nu degrees of freedom.
-- Function: double gsl_ran_tdist (const gsl_rng * R, double NU)
This function returns a random variate from the t-distribution.
The distribution function is,
p(x) dx = {\Gamma((\nu + 1)/2) \over \sqrt{\pi \nu} \Gamma(\nu/2)}
(1 + x^2/\nu)^{-(\nu + 1)/2} dx
for -\infty < x < +\infty.
-- Function: double gsl_ran_tdist_pdf (double X, double NU)
This function computes the probability density p(x) at X for a
t-distribution with NU degrees of freedom, using the formula given
above.
-- Function: double gsl_cdf_tdist_P (double X, double NU)
-- Function: double gsl_cdf_tdist_Q (double X, double NU)
-- Function: double gsl_cdf_tdist_Pinv (double P, double NU)
-- Function: double gsl_cdf_tdist_Qinv (double Q, double NU)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the t-distribution with NU degrees of
freedom.
20.21 The Beta Distribution
===========================
-- Function: double gsl_ran_beta (const gsl_rng * R, double A, double
B)
This function returns a random variate from the beta distribution.
The distribution function is,
p(x) dx = {\Gamma(a+b) \over \Gamma(a) \Gamma(b)} x^{a-1} (1-x)^{b-1} dx
for 0 <= x <= 1.
-- Function: double gsl_ran_beta_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a beta
distribution with parameters A and B, using the formula given
above.
-- Function: double gsl_cdf_beta_P (double X, double A, double B)
-- Function: double gsl_cdf_beta_Q (double X, double A, double B)
-- Function: double gsl_cdf_beta_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_beta_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the beta distribution with parameters A
and B.
20.22 The Logistic Distribution
===============================
-- Function: double gsl_ran_logistic (const gsl_rng * R, double A)
This function returns a random variate from the logistic
distribution. The distribution function is,
p(x) dx = { \exp(-x/a) \over a (1 + \exp(-x/a))^2 } dx
for -\infty < x < +\infty.
-- Function: double gsl_ran_logistic_pdf (double X, double A)
This function computes the probability density p(x) at X for a
logistic distribution with scale parameter A, using the formula
given above.
-- Function: double gsl_cdf_logistic_P (double X, double A)
-- Function: double gsl_cdf_logistic_Q (double X, double A)
-- Function: double gsl_cdf_logistic_Pinv (double P, double A)
-- Function: double gsl_cdf_logistic_Qinv (double Q, double A)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the logistic distribution with scale
parameter A.
20.23 The Pareto Distribution
=============================
-- Function: double gsl_ran_pareto (const gsl_rng * R, double A, double
B)
This function returns a random variate from the Pareto distribution
of order A. The distribution function is,
p(x) dx = (a/b) / (x/b)^{a+1} dx
for x >= b.
-- Function: double gsl_ran_pareto_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a
Pareto distribution with exponent A and scale B, using the formula
given above.
-- Function: double gsl_cdf_pareto_P (double X, double A, double B)
-- Function: double gsl_cdf_pareto_Q (double X, double A, double B)
-- Function: double gsl_cdf_pareto_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_pareto_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Pareto distribution with exponent A
and scale B.
20.24 Spherical Vector Distributions
====================================
The spherical distributions generate random vectors, located on a
spherical surface. They can be used as random directions, for example
in the steps of a random walk.
-- Function: void gsl_ran_dir_2d (const gsl_rng * R, double * X, double
* Y)
-- Function: void gsl_ran_dir_2d_trig_method (const gsl_rng * R, double
* X, double * Y)
This function returns a random direction vector v = (X,Y) in two
dimensions. The vector is normalized such that |v|^2 = x^2 + y^2 =
1. The obvious way to do this is to take a uniform random number
between 0 and 2\pi and let X and Y be the sine and cosine
respectively. Two trig functions would have been expensive in the
old days, but with modern hardware implementations, this is
sometimes the fastest way to go. This is the case for the Pentium
(but not the case for the Sun Sparcstation). One can avoid the
trig evaluations by choosing X and Y in the interior of a unit
circle (choose them at random from the interior of the enclosing
square, and then reject those that are outside the unit circle),
and then dividing by \sqrt{x^2 + y^2}. A much cleverer approach,
attributed to von Neumann (See Knuth, v2, 3rd ed, p140, exercise
23), requires neither trig nor a square root. In this approach, U
and V are chosen at random from the interior of a unit circle, and
then x=(u^2-v^2)/(u^2+v^2) and y=2uv/(u^2+v^2).
-- Function: void gsl_ran_dir_3d (const gsl_rng * R, double * X, double
* Y, double * Z)
This function returns a random direction vector v = (X,Y,Z) in
three dimensions. The vector is normalized such that |v|^2 = x^2 +
y^2 + z^2 = 1. The method employed is due to Robert E. Knop (CACM
13, 326 (1970)), and explained in Knuth, v2, 3rd ed, p136. It uses
the surprising fact that the distribution projected along any axis
is actually uniform (this is only true for 3 dimensions).
-- Function: void gsl_ran_dir_nd (const gsl_rng * R, size_t N, double *
X)
This function returns a random direction vector v =
(x_1,x_2,...,x_n) in N dimensions. The vector is normalized such
that |v|^2 = x_1^2 + x_2^2 + ... + x_n^2 = 1. The method uses the
fact that a multivariate Gaussian distribution is spherically
symmetric. Each component is generated to have a Gaussian
distribution, and then the components are normalized. The method
is described by Knuth, v2, 3rd ed, p135-136, and attributed to G.
W. Brown, Modern Mathematics for the Engineer (1956).
20.25 The Weibull Distribution
==============================
-- Function: double gsl_ran_weibull (const gsl_rng * R, double A,
double B)
This function returns a random variate from the Weibull
distribution. The distribution function is,
p(x) dx = {b \over a^b} x^{b-1} \exp(-(x/a)^b) dx
for x >= 0.
-- Function: double gsl_ran_weibull_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a
Weibull distribution with scale A and exponent B, using the formula
given above.
-- Function: double gsl_cdf_weibull_P (double X, double A, double B)
-- Function: double gsl_cdf_weibull_Q (double X, double A, double B)
-- Function: double gsl_cdf_weibull_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_weibull_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Weibull distribution with scale A
and exponent B.
20.26 The Type-1 Gumbel Distribution
====================================
-- Function: double gsl_ran_gumbel1 (const gsl_rng * R, double A,
double B)
This function returns a random variate from the Type-1 Gumbel
distribution. The Type-1 Gumbel distribution function is,
p(x) dx = a b \exp(-(b \exp(-ax) + ax)) dx
for -\infty < x < \infty.
-- Function: double gsl_ran_gumbel1_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a
Type-1 Gumbel distribution with parameters A and B, using the
formula given above.
-- Function: double gsl_cdf_gumbel1_P (double X, double A, double B)
-- Function: double gsl_cdf_gumbel1_Q (double X, double A, double B)
-- Function: double gsl_cdf_gumbel1_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_gumbel1_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Type-1 Gumbel distribution with
parameters A and B.
20.27 The Type-2 Gumbel Distribution
====================================
-- Function: double gsl_ran_gumbel2 (const gsl_rng * R, double A,
double B)
This function returns a random variate from the Type-2 Gumbel
distribution. The Type-2 Gumbel distribution function is,
p(x) dx = a b x^{-a-1} \exp(-b x^{-a}) dx
for 0 < x < \infty.
-- Function: double gsl_ran_gumbel2_pdf (double X, double A, double B)
This function computes the probability density p(x) at X for a
Type-2 Gumbel distribution with parameters A and B, using the
formula given above.
-- Function: double gsl_cdf_gumbel2_P (double X, double A, double B)
-- Function: double gsl_cdf_gumbel2_Q (double X, double A, double B)
-- Function: double gsl_cdf_gumbel2_Pinv (double P, double A, double B)
-- Function: double gsl_cdf_gumbel2_Qinv (double Q, double A, double B)
These functions compute the cumulative distribution functions P(x),
Q(x) and their inverses for the Type-2 Gumbel distribution with
parameters A and B.
20.28 The Dirichlet Distribution
================================
-- Function: void gsl_ran_dirichlet (const gsl_rng * R, size_t K, const
double ALPHA[], double THETA[])
This function returns an array of K random variates from a
Dirichlet distribution of order K-1. The distribution function is
p(\theta_1, ..., \theta_K) d\theta_1 ... d\theta_K =
(1/Z) \prod_{i=1}^K \theta_i^{\alpha_i - 1} \delta(1 -\sum_{i=1}^K \theta_i) d\theta_1 ... d\theta_K
for theta_i >= 0 and alpha_i > 0. The delta function ensures that
\sum \theta_i = 1. The normalization factor Z is
Z = {\prod_{i=1}^K \Gamma(\alpha_i)} / {\Gamma( \sum_{i=1}^K \alpha_i)}
The random variates are generated by sampling K values from gamma
distributions with parameters a=alpha_i, b=1, and renormalizing.
See A.M. Law, W.D. Kelton, 'Simulation Modeling and Analysis'
(1991).
-- Function: double gsl_ran_dirichlet_pdf (size_t K, const double
ALPHA[], const double THETA[])
This function computes the probability density p(\theta_1, ... ,
\theta_K) at THETA[K] for a Dirichlet distribution with parameters
ALPHA[K], using the formula given above.
-- Function: double gsl_ran_dirichlet_lnpdf (size_t K, const double
ALPHA[], const double THETA[])
This function computes the logarithm of the probability density
p(\theta_1, ... , \theta_K) for a Dirichlet distribution with
parameters ALPHA[K].
20.29 General Discrete Distributions
====================================
Given K discrete events with different probabilities P[k], produce a
random value k consistent with its probability.
The obvious way to do this is to preprocess the probability list by
generating a cumulative probability array with K+1 elements:
C[0] = 0
C[k+1] = C[k]+P[k].
Note that this construction produces C[K]=1. Now choose a uniform
deviate u between 0 and 1, and find the value of k such that C[k] <= u <
C[k+1]. Although this in principle requires of order \log K steps per
random number generation, they are fast steps, and if you use something
like \lfloor uK \rfloor as a starting point, you can often do pretty
well.
But faster methods have been devised. Again, the idea is to
preprocess the probability list, and save the result in some form of
lookup table; then the individual calls for a random discrete event can
go rapidly. An approach invented by G. Marsaglia (Generating discrete
random variables in a computer, Comm ACM 6, 37-38 (1963)) is very
clever, and readers interested in examples of good algorithm design are
directed to this short and well-written paper. Unfortunately, for large
K, Marsaglia's lookup table can be quite large.
A much better approach is due to Alastair J. Walker (An efficient
method for generating discrete random variables with general
distributions, ACM Trans on Mathematical Software 3, 253-256 (1977); see
also Knuth, v2, 3rd ed, p120-121,139). This requires two lookup tables,
one floating point and one integer, but both only of size K. After
preprocessing, the random numbers are generated in O(1) time, even for
large K. The preprocessing suggested by Walker requires O(K^2) effort,
but that is not actually necessary, and the implementation provided here
only takes O(K) effort. In general, more preprocessing leads to faster
generation of the individual random numbers, but a diminishing return is
reached pretty early. Knuth points out that the optimal preprocessing
is combinatorially difficult for large K.
This method can be used to speed up some of the discrete random
number generators below, such as the binomial distribution. To use it
for something like the Poisson Distribution, a modification would have
to be made, since it only takes a finite set of K outcomes.
-- Function: gsl_ran_discrete_t * gsl_ran_discrete_preproc (size_t K,
const double * P)
This function returns a pointer to a structure that contains the
lookup table for the discrete random number generator. The array
P[] contains the probabilities of the discrete events; these array
elements must all be positive, but they needn't add up to one (so
you can think of them more generally as "weights")--the
preprocessor will normalize appropriately. This return value is
used as an argument for the 'gsl_ran_discrete' function below.
-- Function: size_t gsl_ran_discrete (const gsl_rng * R, const
gsl_ran_discrete_t * G)
After the preprocessor, above, has been called, you use this
function to get the discrete random numbers.
-- Function: double gsl_ran_discrete_pdf (size_t K, const
gsl_ran_discrete_t * G)
Returns the probability P[k] of observing the variable K. Since
P[k] is not stored as part of the lookup table, it must be
recomputed; this computation takes O(K), so if K is large and you
care about the original array P[k] used to create the lookup table,
then you should just keep this original array P[k] around.
-- Function: void gsl_ran_discrete_free (gsl_ran_discrete_t * G)
De-allocates the lookup table pointed to by G.
20.30 The Poisson Distribution
==============================
-- Function: unsigned int gsl_ran_poisson (const gsl_rng * R, double
MU)
This function returns a random integer from the Poisson
distribution with mean MU. The probability distribution for
Poisson variates is,
p(k) = {\mu^k \over k!} \exp(-\mu)
for k >= 0.
-- Function: double gsl_ran_poisson_pdf (unsigned int K, double MU)
This function computes the probability p(k) of obtaining K from a
Poisson distribution with mean MU, using the formula given above.
-- Function: double gsl_cdf_poisson_P (unsigned int K, double MU)
-- Function: double gsl_cdf_poisson_Q (unsigned int K, double MU)
These functions compute the cumulative distribution functions P(k),
Q(k) for the Poisson distribution with parameter MU.
20.31 The Bernoulli Distribution
================================
-- Function: unsigned int gsl_ran_bernoulli (const gsl_rng * R, double
P)
This function returns either 0 or 1, the result of a Bernoulli
trial with probability P. The probability distribution for a
Bernoulli trial is,
p(0) = 1 - p
p(1) = p
-- Function: double gsl_ran_bernoulli_pdf (unsigned int K, double P)
This function computes the probability p(k) of obtaining K from a
Bernoulli distribution with probability parameter P, using the
formula given above.
20.32 The Binomial Distribution
===============================
-- Function: unsigned int gsl_ran_binomial (const gsl_rng * R, double
P, unsigned int N)
This function returns a random integer from the binomial
distribution, the number of successes in N independent trials with
probability P. The probability distribution for binomial variates
is,
p(k) = {n! \over k! (n-k)! } p^k (1-p)^{n-k}
for 0 <= k <= n.
-- Function: double gsl_ran_binomial_pdf (unsigned int K, double P,
unsigned int N)
This function computes the probability p(k) of obtaining K from a
binomial distribution with parameters P and N, using the formula
given above.
-- Function: double gsl_cdf_binomial_P (unsigned int K, double P,
unsigned int N)
-- Function: double gsl_cdf_binomial_Q (unsigned int K, double P,
unsigned int N)
These functions compute the cumulative distribution functions P(k),
Q(k) for the binomial distribution with parameters P and N.
20.33 The Multinomial Distribution
==================================
-- Function: void gsl_ran_multinomial (const gsl_rng * R, size_t K,
unsigned int N, const double P[], unsigned int N[])
This function computes a random sample N[] from the multinomial
distribution formed by N trials from an underlying distribution
P[K]. The distribution function for N[] is,
P(n_1, n_2, ..., n_K) =
(N!/(n_1! n_2! ... n_K!)) p_1^n_1 p_2^n_2 ... p_K^n_K
where (n_1, n_2, ..., n_K) are nonnegative integers with
sum_{k=1}^K n_k = N, and (p_1, p_2, ..., p_K) is a probability
distribution with \sum p_i = 1. If the array P[K] is not
normalized then its entries will be treated as weights and
normalized appropriately. The arrays N[] and P[] must both be of
length K.
Random variates are generated using the conditional binomial method
(see C.S. Davis, 'The computer generation of multinomial random
variates', Comp. Stat. Data Anal. 16 (1993) 205-217 for
details).
-- Function: double gsl_ran_multinomial_pdf (size_t K, const double
P[], const unsigned int N[])
This function computes the probability P(n_1, n_2, ..., n_K) of
sampling N[K] from a multinomial distribution with parameters P[K],
using the formula given above.
-- Function: double gsl_ran_multinomial_lnpdf (size_t K, const double
P[], const unsigned int N[])
This function returns the logarithm of the probability for the
multinomial distribution P(n_1, n_2, ..., n_K) with parameters
P[K].
20.34 The Negative Binomial Distribution
========================================
-- Function: unsigned int gsl_ran_negative_binomial (const gsl_rng * R,
double P, double N)
This function returns a random integer from the negative binomial
distribution, the number of failures occurring before N successes
in independent trials with probability P of success. The
probability distribution for negative binomial variates is,
p(k) = {\Gamma(n + k) \over \Gamma(k+1) \Gamma(n) } p^n (1-p)^k
Note that n is not required to be an integer.
-- Function: double gsl_ran_negative_binomial_pdf (unsigned int K,
double P, double N)
This function computes the probability p(k) of obtaining K from a
negative binomial distribution with parameters P and N, using the
formula given above.
-- Function: double gsl_cdf_negative_binomial_P (unsigned int K, double
P, double N)
-- Function: double gsl_cdf_negative_binomial_Q (unsigned int K, double
P, double N)
These functions compute the cumulative distribution functions P(k),
Q(k) for the negative binomial distribution with parameters P and
N.
20.35 The Pascal Distribution
=============================
-- Function: unsigned int gsl_ran_pascal (const gsl_rng * R, double P,
unsigned int N)
This function returns a random integer from the Pascal
distribution. The Pascal distribution is simply a negative
binomial distribution with an integer value of n.
p(k) = {(n + k - 1)! \over k! (n - 1)! } p^n (1-p)^k
for k >= 0
-- Function: double gsl_ran_pascal_pdf (unsigned int K, double P,
unsigned int N)
This function computes the probability p(k) of obtaining K from a
Pascal distribution with parameters P and N, using the formula
given above.
-- Function: double gsl_cdf_pascal_P (unsigned int K, double P,
unsigned int N)
-- Function: double gsl_cdf_pascal_Q (unsigned int K, double P,
unsigned int N)
These functions compute the cumulative distribution functions P(k),
Q(k) for the Pascal distribution with parameters P and N.
20.36 The Geometric Distribution
================================
-- Function: unsigned int gsl_ran_geometric (const gsl_rng * R, double
P)
This function returns a random integer from the geometric
distribution, the number of independent trials with probability P
until the first success. The probability distribution for
geometric variates is,
p(k) = p (1-p)^(k-1)
for k >= 1. Note that the distribution begins with k=1 with this
definition. There is another convention in which the exponent k-1
is replaced by k.
-- Function: double gsl_ran_geometric_pdf (unsigned int K, double P)
This function computes the probability p(k) of obtaining K from a
geometric distribution with probability parameter P, using the
formula given above.
-- Function: double gsl_cdf_geometric_P (unsigned int K, double P)
-- Function: double gsl_cdf_geometric_Q (unsigned int K, double P)
These functions compute the cumulative distribution functions P(k),
Q(k) for the geometric distribution with parameter P.
20.37 The Hypergeometric Distribution
=====================================
-- Function: unsigned int gsl_ran_hypergeometric (const gsl_rng * R,
unsigned int N1, unsigned int N2, unsigned int T)
This function returns a random integer from the hypergeometric
distribution. The probability distribution for hypergeometric
random variates is,
p(k) = C(n_1, k) C(n_2, t - k) / C(n_1 + n_2, t)
where C(a,b) = a!/(b!(a-b)!) and t <= n_1 + n_2. The domain of k
is max(0,t-n_2), ..., min(t,n_1).
If a population contains n_1 elements of "type 1" and n_2 elements
of "type 2" then the hypergeometric distribution gives the
probability of obtaining k elements of "type 1" in t samples from
the population without replacement.
-- Function: double gsl_ran_hypergeometric_pdf (unsigned int K,
unsigned int N1, unsigned int N2, unsigned int T)
This function computes the probability p(k) of obtaining K from a
hypergeometric distribution with parameters N1, N2, T, using the
formula given above.
-- Function: double gsl_cdf_hypergeometric_P (unsigned int K, unsigned
int N1, unsigned int N2, unsigned int T)
-- Function: double gsl_cdf_hypergeometric_Q (unsigned int K, unsigned
int N1, unsigned int N2, unsigned int T)
These functions compute the cumulative distribution functions P(k),
Q(k) for the hypergeometric distribution with parameters N1, N2 and
T.
20.38 The Logarithmic Distribution
==================================
-- Function: unsigned int gsl_ran_logarithmic (const gsl_rng * R,
double P)
This function returns a random integer from the logarithmic
distribution. The probability distribution for logarithmic random
variates is,
p(k) = {-1 \over \log(1-p)} {(p^k \over k)}
for k >= 1.
-- Function: double gsl_ran_logarithmic_pdf (unsigned int K, double P)
This function computes the probability p(k) of obtaining K from a
logarithmic distribution with probability parameter P, using the
formula given above.
20.39 Shuffling and Sampling
============================
The following functions allow the shuffling and sampling of a set of
objects. The algorithms rely on a random number generator as a source
of randomness and a poor quality generator can lead to correlations in
the output. In particular it is important to avoid generators with a
short period. For more information see Knuth, v2, 3rd ed, Section
3.4.2, "Random Sampling and Shuffling".
-- Function: void gsl_ran_shuffle (const gsl_rng * R, void * BASE,
size_t N, size_t SIZE)
This function randomly shuffles the order of N objects, each of
size SIZE, stored in the array BASE[0..N-1]. The output of the
random number generator R is used to produce the permutation. The
algorithm generates all possible n! permutations with equal
probability, assuming a perfect source of random numbers.
The following code shows how to shuffle the numbers from 0 to 51,
int a[52];
for (i = 0; i < 52; i++)
{
a[i] = i;
}
gsl_ran_shuffle (r, a, 52, sizeof (int));
-- Function: int gsl_ran_choose (const gsl_rng * R, void * DEST, size_t
K, void * SRC, size_t N, size_t SIZE)
This function fills the array DEST[k] with K objects taken randomly
from the N elements of the array SRC[0..N-1]. The objects are each
of size SIZE. The output of the random number generator R is used
to make the selection. The algorithm ensures all possible samples
are equally likely, assuming a perfect source of randomness.
The objects are sampled _without_ replacement, thus each object can
only appear once in DEST[k]. It is required that K be less than or
equal to 'n'. The objects in DEST will be in the same relative
order as those in SRC. You will need to call 'gsl_ran_shuffle(r,
dest, n, size)' if you want to randomize the order.
The following code shows how to select a random sample of three
unique numbers from the set 0 to 99,
double a[3], b[100];
for (i = 0; i < 100; i++)
{
b[i] = (double) i;
}
gsl_ran_choose (r, a, 3, b, 100, sizeof (double));
-- Function: void gsl_ran_sample (const gsl_rng * R, void * DEST,
size_t K, void * SRC, size_t N, size_t SIZE)
This function is like 'gsl_ran_choose' but samples K items from the
original array of N items SRC with replacement, so the same object
can appear more than once in the output sequence DEST. There is no
requirement that K be less than N in this case.
20.40 Examples
==============
The following program demonstrates the use of a random number generator
to produce variates from a distribution. It prints 10 samples from the
Poisson distribution with a mean of 3.
#include
#include
#include
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
int i, n = 10;
double mu = 3.0;
/* create a generator chosen by the
environment variable GSL_RNG_TYPE */
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
/* print n random variates chosen from
the poisson distribution with mean
parameter mu */
for (i = 0; i < n; i++)
{
unsigned int k = gsl_ran_poisson (r, mu);
printf (" %u", k);
}
printf ("\n");
gsl_rng_free (r);
return 0;
}
If the library and header files are installed under '/usr/local' (the
default location) then the program can be compiled with these options,
$ gcc -Wall demo.c -lgsl -lgslcblas -lm
Here is the output of the program,
$ ./a.out
2 5 5 2 1 0 3 4 1 1
The variates depend on the seed used by the generator. The seed for the
default generator type 'gsl_rng_default' can be changed with the
'GSL_RNG_SEED' environment variable to produce a different stream of
variates,
$ GSL_RNG_SEED=123 ./a.out
4 5 6 3 3 1 4 2 5 5
The following program generates a random walk in two dimensions.
#include
#include
#include
int
main (void)
{
int i;
double x = 0, y = 0, dx, dy;
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
printf ("%g %g\n", x, y);
for (i = 0; i < 10; i++)
{
gsl_ran_dir_2d (r, &dx, &dy);
x += dx; y += dy;
printf ("%g %g\n", x, y);
}
gsl_rng_free (r);
return 0;
}
Here is some output from the program, four 10-step random walks from the
origin,
The following program computes the upper and lower cumulative
distribution functions for the standard normal distribution at x=2.
#include
#include
int
main (void)
{
double P, Q;
double x = 2.0;
P = gsl_cdf_ugaussian_P (x);
printf ("prob(x < %f) = %f\n", x, P);
Q = gsl_cdf_ugaussian_Q (x);
printf ("prob(x > %f) = %f\n", x, Q);
x = gsl_cdf_ugaussian_Pinv (P);
printf ("Pinv(%f) = %f\n", P, x);
x = gsl_cdf_ugaussian_Qinv (Q);
printf ("Qinv(%f) = %f\n", Q, x);
return 0;
}
Here is the output of the program,
prob(x < 2.000000) = 0.977250
prob(x > 2.000000) = 0.022750
Pinv(0.977250) = 2.000000
Qinv(0.022750) = 2.000000
20.41 References and Further Reading
====================================
For an encyclopaedic coverage of the subject readers are advised to
consult the book 'Non-Uniform Random Variate Generation' by Luc Devroye.
It covers every imaginable distribution and provides hundreds of
algorithms.
Luc Devroye, 'Non-Uniform Random Variate Generation',
Springer-Verlag, ISBN 0-387-96305-7. Available online at
.
The subject of random variate generation is also reviewed by Knuth, who
describes algorithms for all the major distributions.
Donald E. Knuth, 'The Art of Computer Programming: Seminumerical
Algorithms' (Vol 2, 3rd Ed, 1997), Addison-Wesley, ISBN 0201896842.
The Particle Data Group provides a short review of techniques for
generating distributions of random numbers in the "Monte Carlo" section
of its Annual Review of Particle Physics.
'Review of Particle Properties' R.M. Barnett et al., Physical
Review D54, 1 (1996) .
The Review of Particle Physics is available online in postscript and pdf
format.
An overview of methods used to compute cumulative distribution functions
can be found in 'Statistical Computing' by W.J. Kennedy and J.E. Gentle.
Another general reference is 'Elements of Statistical Computing' by R.A.
Thisted.
William E. Kennedy and James E. Gentle, 'Statistical Computing'
(1980), Marcel Dekker, ISBN 0-8247-6898-1.
Ronald A. Thisted, 'Elements of Statistical Computing' (1988),
Chapman & Hall, ISBN 0-412-01371-1.
The cumulative distribution functions for the Gaussian distribution are
based on the following papers,
'Rational Chebyshev Approximations Using Linear Equations', W.J.
Cody, W. Fraser, J.F. Hart. Numerische Mathematik 12, 242-251
(1968).
'Rational Chebyshev Approximations for the Error Function', W.J.
Cody. Mathematics of Computation 23, n107, 631-637 (July 1969).
21 Statistics
*************
This chapter describes the statistical functions in the library. The
basic statistical functions include routines to compute the mean,
variance and standard deviation. More advanced functions allow you to
calculate absolute deviations, skewness, and kurtosis as well as the
median and arbitrary percentiles. The algorithms use recurrence
relations to compute average quantities in a stable way, without large
intermediate values that might overflow.
The functions are available in versions for datasets in the standard
floating-point and integer types. The versions for double precision
floating-point data have the prefix 'gsl_stats' and are declared in the
header file 'gsl_statistics_double.h'. The versions for integer data
have the prefix 'gsl_stats_int' and are declared in the header file
'gsl_statistics_int.h'. All the functions operate on C arrays with a
STRIDE parameter specifying the spacing between elements.
21.1 Mean, Standard Deviation and Variance
==========================================
-- Function: double gsl_stats_mean (const double DATA[], size_t STRIDE,
size_t N)
This function returns the arithmetic mean of DATA, a dataset of
length N with stride STRIDE. The arithmetic mean, or "sample
mean", is denoted by \Hat\mu and defined as,
\Hat\mu = (1/N) \sum x_i
where x_i are the elements of the dataset DATA. For samples drawn
from a gaussian distribution the variance of \Hat\mu is \sigma^2 /
N.
-- Function: double gsl_stats_variance (const double DATA[], size_t
STRIDE, size_t N)
This function returns the estimated, or "sample", variance of DATA,
a dataset of length N with stride STRIDE. The estimated variance
is denoted by \Hat\sigma^2 and is defined by,
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
where x_i are the elements of the dataset DATA. Note that the
normalization factor of 1/(N-1) results from the derivation of
\Hat\sigma^2 as an unbiased estimator of the population variance
\sigma^2. For samples drawn from a Gaussian distribution the
variance of \Hat\sigma^2 itself is 2 \sigma^4 / N.
This function computes the mean via a call to 'gsl_stats_mean'. If
you have already computed the mean then you can pass it directly to
'gsl_stats_variance_m'.
-- Function: double gsl_stats_variance_m (const double DATA[], size_t
STRIDE, size_t N, double MEAN)
This function returns the sample variance of DATA relative to the
given value of MEAN. The function is computed with \Hat\mu
replaced by the value of MEAN that you supply,
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - mean)^2
-- Function: double gsl_stats_sd (const double DATA[], size_t STRIDE,
size_t N)
-- Function: double gsl_stats_sd_m (const double DATA[], size_t STRIDE,
size_t N, double MEAN)
The standard deviation is defined as the square root of the
variance. These functions return the square root of the
corresponding variance functions above.
-- Function: double gsl_stats_tss (const double DATA[], size_t STRIDE,
size_t N)
-- Function: double gsl_stats_tss_m (const double DATA[], size_t
STRIDE, size_t N, double MEAN)
These functions return the total sum of squares (TSS) of DATA about
the mean. For 'gsl_stats_tss_m' the user-supplied value of MEAN is
used, and for 'gsl_stats_tss' it is computed using
'gsl_stats_mean'.
TSS = \sum (x_i - mean)^2
-- Function: double gsl_stats_variance_with_fixed_mean (const double
DATA[], size_t STRIDE, size_t N, double MEAN)
This function computes an unbiased estimate of the variance of DATA
when the population mean MEAN of the underlying distribution is
known _a priori_. In this case the estimator for the variance uses
the factor 1/N and the sample mean \Hat\mu is replaced by the known
population mean \mu,
\Hat\sigma^2 = (1/N) \sum (x_i - \mu)^2
-- Function: double gsl_stats_sd_with_fixed_mean (const double DATA[],
size_t STRIDE, size_t N, double MEAN)
This function calculates the standard deviation of DATA for a fixed
population mean MEAN. The result is the square root of the
corresponding variance function.
21.2 Absolute deviation
=======================
-- Function: double gsl_stats_absdev (const double DATA[], size_t
STRIDE, size_t N)
This function computes the absolute deviation from the mean of
DATA, a dataset of length N with stride STRIDE. The absolute
deviation from the mean is defined as,
absdev = (1/N) \sum |x_i - \Hat\mu|
where x_i are the elements of the dataset DATA. The absolute
deviation from the mean provides a more robust measure of the width
of a distribution than the variance. This function computes the
mean of DATA via a call to 'gsl_stats_mean'.
-- Function: double gsl_stats_absdev_m (const double DATA[], size_t
STRIDE, size_t N, double MEAN)
This function computes the absolute deviation of the dataset DATA
relative to the given value of MEAN,
absdev = (1/N) \sum |x_i - mean|
This function is useful if you have already computed the mean of
DATA (and want to avoid recomputing it), or wish to calculate the
absolute deviation relative to another value (such as zero, or the
median).
21.3 Higher moments (skewness and kurtosis)
===========================================
-- Function: double gsl_stats_skew (const double DATA[], size_t STRIDE,
size_t N)
This function computes the skewness of DATA, a dataset of length N
with stride STRIDE. The skewness is defined as,
skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
where x_i are the elements of the dataset DATA. The skewness
measures the asymmetry of the tails of a distribution.
The function computes the mean and estimated standard deviation of
DATA via calls to 'gsl_stats_mean' and 'gsl_stats_sd'.
-- Function: double gsl_stats_skew_m_sd (const double DATA[], size_t
STRIDE, size_t N, double MEAN, double SD)
This function computes the skewness of the dataset DATA using the
given values of the mean MEAN and standard deviation SD,
skew = (1/N) \sum ((x_i - mean)/sd)^3
These functions are useful if you have already computed the mean
and standard deviation of DATA and want to avoid recomputing them.
-- Function: double gsl_stats_kurtosis (const double DATA[], size_t
STRIDE, size_t N)
This function computes the kurtosis of DATA, a dataset of length N
with stride STRIDE. The kurtosis is defined as,
kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4) - 3
The kurtosis measures how sharply peaked a distribution is,
relative to its width. The kurtosis is normalized to zero for a
Gaussian distribution.
-- Function: double gsl_stats_kurtosis_m_sd (const double DATA[],
size_t STRIDE, size_t N, double MEAN, double SD)
This function computes the kurtosis of the dataset DATA using the
given values of the mean MEAN and standard deviation SD,
kurtosis = ((1/N) \sum ((x_i - mean)/sd)^4) - 3
This function is useful if you have already computed the mean and
standard deviation of DATA and want to avoid recomputing them.
21.4 Autocorrelation
====================
-- Function: double gsl_stats_lag1_autocorrelation (const double
DATA[], const size_t STRIDE, const size_t N)
This function computes the lag-1 autocorrelation of the dataset
DATA.
a_1 = {\sum_{i = 2}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu)
\over
\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}
-- Function: double gsl_stats_lag1_autocorrelation_m (const double
DATA[], const size_t STRIDE, const size_t N, const double
MEAN)
This function computes the lag-1 autocorrelation of the dataset
DATA using the given value of the mean MEAN.
21.5 Covariance
===============
-- Function: double gsl_stats_covariance (const double DATA1[], const
size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
const size_t N)
This function computes the covariance of the datasets DATA1 and
DATA2 which must both be of the same length N.
covar = (1/(n - 1)) \sum_{i = 1}^{n} (x_i - \Hat x) (y_i - \Hat y)
-- Function: double gsl_stats_covariance_m (const double DATA1[], const
size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
const size_t N, const double MEAN1, const double MEAN2)
This function computes the covariance of the datasets DATA1 and
DATA2 using the given values of the means, MEAN1 and MEAN2. This
is useful if you have already computed the means of DATA1 and DATA2
and want to avoid recomputing them.
21.6 Correlation
================
-- Function: double gsl_stats_correlation (const double DATA1[], const
size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
const size_t N)
This function efficiently computes the Pearson correlation
coefficient between the datasets DATA1 and DATA2 which must both be
of the same length N.
r = cov(x, y) / (\Hat\sigma_x \Hat\sigma_y)
= {1/(n-1) \sum (x_i - \Hat x) (y_i - \Hat y)
\over
\sqrt{1/(n-1) \sum (x_i - \Hat x)^2} \sqrt{1/(n-1) \sum (y_i - \Hat y)^2}
}
-- Function: double gsl_stats_spearman (const double DATA1[], const
size_t STRIDE1, const double DATA2[], const size_t STRIDE2,
const size_t N, double WORK[])
This function computes the Spearman rank correlation coefficient
between the datasets DATA1 and DATA2 which must both be of the same
length N. Additional workspace of size 2*N is required in WORK.
The Spearman rank correlation between vectors x and y is equivalent
to the Pearson correlation between the ranked vectors x_R and y_R,
where ranks are defined to be the average of the positions of an
element in the ascending order of the values.
21.7 Weighted Samples
=====================
The functions described in this section allow the computation of
statistics for weighted samples. The functions accept an array of
samples, x_i, with associated weights, w_i. Each sample x_i is
considered as having been drawn from a Gaussian distribution with
variance \sigma_i^2. The sample weight w_i is defined as the reciprocal
of this variance, w_i = 1/\sigma_i^2. Setting a weight to zero
corresponds to removing a sample from a dataset.
-- Function: double gsl_stats_wmean (const double W[], size_t WSTRIDE,
const double DATA[], size_t STRIDE, size_t N)
This function returns the weighted mean of the dataset DATA with
stride STRIDE and length N, using the set of weights W with stride
WSTRIDE and length N. The weighted mean is defined as,
\Hat\mu = (\sum w_i x_i) / (\sum w_i)
-- Function: double gsl_stats_wvariance (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
This function returns the estimated variance of the dataset DATA
with stride STRIDE and length N, using the set of weights W with
stride WSTRIDE and length N. The estimated variance of a weighted
dataset is calculated as,
\Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2)))
\sum w_i (x_i - \Hat\mu)^2
Note that this expression reduces to an unweighted variance with
the familiar 1/(N-1) factor when there are N equal non-zero
weights.
-- Function: double gsl_stats_wvariance_m (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
WMEAN)
This function returns the estimated variance of the weighted
dataset DATA using the given weighted mean WMEAN.
-- Function: double gsl_stats_wsd (const double W[], size_t WSTRIDE,
const double DATA[], size_t STRIDE, size_t N)
The standard deviation is defined as the square root of the
variance. This function returns the square root of the
corresponding variance function 'gsl_stats_wvariance' above.
-- Function: double gsl_stats_wsd_m (const double W[], size_t WSTRIDE,
const double DATA[], size_t STRIDE, size_t N, double WMEAN)
This function returns the square root of the corresponding variance
function 'gsl_stats_wvariance_m' above.
-- Function: double gsl_stats_wvariance_with_fixed_mean (const double
W[], size_t WSTRIDE, const double DATA[], size_t STRIDE,
size_t N, const double MEAN)
This function computes an unbiased estimate of the variance of the
weighted dataset DATA when the population mean MEAN of the
underlying distribution is known _a priori_. In this case the
estimator for the variance replaces the sample mean \Hat\mu by the
known population mean \mu,
\Hat\sigma^2 = (\sum w_i (x_i - \mu)^2) / (\sum w_i)
-- Function: double gsl_stats_wsd_with_fixed_mean (const double W[],
size_t WSTRIDE, const double DATA[], size_t STRIDE, size_t N,
const double MEAN)
The standard deviation is defined as the square root of the
variance. This function returns the square root of the
corresponding variance function above.
-- Function: double gsl_stats_wtss (const double W[], const size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
-- Function: double gsl_stats_wtss_m (const double W[], const size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
WMEAN)
These functions return the weighted total sum of squares (TSS) of
DATA about the weighted mean. For 'gsl_stats_wtss_m' the
user-supplied value of WMEAN is used, and for 'gsl_stats_wtss' it
is computed using 'gsl_stats_wmean'.
TSS = \sum w_i (x_i - wmean)^2
-- Function: double gsl_stats_wabsdev (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
This function computes the weighted absolute deviation from the
weighted mean of DATA. The absolute deviation from the mean is
defined as,
absdev = (\sum w_i |x_i - \Hat\mu|) / (\sum w_i)
-- Function: double gsl_stats_wabsdev_m (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
WMEAN)
This function computes the absolute deviation of the weighted
dataset DATA about the given weighted mean WMEAN.
-- Function: double gsl_stats_wskew (const double W[], size_t WSTRIDE,
const double DATA[], size_t STRIDE, size_t N)
This function computes the weighted skewness of the dataset DATA.
skew = (\sum w_i ((x_i - \Hat x)/\Hat \sigma)^3) / (\sum w_i)
-- Function: double gsl_stats_wskew_m_sd (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
WMEAN, double WSD)
This function computes the weighted skewness of the dataset DATA
using the given values of the weighted mean and weighted standard
deviation, WMEAN and WSD.
-- Function: double gsl_stats_wkurtosis (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N)
This function computes the weighted kurtosis of the dataset DATA.
kurtosis = ((\sum w_i ((x_i - \Hat x)/\Hat \sigma)^4) / (\sum w_i)) - 3
-- Function: double gsl_stats_wkurtosis_m_sd (const double W[], size_t
WSTRIDE, const double DATA[], size_t STRIDE, size_t N, double
WMEAN, double WSD)
This function computes the weighted kurtosis of the dataset DATA
using the given values of the weighted mean and weighted standard
deviation, WMEAN and WSD.
21.8 Maximum and Minimum values
===============================
The following functions find the maximum and minimum values of a dataset
(or their indices). If the data contains 'NaN's then a 'NaN' will be
returned, since the maximum or minimum value is undefined. For
functions which return an index, the location of the first 'NaN' in the
array is returned.
-- Function: double gsl_stats_max (const double DATA[], size_t STRIDE,
size_t N)
This function returns the maximum value in DATA, a dataset of
length N with stride STRIDE. The maximum value is defined as the
value of the element x_i which satisfies x_i >= x_j for all j.
If you want instead to find the element with the largest absolute
magnitude you will need to apply 'fabs' or 'abs' to your data
before calling this function.
-- Function: double gsl_stats_min (const double DATA[], size_t STRIDE,
size_t N)
This function returns the minimum value in DATA, a dataset of
length N with stride STRIDE. The minimum value is defined as the
value of the element x_i which satisfies x_i <= x_j for all j.
If you want instead to find the element with the smallest absolute
magnitude you will need to apply 'fabs' or 'abs' to your data
before calling this function.
-- Function: void gsl_stats_minmax (double * MIN, double * MAX, const
double DATA[], size_t STRIDE, size_t N)
This function finds both the minimum and maximum values MIN, MAX in
DATA in a single pass.
-- Function: size_t gsl_stats_max_index (const double DATA[], size_t
STRIDE, size_t N)
This function returns the index of the maximum value in DATA, a
dataset of length N with stride STRIDE. The maximum value is
defined as the value of the element x_i which satisfies x_i >= x_j
for all j. When there are several equal maximum elements then the
first one is chosen.
-- Function: size_t gsl_stats_min_index (const double DATA[], size_t
STRIDE, size_t N)
This function returns the index of the minimum value in DATA, a
dataset of length N with stride STRIDE. The minimum value is
defined as the value of the element x_i which satisfies x_i >= x_j
for all j. When there are several equal minimum elements then the
first one is chosen.
-- Function: void gsl_stats_minmax_index (size_t * MIN_INDEX, size_t *
MAX_INDEX, const double DATA[], size_t STRIDE, size_t N)
This function returns the indexes MIN_INDEX, MAX_INDEX of the
minimum and maximum values in DATA in a single pass.
21.9 Median and Percentiles
===========================
The median and percentile functions described in this section operate on
sorted data. For convenience we use "quantiles", measured on a scale of
0 to 1, instead of percentiles (which use a scale of 0 to 100).
-- Function: double gsl_stats_median_from_sorted_data (const double
SORTED_DATA[], size_t STRIDE, size_t N)
This function returns the median value of SORTED_DATA, a dataset of
length N with stride STRIDE. The elements of the array must be in
ascending numerical order. There are no checks to see whether the
data are sorted, so the function 'gsl_sort' should always be used
first.
When the dataset has an odd number of elements the median is the
value of element (n-1)/2. When the dataset has an even number of
elements the median is the mean of the two nearest middle values,
elements (n-1)/2 and n/2. Since the algorithm for computing the
median involves interpolation this function always returns a
floating-point number, even for integer data types.
-- Function: double gsl_stats_quantile_from_sorted_data (const double
SORTED_DATA[], size_t STRIDE, size_t N, double F)
This function returns a quantile value of SORTED_DATA, a
double-precision array of length N with stride STRIDE. The
elements of the array must be in ascending numerical order. The
quantile is determined by the F, a fraction between 0 and 1. For
example, to compute the value of the 75th percentile F should have
the value 0.75.
There are no checks to see whether the data are sorted, so the
function 'gsl_sort' should always be used first.
The quantile is found by interpolation, using the formula
quantile = (1 - \delta) x_i + \delta x_{i+1}
where i is 'floor'((n - 1)f) and \delta is (n-1)f - i.
Thus the minimum value of the array ('data[0*stride]') is given by
F equal to zero, the maximum value ('data[(n-1)*stride]') is given
by F equal to one and the median value is given by F equal to 0.5.
Since the algorithm for computing quantiles involves interpolation
this function always returns a floating-point number, even for
integer data types.
21.10 Examples
==============
Here is a basic example of how to use the statistical functions:
#include
#include
int
main(void)
{
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double mean, variance, largest, smallest;
mean = gsl_stats_mean(data, 1, 5);
variance = gsl_stats_variance(data, 1, 5);
largest = gsl_stats_max(data, 1, 5);
smallest = gsl_stats_min(data, 1, 5);
printf ("The dataset is %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
printf ("The sample mean is %g\n", mean);
printf ("The estimated variance is %g\n", variance);
printf ("The largest value is %g\n", largest);
printf ("The smallest value is %g\n", smallest);
return 0;
}
The program should produce the following output,
The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
The sample mean is 16.54
The estimated variance is 5.373
The largest value is 18.3
The smallest value is 12.6
Here is an example using sorted data,
#include
#include
#include
int
main(void)
{
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double median, upperq, lowerq;
printf ("Original dataset: %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
gsl_sort (data, 1, 5);
printf ("Sorted dataset: %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
median
= gsl_stats_median_from_sorted_data (data,
1, 5);
upperq
= gsl_stats_quantile_from_sorted_data (data,
1, 5,
0.75);
lowerq
= gsl_stats_quantile_from_sorted_data (data,
1, 5,
0.25);
printf ("The median is %g\n", median);
printf ("The upper quartile is %g\n", upperq);
printf ("The lower quartile is %g\n", lowerq);
return 0;
}
This program should produce the following output,
Original dataset: 17.2, 18.1, 16.5, 18.3, 12.6
Sorted dataset: 12.6, 16.5, 17.2, 18.1, 18.3
The median is 17.2
The upper quartile is 18.1
The lower quartile is 16.5
21.11 References and Further Reading
====================================
The standard reference for almost any topic in statistics is the
multi-volume 'Advanced Theory of Statistics' by Kendall and Stuart.
Maurice Kendall, Alan Stuart, and J. Keith Ord. 'The Advanced
Theory of Statistics' (multiple volumes) reprinted as 'Kendall's
Advanced Theory of Statistics'. Wiley, ISBN 047023380X.
Many statistical concepts can be more easily understood by a Bayesian
approach. The following book by Gelman, Carlin, Stern and Rubin gives a
comprehensive coverage of the subject.
Andrew Gelman, John B. Carlin, Hal S. Stern, Donald B. Rubin.
'Bayesian Data Analysis'. Chapman & Hall, ISBN 0412039915.
For physicists the Particle Data Group provides useful reviews of
Probability and Statistics in the "Mathematical Tools" section of its
Annual Review of Particle Physics.
'Review of Particle Properties' R.M. Barnett et al., Physical
Review D54, 1 (1996)
The Review of Particle Physics is available online at the website
.
22 Running Statistics
*********************
This chapter describes routines for computing running statistics, also
known as online statistics, of data. These routines are suitable for
handling large datasets for which it may be inconvenient or impractical
to store in memory all at once. The data can be processed in a single
pass, one point at a time. Each time a data point is added to the
accumulator, internal parameters are updated in order to compute the
current mean, variance, standard deviation, skewness, and kurtosis.
These statistics are exact, and are updated with numerically stable
single-pass algorithms. The median and arbitrary quantiles are also
available, however these calculations use algorithms which provide
approximations, and grow more accurate as more data is added to the
accumulator.
The functions described in this chapter are declared in the header
file 'gsl_rstat.h'.
22.1 Initializing the Accumulator
=================================
-- Function: gsl_rstat_workspace * gsl_rstat_alloc (void)
This function allocates a workspace for computing running
statistics. The size of the workspace is O(1).
-- Function: void gsl_rstat_free (gsl_rstat_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_rstat_reset (gsl_rstat_workspace * W)
This function resets the workspace W to its initial state, so it
can begin working on a new set of data.
22.2 Adding Data to the Accumulator
===================================
-- Function: int gsl_rstat_add (const double X, gsl_rstat_workspace *
W)
This function adds the data point X to the statistical accumulator,
updating calculations of the mean, variance, standard deviation,
skewness, kurtosis, and median.
-- Function: size_t gsl_rstat_n (gsl_rstat_workspace * W)
This function returns the number of data so far added to the
accumulator.
22.3 Current Statistics
=======================
-- Function: double gsl_rstat_min (gsl_rstat_workspace * W)
This function returns the minimum value added to the accumulator.
-- Function: double gsl_rstat_max (gsl_rstat_workspace * W)
This function returns the maximum value added to the accumulator.
-- Function: double gsl_rstat_mean (gsl_rstat_workspace * W)
This function returns the mean of all data added to the
accumulator, defined as
\Hat\mu = (1/N) \sum x_i
-- Function: double gsl_rstat_variance (gsl_rstat_workspace * W)
This function returns the variance of all data added to the
accumulator, defined as
\Hat\sigma^2 = (1/(N-1)) \sum (x_i - \Hat\mu)^2
-- Function: double gsl_rstat_sd (gsl_rstat_workspace * W)
This function returns the standard deviation of all data added to
the accumulator, defined as the square root of the variance given
above.
-- Function: double gsl_rstat_sd_mean (gsl_rstat_workspace * W)
This function returns the standard deviation of the mean, defined
as
sd_mean = \Hat\sigma / \sqrt{N}
-- Function: double gsl_rstat_rms (gsl_rstat_workspace * W)
This function returns the root mean square of all data added to the
accumulator, defined as
rms = \sqrt ( 1/N \sum x_i^2 )
-- Function: double gsl_rstat_skew (gsl_rstat_workspace * W)
This function returns the skewness of all data added to the
accumulator, defined as
skew = (1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^3
-- Function: double gsl_rstat_kurtosis (gsl_rstat_workspace * W)
This function returns the kurtosis of all data added to the
accumulator, defined as
kurtosis = ((1/N) \sum ((x_i - \Hat\mu)/\Hat\sigma)^4) - 3
-- Function: double gsl_rstat_median (gsl_rstat_workspace * W)
This function returns an estimate of the median of the data added
to the accumulator.
22.4 Quantiles
==============
The functions in this section estimate quantiles dynamically without
storing the entire dataset, using the algorithm of Jain and Chlamtec,
1985. Only five points (markers) are stored which represent the minimum
and maximum of the data, as well as current estimates of the p/2-, p-,
and (1+p)/2-quantiles. Each time a new data point is added, the marker
positions and heights are updated.
-- Function: gsl_rstat_quantile_workspace * gsl_rstat_quantile_alloc
(const double P)
This function allocates a workspace for the dynamic estimation of
P-quantiles, where P is between 0 and 1. The median corresponds to
p = 0.5. The size of the workspace is O(1).
-- Function: void gsl_rstat_quantile_free (gsl_rstat_quantile_workspace
* W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_rstat_quantile_reset (gsl_rstat_quantile_workspace
* W)
This function resets the workspace W to its initial state, so it
can begin working on a new set of data.
-- Function: int gsl_rstat_quantile_add (const double X,
gsl_rstat_quantile_workspace * W)
This function updates the estimate of the p-quantile with the new
data point X.
-- Function: double gsl_rstat_quantile_get
(gsl_rstat_quantile_workspace * W)
This function returns the current estimate of the p-quantile.
22.5 Examples
=============
Here is a basic example of how to use the statistical functions:
#include
#include
int
main(void)
{
double data[5] = {17.2, 18.1, 16.5, 18.3, 12.6};
double mean, variance, largest, smallest, sd,
rms, sd_mean, median, skew, kurtosis;
gsl_rstat_workspace *rstat_p = gsl_rstat_alloc();
size_t i, n;
/* add data to rstat accumulator */
for (i = 0; i < 5; ++i)
gsl_rstat_add(data[i], rstat_p);
mean = gsl_rstat_mean(rstat_p);
variance = gsl_rstat_variance(rstat_p);
largest = gsl_rstat_max(rstat_p);
smallest = gsl_rstat_min(rstat_p);
median = gsl_rstat_median(rstat_p);
sd = gsl_rstat_sd(rstat_p);
sd_mean = gsl_rstat_sd_mean(rstat_p);
skew = gsl_rstat_skew(rstat_p);
rms = gsl_rstat_rms(rstat_p);
kurtosis = gsl_rstat_kurtosis(rstat_p);
n = gsl_rstat_n(rstat_p);
printf ("The dataset is %g, %g, %g, %g, %g\n",
data[0], data[1], data[2], data[3], data[4]);
printf ("The sample mean is %g\n", mean);
printf ("The estimated variance is %g\n", variance);
printf ("The largest value is %g\n", largest);
printf ("The smallest value is %g\n", smallest);
printf( "The median is %g\n", median);
printf( "The standard deviation is %g\n", sd);
printf( "The root mean square is %g\n", rms);
printf( "The standard devation of the mean is %g\n", sd_mean);
printf( "The skew is %g\n", skew);
printf( "The kurtosis %g\n", kurtosis);
printf( "There are %zu items in the accumulator\n", n);
gsl_rstat_reset(rstat_p);
n = gsl_rstat_n(rstat_p);
printf( "There are %zu items in the accumulator\n", n);
gsl_rstat_free(rstat_p);
return 0;
}
The program should produce the following output,
The dataset is 17.2, 18.1, 16.5, 18.3, 12.6
The sample mean is 16.54
The estimated variance is 5.373
The largest value is 18.3
The smallest value is 12.6
The median is 16.5
The standard deviation is 2.31797
The root mean square is 16.6694
The standard devation of the mean is 1.03663
The skew is -0.829058
The kurtosis -1.2217
There are 5 items in the accumulator
There are 0 items in the accumulator
This next program estimates the lower quartile, median and upper
quartile from 10,000 samples of a random Rayleigh distribution, using
the P^2 algorithm of Jain and Chlamtec. For comparison, the exact
values are also computed from the sorted dataset.
#include
#include
#include
#include
#include
#include
#include
int
main(void)
{
const size_t N = 10000;
double *data = malloc(N * sizeof(double));
gsl_rstat_quantile_workspace *work_25 = gsl_rstat_quantile_alloc(0.25);
gsl_rstat_quantile_workspace *work_50 = gsl_rstat_quantile_alloc(0.5);
gsl_rstat_quantile_workspace *work_75 = gsl_rstat_quantile_alloc(0.75);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
double exact_p25, exact_p50, exact_p75;
double val_p25, val_p50, val_p75;
size_t i;
/* add data to quantile accumulators; also store data for exact
* comparisons */
for (i = 0; i < N; ++i)
{
data[i] = gsl_ran_rayleigh(r, 1.0);
gsl_rstat_quantile_add(data[i], work_25);
gsl_rstat_quantile_add(data[i], work_50);
gsl_rstat_quantile_add(data[i], work_75);
}
/* exact values */
gsl_sort(data, 1, N);
exact_p25 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.25);
exact_p50 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.5);
exact_p75 = gsl_stats_quantile_from_sorted_data(data, 1, N, 0.75);
/* estimated values */
val_p25 = gsl_rstat_quantile_get(work_25);
val_p50 = gsl_rstat_quantile_get(work_50);
val_p75 = gsl_rstat_quantile_get(work_75);
printf ("The dataset is %g, %g, %g, %g, %g, ...\n",
data[0], data[1], data[2], data[3], data[4]);
printf ("0.25 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
exact_p25, val_p25, (val_p25 - exact_p25) / exact_p25);
printf ("0.50 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
exact_p50, val_p50, (val_p50 - exact_p50) / exact_p50);
printf ("0.75 quartile: exact = %.5f, estimated = %.5f, error = %.6e\n",
exact_p75, val_p75, (val_p75 - exact_p75) / exact_p75);
gsl_rstat_quantile_free(work_25);
gsl_rstat_quantile_free(work_50);
gsl_rstat_quantile_free(work_75);
gsl_rng_free(r);
free(data);
return 0;
}
The program should produce the following output,
The dataset is 0.00645272, 0.0074002, 0.0120706, 0.0207256, 0.0227282, ...
0.25 quartile: exact = 0.75766, estimated = 0.75580, error = -2.450209e-03
0.50 quartile: exact = 1.17508, estimated = 1.17438, error = -5.995912e-04
0.75 quartile: exact = 1.65347, estimated = 1.65696, error = 2.110571e-03
22.6 References and Further Reading
===================================
The algorithm used to dynamically estimate p-quantiles is described in
the paper,
R. Jain and I. Chlamtac. 'The P^2 algorithm for dynamic
calculation of quantiles and histograms without storing
observations', Communications of the ACM, Volume 28 (October),
Number 10, 1985, p. 1076-1085.
23 Histograms
*************
This chapter describes functions for creating histograms. Histograms
provide a convenient way of summarizing the distribution of a set of
data. A histogram consists of a set of "bins" which count the number of
events falling into a given range of a continuous variable x. In GSL
the bins of a histogram contain floating-point numbers, so they can be
used to record both integer and non-integer distributions. The bins can
use arbitrary sets of ranges (uniformly spaced bins are the default).
Both one and two-dimensional histograms are supported.
Once a histogram has been created it can also be converted into a
probability distribution function. The library provides efficient
routines for selecting random samples from probability distributions.
This can be useful for generating simulations based on real data.
The functions are declared in the header files 'gsl_histogram.h' and
'gsl_histogram2d.h'.
23.1 The histogram struct
=========================
A histogram is defined by the following struct,
-- Data Type: gsl_histogram
'size_t n'
This is the number of histogram bins
'double * range'
The ranges of the bins are stored in an array of N+1 elements
pointed to by RANGE.
'double * bin'
The counts for each bin are stored in an array of N elements
pointed to by BIN. The bins are floating-point numbers, so
you can increment them by non-integer values if necessary.
The range for BIN[i] is given by RANGE[i] to RANGE[i+1]. For n bins
there are n+1 entries in the array RANGE. Each bin is inclusive at the
lower end and exclusive at the upper end. Mathematically this means
that the bins are defined by the following inequality,
bin[i] corresponds to range[i] <= x < range[i+1]
Here is a diagram of the correspondence between ranges and bins on the
number-line for x,
[ bin[0] )[ bin[1] )[ bin[2] )[ bin[3] )[ bin[4] )
---|---------|---------|---------|---------|---------|--- x
r[0] r[1] r[2] r[3] r[4] r[5]
In this picture the values of the RANGE array are denoted by r. On the
left-hand side of each bin the square bracket '[' denotes an inclusive
lower bound (r <= x), and the round parentheses ')' on the right-hand
side denote an exclusive upper bound (x < r). Thus any samples which
fall on the upper end of the histogram are excluded. If you want to
include this value for the last bin you will need to add an extra bin to
your histogram.
The 'gsl_histogram' struct and its associated functions are defined
in the header file 'gsl_histogram.h'.
23.2 Histogram allocation
=========================
The functions for allocating memory to a histogram follow the style of
'malloc' and 'free'. In addition they also perform their own error
checking. If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of 'GSL_ENOMEM') in addition to returning a null pointer. Thus
if you use the library error handler to abort your program then it isn't
necessary to check every histogram 'alloc'.
-- Function: gsl_histogram * gsl_histogram_alloc (size_t N)
This function allocates memory for a histogram with N bins, and
returns a pointer to a newly created 'gsl_histogram' struct. If
insufficient memory is available a null pointer is returned and the
error handler is invoked with an error code of 'GSL_ENOMEM'. The
bins and ranges are not initialized, and should be prepared using
one of the range-setting functions below in order to make the
histogram ready for use.
-- Function: int gsl_histogram_set_ranges (gsl_histogram * H, const
double RANGE[], size_t SIZE)
This function sets the ranges of the existing histogram H using the
array RANGE of size SIZE. The values of the histogram bins are
reset to zero. The 'range' array should contain the desired bin
limits. The ranges can be arbitrary, subject to the restriction
that they are monotonically increasing.
The following example shows how to create a histogram with
logarithmic bins with ranges [1,10), [10,100) and [100,1000).
gsl_histogram * h = gsl_histogram_alloc (3);
/* bin[0] covers the range 1 <= x < 10 */
/* bin[1] covers the range 10 <= x < 100 */
/* bin[2] covers the range 100 <= x < 1000 */
double range[4] = { 1.0, 10.0, 100.0, 1000.0 };
gsl_histogram_set_ranges (h, range, 4);
Note that the size of the RANGE array should be defined to be one
element bigger than the number of bins. The additional element is
required for the upper value of the final bin.
-- Function: int gsl_histogram_set_ranges_uniform (gsl_histogram * H,
double XMIN, double XMAX)
This function sets the ranges of the existing histogram H to cover
the range XMIN to XMAX uniformly. The values of the histogram bins
are reset to zero. The bin ranges are shown in the table below,
bin[0] corresponds to xmin <= x < xmin + d
bin[1] corresponds to xmin + d <= x < xmin + 2 d
......
bin[n-1] corresponds to xmin + (n-1)d <= x < xmax
where d is the bin spacing, d = (xmax-xmin)/n.
-- Function: void gsl_histogram_free (gsl_histogram * H)
This function frees the histogram H and all of the memory
associated with it.
23.3 Copying Histograms
=======================
-- Function: int gsl_histogram_memcpy (gsl_histogram * DEST, const
gsl_histogram * SRC)
This function copies the histogram SRC into the pre-existing
histogram DEST, making DEST into an exact copy of SRC. The two
histograms must be of the same size.
-- Function: gsl_histogram * gsl_histogram_clone (const gsl_histogram *
SRC)
This function returns a pointer to a newly created histogram which
is an exact copy of the histogram SRC.
23.4 Updating and accessing histogram elements
==============================================
There are two ways to access histogram bins, either by specifying an x
coordinate or by using the bin-index directly. The functions for
accessing the histogram through x coordinates use a binary search to
identify the bin which covers the appropriate range.
-- Function: int gsl_histogram_increment (gsl_histogram * H, double X)
This function updates the histogram H by adding one (1.0) to the
bin whose range contains the coordinate X.
If X lies in the valid range of the histogram then the function
returns zero to indicate success. If X is less than the lower
limit of the histogram then the function returns 'GSL_EDOM', and
none of bins are modified. Similarly, if the value of X is greater
than or equal to the upper limit of the histogram then the function
returns 'GSL_EDOM', and none of the bins are modified. The error
handler is not called, however, since it is often necessary to
compute histograms for a small range of a larger dataset, ignoring
the values outside the range of interest.
-- Function: int gsl_histogram_accumulate (gsl_histogram * H, double X,
double WEIGHT)
This function is similar to 'gsl_histogram_increment' but increases
the value of the appropriate bin in the histogram H by the
floating-point number WEIGHT.
-- Function: double gsl_histogram_get (const gsl_histogram * H, size_t
I)
This function returns the contents of the I-th bin of the histogram
H. If I lies outside the valid range of indices for the histogram
then the error handler is called with an error code of 'GSL_EDOM'
and the function returns 0.
-- Function: int gsl_histogram_get_range (const gsl_histogram * H,
size_t I, double * LOWER, double * UPPER)
This function finds the upper and lower range limits of the I-th
bin of the histogram H. If the index I is valid then the
corresponding range limits are stored in LOWER and UPPER. The
lower limit is inclusive (i.e. events with this coordinate are
included in the bin) and the upper limit is exclusive (i.e. events
with the coordinate of the upper limit are excluded and fall in the
neighboring higher bin, if it exists). The function returns 0 to
indicate success. If I lies outside the valid range of indices for
the histogram then the error handler is called and the function
returns an error code of 'GSL_EDOM'.
-- Function: double gsl_histogram_max (const gsl_histogram * H)
-- Function: double gsl_histogram_min (const gsl_histogram * H)
-- Function: size_t gsl_histogram_bins (const gsl_histogram * H)
These functions return the maximum upper and minimum lower range
limits and the number of bins of the histogram H. They provide a
way of determining these values without accessing the
'gsl_histogram' struct directly.
-- Function: void gsl_histogram_reset (gsl_histogram * H)
This function resets all the bins in the histogram H to zero.
23.5 Searching histogram ranges
===============================
The following functions are used by the access and update routines to
locate the bin which corresponds to a given x coordinate.
-- Function: int gsl_histogram_find (const gsl_histogram * H, double X,
size_t * I)
This function finds and sets the index I to the bin number which
covers the coordinate X in the histogram H. The bin is located
using a binary search. The search includes an optimization for
histograms with uniform range, and will return the correct bin
immediately in this case. If X is found in the range of the
histogram then the function sets the index I and returns
'GSL_SUCCESS'. If X lies outside the valid range of the histogram
then the function returns 'GSL_EDOM' and the error handler is
invoked.
23.6 Histogram Statistics
=========================
-- Function: double gsl_histogram_max_val (const gsl_histogram * H)
This function returns the maximum value contained in the histogram
bins.
-- Function: size_t gsl_histogram_max_bin (const gsl_histogram * H)
This function returns the index of the bin containing the maximum
value. In the case where several bins contain the same maximum
value the smallest index is returned.
-- Function: double gsl_histogram_min_val (const gsl_histogram * H)
This function returns the minimum value contained in the histogram
bins.
-- Function: size_t gsl_histogram_min_bin (const gsl_histogram * H)
This function returns the index of the bin containing the minimum
value. In the case where several bins contain the same maximum
value the smallest index is returned.
-- Function: double gsl_histogram_mean (const gsl_histogram * H)
This function returns the mean of the histogrammed variable, where
the histogram is regarded as a probability distribution. Negative
bin values are ignored for the purposes of this calculation. The
accuracy of the result is limited by the bin width.
-- Function: double gsl_histogram_sigma (const gsl_histogram * H)
This function returns the standard deviation of the histogrammed
variable, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of
this calculation. The accuracy of the result is limited by the bin
width.
-- Function: double gsl_histogram_sum (const gsl_histogram * H)
This function returns the sum of all bin values. Negative bin
values are included in the sum.
23.7 Histogram Operations
=========================
-- Function: int gsl_histogram_equal_bins_p (const gsl_histogram * H1,
const gsl_histogram * H2)
This function returns 1 if the all of the individual bin ranges of
the two histograms are identical, and 0 otherwise.
-- Function: int gsl_histogram_add (gsl_histogram * H1, const
gsl_histogram * H2)
This function adds the contents of the bins in histogram H2 to the
corresponding bins of histogram H1, i.e. h'_1(i) = h_1(i) +
h_2(i). The two histograms must have identical bin ranges.
-- Function: int gsl_histogram_sub (gsl_histogram * H1, const
gsl_histogram * H2)
This function subtracts the contents of the bins in histogram H2
from the corresponding bins of histogram H1, i.e. h'_1(i) = h_1(i)
- h_2(i). The two histograms must have identical bin ranges.
-- Function: int gsl_histogram_mul (gsl_histogram * H1, const
gsl_histogram * H2)
This function multiplies the contents of the bins of histogram H1
by the contents of the corresponding bins in histogram H2, i.e.
h'_1(i) = h_1(i) * h_2(i). The two histograms must have identical
bin ranges.
-- Function: int gsl_histogram_div (gsl_histogram * H1, const
gsl_histogram * H2)
This function divides the contents of the bins of histogram H1 by
the contents of the corresponding bins in histogram H2, i.e.
h'_1(i) = h_1(i) / h_2(i). The two histograms must have identical
bin ranges.
-- Function: int gsl_histogram_scale (gsl_histogram * H, double SCALE)
This function multiplies the contents of the bins of histogram H by
the constant SCALE, i.e. h'_1(i) = h_1(i) * scale.
-- Function: int gsl_histogram_shift (gsl_histogram * H, double OFFSET)
This function shifts the contents of the bins of histogram H by the
constant OFFSET, i.e. h'_1(i) = h_1(i) + offset.
23.8 Reading and writing histograms
===================================
The library provides functions for reading and writing histograms to a
file as binary data or formatted text.
-- Function: int gsl_histogram_fwrite (FILE * STREAM, const
gsl_histogram * H)
This function writes the ranges and bins of the histogram H to the
stream STREAM in binary format. The return value is 0 for success
and 'GSL_EFAILED' if there was a problem writing to the file.
Since the data is written in the native binary format it may not be
portable between different architectures.
-- Function: int gsl_histogram_fread (FILE * STREAM, gsl_histogram * H)
This function reads into the histogram H from the open stream
STREAM in binary format. The histogram H must be preallocated with
the correct size since the function uses the number of bins in H to
determine how many bytes to read. The return value is 0 for
success and 'GSL_EFAILED' if there was a problem reading from the
file. The data is assumed to have been written in the native
binary format on the same architecture.
-- Function: int gsl_histogram_fprintf (FILE * STREAM, const
gsl_histogram * H, const char * RANGE_FORMAT, const char *
BIN_FORMAT)
This function writes the ranges and bins of the histogram H
line-by-line to the stream STREAM using the format specifiers
RANGE_FORMAT and BIN_FORMAT. These should be one of the '%g', '%e'
or '%f' formats for floating point numbers. The function returns 0
for success and 'GSL_EFAILED' if there was a problem writing to the
file. The histogram output is formatted in three columns, and the
columns are separated by spaces, like this,
range[0] range[1] bin[0]
range[1] range[2] bin[1]
range[2] range[3] bin[2]
....
range[n-1] range[n] bin[n-1]
The values of the ranges are formatted using RANGE_FORMAT and the
value of the bins are formatted using BIN_FORMAT. Each line
contains the lower and upper limit of the range of the bins and the
value of the bin itself. Since the upper limit of one bin is the
lower limit of the next there is duplication of these values
between lines but this allows the histogram to be manipulated with
line-oriented tools.
-- Function: int gsl_histogram_fscanf (FILE * STREAM, gsl_histogram *
H)
This function reads formatted data from the stream STREAM into the
histogram H. The data is assumed to be in the three-column format
used by 'gsl_histogram_fprintf'. The histogram H must be
preallocated with the correct length since the function uses the
size of H to determine how many numbers to read. The function
returns 0 for success and 'GSL_EFAILED' if there was a problem
reading from the file.
23.9 Resampling from histograms
===============================
A histogram made by counting events can be regarded as a measurement of
a probability distribution. Allowing for statistical error, the height
of each bin represents the probability of an event where the value of x
falls in the range of that bin. The probability distribution function
has the one-dimensional form p(x)dx where,
p(x) = n_i/ (N w_i)
In this equation n_i is the number of events in the bin which contains
x, w_i is the width of the bin and N is the total number of events. The
distribution of events within each bin is assumed to be uniform.
23.10 The histogram probability distribution struct
===================================================
The probability distribution function for a histogram consists of a set
of "bins" which measure the probability of an event falling into a given
range of a continuous variable x. A probability distribution function
is defined by the following struct, which actually stores the cumulative
probability distribution function. This is the natural quantity for
generating samples via the inverse transform method, because there is a
one-to-one mapping between the cumulative probability distribution and
the range [0,1]. It can be shown that by taking a uniform random number
in this range and finding its corresponding coordinate in the cumulative
probability distribution we obtain samples with the desired probability
distribution.
-- Data Type: gsl_histogram_pdf
'size_t n'
This is the number of bins used to approximate the probability
distribution function.
'double * range'
The ranges of the bins are stored in an array of N+1 elements
pointed to by RANGE.
'double * sum'
The cumulative probability for the bins is stored in an array
of N elements pointed to by SUM.
The following functions allow you to create a 'gsl_histogram_pdf' struct
which represents this probability distribution and generate random
samples from it.
-- Function: gsl_histogram_pdf * gsl_histogram_pdf_alloc (size_t N)
This function allocates memory for a probability distribution with
N bins and returns a pointer to a newly initialized
'gsl_histogram_pdf' struct. If insufficient memory is available a
null pointer is returned and the error handler is invoked with an
error code of 'GSL_ENOMEM'.
-- Function: int gsl_histogram_pdf_init (gsl_histogram_pdf * P, const
gsl_histogram * H)
This function initializes the probability distribution P with the
contents of the histogram H. If any of the bins of H are negative
then the error handler is invoked with an error code of 'GSL_EDOM'
because a probability distribution cannot contain negative values.
-- Function: void gsl_histogram_pdf_free (gsl_histogram_pdf * P)
This function frees the probability distribution function P and all
of the memory associated with it.
-- Function: double gsl_histogram_pdf_sample (const gsl_histogram_pdf *
P, double R)
This function uses R, a uniform random number between zero and one,
to compute a single random sample from the probability distribution
P. The algorithm used to compute the sample s is given by the
following formula,
s = range[i] + delta * (range[i+1] - range[i])
where i is the index which satisfies sum[i] <= r < sum[i+1] and
delta is (r - sum[i])/(sum[i+1] - sum[i]).
23.11 Example programs for histograms
=====================================
The following program shows how to make a simple histogram of a column
of numerical data supplied on 'stdin'. The program takes three
arguments, specifying the upper and lower bounds of the histogram and
the number of bins. It then reads numbers from 'stdin', one line at a
time, and adds them to the histogram. When there is no more data to
read it prints out the accumulated histogram using
'gsl_histogram_fprintf'.
#include
#include
#include
int
main (int argc, char **argv)
{
double a, b;
size_t n;
if (argc != 4)
{
printf ("Usage: gsl-histogram xmin xmax n\n"
"Computes a histogram of the data "
"on stdin using n bins from xmin "
"to xmax\n");
exit (0);
}
a = atof (argv[1]);
b = atof (argv[2]);
n = atoi (argv[3]);
{
double x;
gsl_histogram * h = gsl_histogram_alloc (n);
gsl_histogram_set_ranges_uniform (h, a, b);
while (fscanf (stdin, "%lg", &x) == 1)
{
gsl_histogram_increment (h, x);
}
gsl_histogram_fprintf (stdout, h, "%g", "%g");
gsl_histogram_free (h);
}
exit (0);
}
Here is an example of the program in use. We generate 10000 random
samples from a Cauchy distribution with a width of 30 and histogram them
over the range -100 to 100, using 200 bins.
$ gsl-randist 0 10000 cauchy 30
| gsl-histogram -100 100 200 > histogram.dat
A plot of the resulting histogram shows the familiar shape of the Cauchy
distribution and the fluctuations caused by the finite sample size.
$ awk '{print $1, $3 ; print $2, $3}' histogram.dat
| graph -T X
23.12 Two dimensional histograms
================================
A two dimensional histogram consists of a set of "bins" which count the
number of events falling in a given area of the (x,y) plane. The
simplest way to use a two dimensional histogram is to record
two-dimensional position information, n(x,y). Another possibility is to
form a "joint distribution" by recording related variables. For example
a detector might record both the position of an event (x) and the amount
of energy it deposited E. These could be histogrammed as the joint
distribution n(x,E).
23.13 The 2D histogram struct
=============================
Two dimensional histograms are defined by the following struct,
-- Data Type: gsl_histogram2d
'size_t nx, ny'
This is the number of histogram bins in the x and y
directions.
'double * xrange'
The ranges of the bins in the x-direction are stored in an
array of NX + 1 elements pointed to by XRANGE.
'double * yrange'
The ranges of the bins in the y-direction are stored in an
array of NY + 1 elements pointed to by YRANGE.
'double * bin'
The counts for each bin are stored in an array pointed to by
BIN. The bins are floating-point numbers, so you can
increment them by non-integer values if necessary. The array
BIN stores the two dimensional array of bins in a single block
of memory according to the mapping 'bin(i,j)' = 'bin[i * ny +
j]'.
The range for 'bin(i,j)' is given by 'xrange[i]' to 'xrange[i+1]' in the
x-direction and 'yrange[j]' to 'yrange[j+1]' in the y-direction. Each
bin is inclusive at the lower end and exclusive at the upper end.
Mathematically this means that the bins are defined by the following
inequality,
bin(i,j) corresponds to xrange[i] <= x < xrange[i+1]
and yrange[j] <= y < yrange[j+1]
Note that any samples which fall on the upper sides of the histogram are
excluded. If you want to include these values for the side bins you
will need to add an extra row or column to your histogram.
The 'gsl_histogram2d' struct and its associated functions are defined
in the header file 'gsl_histogram2d.h'.
23.14 2D Histogram allocation
=============================
The functions for allocating memory to a 2D histogram follow the style
of 'malloc' and 'free'. In addition they also perform their own error
checking. If there is insufficient memory available to allocate a
histogram then the functions call the error handler (with an error
number of 'GSL_ENOMEM') in addition to returning a null pointer. Thus
if you use the library error handler to abort your program then it isn't
necessary to check every 2D histogram 'alloc'.
-- Function: gsl_histogram2d * gsl_histogram2d_alloc (size_t NX, size_t
NY)
This function allocates memory for a two-dimensional histogram with
NX bins in the x direction and NY bins in the y direction. The
function returns a pointer to a newly created 'gsl_histogram2d'
struct. If insufficient memory is available a null pointer is
returned and the error handler is invoked with an error code of
'GSL_ENOMEM'. The bins and ranges must be initialized with one of
the functions below before the histogram is ready for use.
-- Function: int gsl_histogram2d_set_ranges (gsl_histogram2d * H, const
double XRANGE[], size_t XSIZE, const double YRANGE[], size_t
YSIZE)
This function sets the ranges of the existing histogram H using the
arrays XRANGE and YRANGE of size XSIZE and YSIZE respectively. The
values of the histogram bins are reset to zero.
-- Function: int gsl_histogram2d_set_ranges_uniform (gsl_histogram2d *
H, double XMIN, double XMAX, double YMIN, double YMAX)
This function sets the ranges of the existing histogram H to cover
the ranges XMIN to XMAX and YMIN to YMAX uniformly. The values of
the histogram bins are reset to zero.
-- Function: void gsl_histogram2d_free (gsl_histogram2d * H)
This function frees the 2D histogram H and all of the memory
associated with it.
23.15 Copying 2D Histograms
===========================
-- Function: int gsl_histogram2d_memcpy (gsl_histogram2d * DEST, const
gsl_histogram2d * SRC)
This function copies the histogram SRC into the pre-existing
histogram DEST, making DEST into an exact copy of SRC. The two
histograms must be of the same size.
-- Function: gsl_histogram2d * gsl_histogram2d_clone (const
gsl_histogram2d * SRC)
This function returns a pointer to a newly created histogram which
is an exact copy of the histogram SRC.
23.16 Updating and accessing 2D histogram elements
==================================================
You can access the bins of a two-dimensional histogram either by
specifying a pair of (x,y) coordinates or by using the bin indices (i,j)
directly. The functions for accessing the histogram through (x,y)
coordinates use binary searches in the x and y directions to identify
the bin which covers the appropriate range.
-- Function: int gsl_histogram2d_increment (gsl_histogram2d * H, double
X, double Y)
This function updates the histogram H by adding one (1.0) to the
bin whose x and y ranges contain the coordinates (X,Y).
If the point (x,y) lies inside the valid ranges of the histogram
then the function returns zero to indicate success. If (x,y) lies
outside the limits of the histogram then the function returns
'GSL_EDOM', and none of the bins are modified. The error handler
is not called, since it is often necessary to compute histograms
for a small range of a larger dataset, ignoring any coordinates
outside the range of interest.
-- Function: int gsl_histogram2d_accumulate (gsl_histogram2d * H,
double X, double Y, double WEIGHT)
This function is similar to 'gsl_histogram2d_increment' but
increases the value of the appropriate bin in the histogram H by
the floating-point number WEIGHT.
-- Function: double gsl_histogram2d_get (const gsl_histogram2d * H,
size_t I, size_t J)
This function returns the contents of the (I,J)-th bin of the
histogram H. If (I,J) lies outside the valid range of indices for
the histogram then the error handler is called with an error code
of 'GSL_EDOM' and the function returns 0.
-- Function: int gsl_histogram2d_get_xrange (const gsl_histogram2d * H,
size_t I, double * XLOWER, double * XUPPER)
-- Function: int gsl_histogram2d_get_yrange (const gsl_histogram2d * H,
size_t J, double * YLOWER, double * YUPPER)
These functions find the upper and lower range limits of the I-th
and J-th bins in the x and y directions of the histogram H. The
range limits are stored in XLOWER and XUPPER or YLOWER and YUPPER.
The lower limits are inclusive (i.e. events with these coordinates
are included in the bin) and the upper limits are exclusive (i.e.
events with the value of the upper limit are not included and fall
in the neighboring higher bin, if it exists). The functions return
0 to indicate success. If I or J lies outside the valid range of
indices for the histogram then the error handler is called with an
error code of 'GSL_EDOM'.
-- Function: double gsl_histogram2d_xmax (const gsl_histogram2d * H)
-- Function: double gsl_histogram2d_xmin (const gsl_histogram2d * H)
-- Function: size_t gsl_histogram2d_nx (const gsl_histogram2d * H)
-- Function: double gsl_histogram2d_ymax (const gsl_histogram2d * H)
-- Function: double gsl_histogram2d_ymin (const gsl_histogram2d * H)
-- Function: size_t gsl_histogram2d_ny (const gsl_histogram2d * H)
These functions return the maximum upper and minimum lower range
limits and the number of bins for the x and y directions of the
histogram H. They provide a way of determining these values
without accessing the 'gsl_histogram2d' struct directly.
-- Function: void gsl_histogram2d_reset (gsl_histogram2d * H)
This function resets all the bins of the histogram H to zero.
23.17 Searching 2D histogram ranges
===================================
The following functions are used by the access and update routines to
locate the bin which corresponds to a given (x,y) coordinate.
-- Function: int gsl_histogram2d_find (const gsl_histogram2d * H,
double X, double Y, size_t * I, size_t * J)
This function finds and sets the indices I and J to the bin which
covers the coordinates (X,Y). The bin is located using a binary
search. The search includes an optimization for histograms with
uniform ranges, and will return the correct bin immediately in this
case. If (x,y) is found then the function sets the indices (I,J)
and returns 'GSL_SUCCESS'. If (x,y) lies outside the valid range
of the histogram then the function returns 'GSL_EDOM' and the error
handler is invoked.
23.18 2D Histogram Statistics
=============================
-- Function: double gsl_histogram2d_max_val (const gsl_histogram2d * H)
This function returns the maximum value contained in the histogram
bins.
-- Function: void gsl_histogram2d_max_bin (const gsl_histogram2d * H,
size_t * I, size_t * J)
This function finds the indices of the bin containing the maximum
value in the histogram H and stores the result in (I,J). In the
case where several bins contain the same maximum value the first
bin found is returned.
-- Function: double gsl_histogram2d_min_val (const gsl_histogram2d * H)
This function returns the minimum value contained in the histogram
bins.
-- Function: void gsl_histogram2d_min_bin (const gsl_histogram2d * H,
size_t * I, size_t * J)
This function finds the indices of the bin containing the minimum
value in the histogram H and stores the result in (I,J). In the
case where several bins contain the same maximum value the first
bin found is returned.
-- Function: double gsl_histogram2d_xmean (const gsl_histogram2d * H)
This function returns the mean of the histogrammed x variable,
where the histogram is regarded as a probability distribution.
Negative bin values are ignored for the purposes of this
calculation.
-- Function: double gsl_histogram2d_ymean (const gsl_histogram2d * H)
This function returns the mean of the histogrammed y variable,
where the histogram is regarded as a probability distribution.
Negative bin values are ignored for the purposes of this
calculation.
-- Function: double gsl_histogram2d_xsigma (const gsl_histogram2d * H)
This function returns the standard deviation of the histogrammed x
variable, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of
this calculation.
-- Function: double gsl_histogram2d_ysigma (const gsl_histogram2d * H)
This function returns the standard deviation of the histogrammed y
variable, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of
this calculation.
-- Function: double gsl_histogram2d_cov (const gsl_histogram2d * H)
This function returns the covariance of the histogrammed x and y
variables, where the histogram is regarded as a probability
distribution. Negative bin values are ignored for the purposes of
this calculation.
-- Function: double gsl_histogram2d_sum (const gsl_histogram2d * H)
This function returns the sum of all bin values. Negative bin
values are included in the sum.
23.19 2D Histogram Operations
=============================
-- Function: int gsl_histogram2d_equal_bins_p (const gsl_histogram2d *
H1, const gsl_histogram2d * H2)
This function returns 1 if all the individual bin ranges of the two
histograms are identical, and 0 otherwise.
-- Function: int gsl_histogram2d_add (gsl_histogram2d * H1, const
gsl_histogram2d * H2)
This function adds the contents of the bins in histogram H2 to the
corresponding bins of histogram H1, i.e. h'_1(i,j) = h_1(i,j) +
h_2(i,j). The two histograms must have identical bin ranges.
-- Function: int gsl_histogram2d_sub (gsl_histogram2d * H1, const
gsl_histogram2d * H2)
This function subtracts the contents of the bins in histogram H2
from the corresponding bins of histogram H1, i.e. h'_1(i,j) =
h_1(i,j) - h_2(i,j). The two histograms must have identical bin
ranges.
-- Function: int gsl_histogram2d_mul (gsl_histogram2d * H1, const
gsl_histogram2d * H2)
This function multiplies the contents of the bins of histogram H1
by the contents of the corresponding bins in histogram H2, i.e.
h'_1(i,j) = h_1(i,j) * h_2(i,j). The two histograms must have
identical bin ranges.
-- Function: int gsl_histogram2d_div (gsl_histogram2d * H1, const
gsl_histogram2d * H2)
This function divides the contents of the bins of histogram H1 by
the contents of the corresponding bins in histogram H2, i.e.
h'_1(i,j) = h_1(i,j) / h_2(i,j). The two histograms must have
identical bin ranges.
-- Function: int gsl_histogram2d_scale (gsl_histogram2d * H, double
SCALE)
This function multiplies the contents of the bins of histogram H by
the constant SCALE, i.e. h'_1(i,j) = h_1(i,j) scale.
-- Function: int gsl_histogram2d_shift (gsl_histogram2d * H, double
OFFSET)
This function shifts the contents of the bins of histogram H by the
constant OFFSET, i.e. h'_1(i,j) = h_1(i,j) + offset.
23.20 Reading and writing 2D histograms
=======================================
The library provides functions for reading and writing two dimensional
histograms to a file as binary data or formatted text.
-- Function: int gsl_histogram2d_fwrite (FILE * STREAM, const
gsl_histogram2d * H)
This function writes the ranges and bins of the histogram H to the
stream STREAM in binary format. The return value is 0 for success
and 'GSL_EFAILED' if there was a problem writing to the file.
Since the data is written in the native binary format it may not be
portable between different architectures.
-- Function: int gsl_histogram2d_fread (FILE * STREAM, gsl_histogram2d
* H)
This function reads into the histogram H from the stream STREAM in
binary format. The histogram H must be preallocated with the
correct size since the function uses the number of x and y bins in
H to determine how many bytes to read. The return value is 0 for
success and 'GSL_EFAILED' if there was a problem reading from the
file. The data is assumed to have been written in the native
binary format on the same architecture.
-- Function: int gsl_histogram2d_fprintf (FILE * STREAM, const
gsl_histogram2d * H, const char * RANGE_FORMAT, const char *
BIN_FORMAT)
This function writes the ranges and bins of the histogram H
line-by-line to the stream STREAM using the format specifiers
RANGE_FORMAT and BIN_FORMAT. These should be one of the '%g', '%e'
or '%f' formats for floating point numbers. The function returns 0
for success and 'GSL_EFAILED' if there was a problem writing to the
file. The histogram output is formatted in five columns, and the
columns are separated by spaces, like this,
xrange[0] xrange[1] yrange[0] yrange[1] bin(0,0)
xrange[0] xrange[1] yrange[1] yrange[2] bin(0,1)
xrange[0] xrange[1] yrange[2] yrange[3] bin(0,2)
....
xrange[0] xrange[1] yrange[ny-1] yrange[ny] bin(0,ny-1)
xrange[1] xrange[2] yrange[0] yrange[1] bin(1,0)
xrange[1] xrange[2] yrange[1] yrange[2] bin(1,1)
xrange[1] xrange[2] yrange[1] yrange[2] bin(1,2)
....
xrange[1] xrange[2] yrange[ny-1] yrange[ny] bin(1,ny-1)
....
xrange[nx-1] xrange[nx] yrange[0] yrange[1] bin(nx-1,0)
xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,1)
xrange[nx-1] xrange[nx] yrange[1] yrange[2] bin(nx-1,2)
....
xrange[nx-1] xrange[nx] yrange[ny-1] yrange[ny] bin(nx-1,ny-1)
Each line contains the lower and upper limits of the bin and the
contents of the bin. Since the upper limits of the each bin are
the lower limits of the neighboring bins there is duplication of
these values but this allows the histogram to be manipulated with
line-oriented tools.
-- Function: int gsl_histogram2d_fscanf (FILE * STREAM, gsl_histogram2d
* H)
This function reads formatted data from the stream STREAM into the
histogram H. The data is assumed to be in the five-column format
used by 'gsl_histogram2d_fprintf'. The histogram H must be
preallocated with the correct lengths since the function uses the
sizes of H to determine how many numbers to read. The function
returns 0 for success and 'GSL_EFAILED' if there was a problem
reading from the file.
23.21 Resampling from 2D histograms
===================================
As in the one-dimensional case, a two-dimensional histogram made by
counting events can be regarded as a measurement of a probability
distribution. Allowing for statistical error, the height of each bin
represents the probability of an event where (x,y) falls in the range of
that bin. For a two-dimensional histogram the probability distribution
takes the form p(x,y) dx dy where,
p(x,y) = n_{ij}/ (N A_{ij})
In this equation n_{ij} is the number of events in the bin which
contains (x,y), A_{ij} is the area of the bin and N is the total number
of events. The distribution of events within each bin is assumed to be
uniform.
-- Data Type: gsl_histogram2d_pdf
'size_t nx, ny'
This is the number of histogram bins used to approximate the
probability distribution function in the x and y directions.
'double * xrange'
The ranges of the bins in the x-direction are stored in an
array of NX + 1 elements pointed to by XRANGE.
'double * yrange'
The ranges of the bins in the y-direction are stored in an
array of NY + 1 pointed to by YRANGE.
'double * sum'
The cumulative probability for the bins is stored in an array
of NX*NY elements pointed to by SUM.
The following functions allow you to create a 'gsl_histogram2d_pdf'
struct which represents a two dimensional probability distribution and
generate random samples from it.
-- Function: gsl_histogram2d_pdf * gsl_histogram2d_pdf_alloc (size_t
NX, size_t NY)
This function allocates memory for a two-dimensional probability
distribution of size NX-by-NY and returns a pointer to a newly
initialized 'gsl_histogram2d_pdf' struct. If insufficient memory
is available a null pointer is returned and the error handler is
invoked with an error code of 'GSL_ENOMEM'.
-- Function: int gsl_histogram2d_pdf_init (gsl_histogram2d_pdf * P,
const gsl_histogram2d * H)
This function initializes the two-dimensional probability
distribution calculated P from the histogram H. If any of the bins
of H are negative then the error handler is invoked with an error
code of 'GSL_EDOM' because a probability distribution cannot
contain negative values.
-- Function: void gsl_histogram2d_pdf_free (gsl_histogram2d_pdf * P)
This function frees the two-dimensional probability distribution
function P and all of the memory associated with it.
-- Function: int gsl_histogram2d_pdf_sample (const gsl_histogram2d_pdf
* P, double R1, double R2, double * X, double * Y)
This function uses two uniform random numbers between zero and one,
R1 and R2, to compute a single random sample from the
two-dimensional probability distribution P.
23.22 Example programs for 2D histograms
========================================
This program demonstrates two features of two-dimensional histograms.
First a 10-by-10 two-dimensional histogram is created with x and y
running from 0 to 1. Then a few sample points are added to the
histogram, at (0.3,0.3) with a height of 1, at (0.8,0.1) with a height
of 5 and at (0.7,0.9) with a height of 0.5. This histogram with three
events is used to generate a random sample of 1000 simulated events,
which are printed out.
#include
#include
#include
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
gsl_histogram2d * h = gsl_histogram2d_alloc (10, 10);
gsl_histogram2d_set_ranges_uniform (h,
0.0, 1.0,
0.0, 1.0);
gsl_histogram2d_accumulate (h, 0.3, 0.3, 1);
gsl_histogram2d_accumulate (h, 0.8, 0.1, 5);
gsl_histogram2d_accumulate (h, 0.7, 0.9, 0.5);
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
int i;
gsl_histogram2d_pdf * p
= gsl_histogram2d_pdf_alloc (h->nx, h->ny);
gsl_histogram2d_pdf_init (p, h);
for (i = 0; i < 1000; i++) {
double x, y;
double u = gsl_rng_uniform (r);
double v = gsl_rng_uniform (r);
gsl_histogram2d_pdf_sample (p, u, v, &x, &y);
printf ("%g %g\n", x, y);
}
gsl_histogram2d_pdf_free (p);
}
gsl_histogram2d_free (h);
gsl_rng_free (r);
return 0;
}
24 N-tuples
***********
This chapter describes functions for creating and manipulating
"ntuples", sets of values associated with events. The ntuples are
stored in files. Their values can be extracted in any combination and
"booked" in a histogram using a selection function.
The values to be stored are held in a user-defined data structure,
and an ntuple is created associating this data structure with a file.
The values are then written to the file (normally inside a loop) using
the ntuple functions described below.
A histogram can be created from ntuple data by providing a selection
function and a value function. The selection function specifies whether
an event should be included in the subset to be analyzed or not. The
value function computes the entry to be added to the histogram for each
event.
All the ntuple functions are defined in the header file
'gsl_ntuple.h'
24.1 The ntuple struct
======================
Ntuples are manipulated using the 'gsl_ntuple' struct. This struct
contains information on the file where the ntuple data is stored, a
pointer to the current ntuple data row and the size of the user-defined
ntuple data struct.
typedef struct {
FILE * file;
void * ntuple_data;
size_t size;
} gsl_ntuple;
24.2 Creating ntuples
=====================
-- Function: gsl_ntuple * gsl_ntuple_create (char * FILENAME, void *
NTUPLE_DATA, size_t SIZE)
This function creates a new write-only ntuple file FILENAME for
ntuples of size SIZE and returns a pointer to the newly created
ntuple struct. Any existing file with the same name is truncated
to zero length and overwritten. A pointer to memory for the
current ntuple row NTUPLE_DATA must be supplied--this is used to
copy ntuples in and out of the file.
24.3 Opening an existing ntuple file
====================================
-- Function: gsl_ntuple * gsl_ntuple_open (char * FILENAME, void *
NTUPLE_DATA, size_t SIZE)
This function opens an existing ntuple file FILENAME for reading
and returns a pointer to a corresponding ntuple struct. The
ntuples in the file must have size SIZE. A pointer to memory for
the current ntuple row NTUPLE_DATA must be supplied--this is used
to copy ntuples in and out of the file.
24.4 Writing ntuples
====================
-- Function: int gsl_ntuple_write (gsl_ntuple * NTUPLE)
This function writes the current ntuple NTUPLE->NTUPLE_DATA of size
NTUPLE->SIZE to the corresponding file.
-- Function: int gsl_ntuple_bookdata (gsl_ntuple * NTUPLE)
This function is a synonym for 'gsl_ntuple_write'.
24.5 Reading ntuples
====================
-- Function: int gsl_ntuple_read (gsl_ntuple * NTUPLE)
This function reads the current row of the ntuple file for NTUPLE
and stores the values in NTUPLE->DATA.
24.6 Closing an ntuple file
===========================
-- Function: int gsl_ntuple_close (gsl_ntuple * NTUPLE)
This function closes the ntuple file NTUPLE and frees its
associated allocated memory.
24.7 Histogramming ntuple values
================================
Once an ntuple has been created its contents can be histogrammed in
various ways using the function 'gsl_ntuple_project'. Two user-defined
functions must be provided, a function to select events and a function
to compute scalar values. The selection function and the value function
both accept the ntuple row as a first argument and other parameters as a
second argument.
The "selection function" determines which ntuple rows are selected
for histogramming. It is defined by the following struct,
typedef struct {
int (* function) (void * ntuple_data, void * params);
void * params;
} gsl_ntuple_select_fn;
The struct component FUNCTION should return a non-zero value for each
ntuple row that is to be included in the histogram.
The "value function" computes scalar values for those ntuple rows
selected by the selection function,
typedef struct {
double (* function) (void * ntuple_data, void * params);
void * params;
} gsl_ntuple_value_fn;
In this case the struct component FUNCTION should return the value to be
added to the histogram for the ntuple row.
-- Function: int gsl_ntuple_project (gsl_histogram * H, gsl_ntuple *
NTUPLE, gsl_ntuple_value_fn * VALUE_FUNC, gsl_ntuple_select_fn
* SELECT_FUNC)
This function updates the histogram H from the ntuple NTUPLE using
the functions VALUE_FUNC and SELECT_FUNC. For each ntuple row
where the selection function SELECT_FUNC is non-zero the
corresponding value of that row is computed using the function
VALUE_FUNC and added to the histogram. Those ntuple rows where
SELECT_FUNC returns zero are ignored. New entries are added to the
histogram, so subsequent calls can be used to accumulate further
data in the same histogram.
24.8 Examples
=============
The following example programs demonstrate the use of ntuples in
managing a large dataset. The first program creates a set of 10,000
simulated "events", each with 3 associated values (x,y,z). These are
generated from a Gaussian distribution with unit variance, for
demonstration purposes, and written to the ntuple file 'test.dat'.
#include
#include
#include
struct data
{
double x;
double y;
double z;
};
int
main (void)
{
const gsl_rng_type * T;
gsl_rng * r;
struct data ntuple_row;
int i;
gsl_ntuple *ntuple
= gsl_ntuple_create ("test.dat", &ntuple_row,
sizeof (ntuple_row));
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (i = 0; i < 10000; i++)
{
ntuple_row.x = gsl_ran_ugaussian (r);
ntuple_row.y = gsl_ran_ugaussian (r);
ntuple_row.z = gsl_ran_ugaussian (r);
gsl_ntuple_write (ntuple);
}
gsl_ntuple_close (ntuple);
gsl_rng_free (r);
return 0;
}
The next program analyses the ntuple data in the file 'test.dat'. The
analysis procedure is to compute the squared-magnitude of each event,
E^2=x^2+y^2+z^2, and select only those which exceed a lower limit of
1.5. The selected events are then histogrammed using their E^2 values.
#include
#include
#include
struct data
{
double x;
double y;
double z;
};
int sel_func (void *ntuple_data, void *params);
double val_func (void *ntuple_data, void *params);
int
main (void)
{
struct data ntuple_row;
gsl_ntuple *ntuple
= gsl_ntuple_open ("test.dat", &ntuple_row,
sizeof (ntuple_row));
double lower = 1.5;
gsl_ntuple_select_fn S;
gsl_ntuple_value_fn V;
gsl_histogram *h = gsl_histogram_alloc (100);
gsl_histogram_set_ranges_uniform(h, 0.0, 10.0);
S.function = &sel_func;
S.params = &lower;
V.function = &val_func;
V.params = 0;
gsl_ntuple_project (h, ntuple, &V, &S);
gsl_histogram_fprintf (stdout, h, "%f", "%f");
gsl_histogram_free (h);
gsl_ntuple_close (ntuple);
return 0;
}
int
sel_func (void *ntuple_data, void *params)
{
struct data * data = (struct data *) ntuple_data;
double x, y, z, E2, scale;
scale = *(double *) params;
x = data->x;
y = data->y;
z = data->z;
E2 = x * x + y * y + z * z;
return E2 > scale;
}
double
val_func (void *ntuple_data, void *params)
{
(void)(params); /* avoid unused parameter warning */
struct data * data = (struct data *) ntuple_data;
double x, y, z;
x = data->x;
y = data->y;
z = data->z;
return x * x + y * y + z * z;
}
The following plot shows the distribution of the selected events.
Note the cut-off at the lower bound.
24.9 References and Further Reading
===================================
Further information on the use of ntuples can be found in the
documentation for the CERN packages PAW and HBOOK (available online).
25 Monte Carlo Integration
**************************
This chapter describes routines for multidimensional Monte Carlo
integration. These include the traditional Monte Carlo method and
adaptive algorithms such as VEGAS and MISER which use importance
sampling and stratified sampling techniques. Each algorithm computes an
estimate of a multidimensional definite integral of the form,
I = \int_xl^xu dx \int_yl^yu dy ... f(x, y, ...)
over a hypercubic region ((x_l,x_u), (y_l,y_u), ...) using a fixed
number of function calls. The routines also provide a statistical
estimate of the error on the result. This error estimate should be
taken as a guide rather than as a strict error bound--random sampling of
the region may not uncover all the important features of the function,
resulting in an underestimate of the error.
The functions are defined in separate header files for each routine,
'gsl_monte_plain.h', 'gsl_monte_miser.h' and 'gsl_monte_vegas.h'.
25.1 Interface
==============
All of the Monte Carlo integration routines use the same general form of
interface. There is an allocator to allocate memory for control
variables and workspace, a routine to initialize those control
variables, the integrator itself, and a function to free the space when
done.
Each integration function requires a random number generator to be
supplied, and returns an estimate of the integral and its standard
deviation. The accuracy of the result is determined by the number of
function calls specified by the user. If a known level of accuracy is
required this can be achieved by calling the integrator several times
and averaging the individual results until the desired accuracy is
obtained.
Random sample points used within the Monte Carlo routines are always
chosen strictly within the integration region, so that endpoint
singularities are automatically avoided.
The function to be integrated has its own datatype, defined in the
header file 'gsl_monte.h'.
-- Data Type: gsl_monte_function
This data type defines a general function with parameters for Monte
Carlo integration.
'double (* f) (double * X, size_t DIM, void * PARAMS)'
this function should return the value f(x,params) for the
argument X and parameters PARAMS, where X is an array of size
DIM giving the coordinates of the point where the function is
to be evaluated.
'size_t dim'
the number of dimensions for X.
'void * params'
a pointer to the parameters of the function.
Here is an example for a quadratic function in two dimensions,
f(x,y) = a x^2 + b x y + c y^2
with a = 3, b = 2, c = 1. The following code defines a
'gsl_monte_function' 'F' which you could pass to an integrator:
struct my_f_params { double a; double b; double c; };
double
my_f (double x[], size_t dim, void * p) {
struct my_f_params * fp = (struct my_f_params *)p;
if (dim != 2)
{
fprintf (stderr, "error: dim != 2");
abort ();
}
return fp->a * x[0] * x[0]
+ fp->b * x[0] * x[1]
+ fp->c * x[1] * x[1];
}
gsl_monte_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };
F.f = &my_f;
F.dim = 2;
F.params = ¶ms;
The function f(x) can be evaluated using the following macro,
#define GSL_MONTE_FN_EVAL(F,x)
(*((F)->f))(x,(F)->dim,(F)->params)
25.2 PLAIN Monte Carlo
======================
The plain Monte Carlo algorithm samples points randomly from the
integration region to estimate the integral and its error. Using this
algorithm the estimate of the integral E(f; N) for N randomly
distributed points x_i is given by,
E(f; N) = = V = (V / N) \sum_i^N f(x_i)
where V is the volume of the integration region. The error on this
estimate \sigma(E;N) is calculated from the estimated variance of the
mean,
\sigma^2 (E; N) = (V^2 / N^2) \sum_i^N (f(x_i) - )^2.
For large N this variance decreases asymptotically as \Var(f)/N, where
\Var(f) is the true variance of the function over the integration
region. The error estimate itself should decrease as
\sigma(f)/\sqrt{N}. The familiar law of errors decreasing as 1/\sqrt{N}
applies--to reduce the error by a factor of 10 requires a 100-fold
increase in the number of sample points.
The functions described in this section are declared in the header
file 'gsl_monte_plain.h'.
-- Function: gsl_monte_plain_state * gsl_monte_plain_alloc (size_t DIM)
This function allocates and initializes a workspace for Monte Carlo
integration in DIM dimensions.
-- Function: int gsl_monte_plain_init (gsl_monte_plain_state* S)
This function initializes a previously allocated integration state.
This allows an existing workspace to be reused for different
integrations.
-- Function: int gsl_monte_plain_integrate (gsl_monte_function * F,
const double XL[], const double XU[], size_t DIM, size_t
CALLS, gsl_rng * R, gsl_monte_plain_state * S, double *
RESULT, double * ABSERR)
This routines uses the plain Monte Carlo algorithm to integrate the
function F over the DIM-dimensional hypercubic region defined by
the lower and upper limits in the arrays XL and XU, each of size
DIM. The integration uses a fixed number of function calls CALLS,
and obtains random sampling points using the random number
generator R. A previously allocated workspace S must be supplied.
The result of the integration is returned in RESULT, with an
estimated absolute error ABSERR.
-- Function: void gsl_monte_plain_free (gsl_monte_plain_state * S)
This function frees the memory associated with the integrator state
S.
25.3 MISER
==========
The MISER algorithm of Press and Farrar is based on recursive stratified
sampling. This technique aims to reduce the overall integration error
by concentrating integration points in the regions of highest variance.
The idea of stratified sampling begins with the observation that for
two disjoint regions a and b with Monte Carlo estimates of the integral
E_a(f) and E_b(f) and variances \sigma_a^2(f) and \sigma_b^2(f), the
variance \Var(f) of the combined estimate E(f) = (1/2) (E_a(f) + E_b(f))
is given by,
\Var(f) = (\sigma_a^2(f) / 4 N_a) + (\sigma_b^2(f) / 4 N_b).
It can be shown that this variance is minimized by distributing the
points such that,
N_a / (N_a + N_b) = \sigma_a / (\sigma_a + \sigma_b).
Hence the smallest error estimate is obtained by allocating sample
points in proportion to the standard deviation of the function in each
sub-region.
The MISER algorithm proceeds by bisecting the integration region
along one coordinate axis to give two sub-regions at each step. The
direction is chosen by examining all d possible bisections and selecting
the one which will minimize the combined variance of the two
sub-regions. The variance in the sub-regions is estimated by sampling
with a fraction of the total number of points available to the current
step. The same procedure is then repeated recursively for each of the
two half-spaces from the best bisection. The remaining sample points
are allocated to the sub-regions using the formula for N_a and N_b.
This recursive allocation of integration points continues down to a
user-specified depth where each sub-region is integrated using a plain
Monte Carlo estimate. These individual values and their error estimates
are then combined upwards to give an overall result and an estimate of
its error.
The functions described in this section are declared in the header
file 'gsl_monte_miser.h'.
-- Function: gsl_monte_miser_state * gsl_monte_miser_alloc (size_t DIM)
This function allocates and initializes a workspace for Monte Carlo
integration in DIM dimensions. The workspace is used to maintain
the state of the integration.
-- Function: int gsl_monte_miser_init (gsl_monte_miser_state* S)
This function initializes a previously allocated integration state.
This allows an existing workspace to be reused for different
integrations.
-- Function: int gsl_monte_miser_integrate (gsl_monte_function * F,
const double XL[], const double XU[], size_t DIM, size_t
CALLS, gsl_rng * R, gsl_monte_miser_state * S, double *
RESULT, double * ABSERR)
This routines uses the MISER Monte Carlo algorithm to integrate the
function F over the DIM-dimensional hypercubic region defined by
the lower and upper limits in the arrays XL and XU, each of size
DIM. The integration uses a fixed number of function calls CALLS,
and obtains random sampling points using the random number
generator R. A previously allocated workspace S must be supplied.
The result of the integration is returned in RESULT, with an
estimated absolute error ABSERR.
-- Function: void gsl_monte_miser_free (gsl_monte_miser_state * S)
This function frees the memory associated with the integrator state
S.
The MISER algorithm has several configurable parameters which can be
changed using the following two functions.(1)
-- Function: void gsl_monte_miser_params_get (const
gsl_monte_miser_state * S, gsl_monte_miser_params * PARAMS)
This function copies the parameters of the integrator state into
the user-supplied PARAMS structure.
-- Function: void gsl_monte_miser_params_set (gsl_monte_miser_state *
S, const gsl_monte_miser_params * PARAMS)
This function sets the integrator parameters based on values
provided in the PARAMS structure.
Typically the values of the parameters are first read using
'gsl_monte_miser_params_get', the necessary changes are made to the
fields of the PARAMS structure, and the values are copied back into the
integrator state using 'gsl_monte_miser_params_set'. The functions use
the 'gsl_monte_miser_params' structure which contains the following
fields:
-- Variable: double estimate_frac
This parameter specifies the fraction of the currently available
number of function calls which are allocated to estimating the
variance at each recursive step. The default value is 0.1.
-- Variable: size_t min_calls
This parameter specifies the minimum number of function calls
required for each estimate of the variance. If the number of
function calls allocated to the estimate using ESTIMATE_FRAC falls
below MIN_CALLS then MIN_CALLS are used instead. This ensures that
each estimate maintains a reasonable level of accuracy. The
default value of MIN_CALLS is '16 * dim'.
-- Variable: size_t min_calls_per_bisection
This parameter specifies the minimum number of function calls
required to proceed with a bisection step. When a recursive step
has fewer calls available than MIN_CALLS_PER_BISECTION it performs
a plain Monte Carlo estimate of the current sub-region and
terminates its branch of the recursion. The default value of this
parameter is '32 * min_calls'.
-- Variable: double alpha
This parameter controls how the estimated variances for the two
sub-regions of a bisection are combined when allocating points.
With recursive sampling the overall variance should scale better
than 1/N, since the values from the sub-regions will be obtained
using a procedure which explicitly minimizes their variance. To
accommodate this behavior the MISER algorithm allows the total
variance to depend on a scaling parameter \alpha,
\Var(f) = {\sigma_a \over N_a^\alpha} + {\sigma_b \over N_b^\alpha}.
The authors of the original paper describing MISER recommend the
value \alpha = 2 as a good choice, obtained from numerical
experiments, and this is used as the default value in this
implementation.
-- Variable: double dither
This parameter introduces a random fractional variation of size
DITHER into each bisection, which can be used to break the symmetry
of integrands which are concentrated near the exact center of the
hypercubic integration region. The default value of dither is
zero, so no variation is introduced. If needed, a typical value of
DITHER is 0.1.
---------- Footnotes ----------
(1) The previous method of accessing these fields directly through
the 'gsl_monte_miser_state' struct is now deprecated.
25.4 VEGAS
==========
The VEGAS algorithm of Lepage is based on importance sampling. It
samples points from the probability distribution described by the
function |f|, so that the points are concentrated in the regions that
make the largest contribution to the integral.
In general, if the Monte Carlo integral of f is sampled with points
distributed according to a probability distribution described by the
function g, we obtain an estimate E_g(f; N),
E_g(f; N) = E(f/g; N)
with a corresponding variance,
\Var_g(f; N) = \Var(f/g; N).
If the probability distribution is chosen as g = |f|/I(|f|) then it can
be shown that the variance V_g(f; N) vanishes, and the error in the
estimate will be zero. In practice it is not possible to sample from
the exact distribution g for an arbitrary function, so importance
sampling algorithms aim to produce efficient approximations to the
desired distribution.
The VEGAS algorithm approximates the exact distribution by making a
number of passes over the integration region while histogramming the
function f. Each histogram is used to define a sampling distribution
for the next pass. Asymptotically this procedure converges to the
desired distribution. In order to avoid the number of histogram bins
growing like K^d the probability distribution is approximated by a
separable function: g(x_1, x_2, ...) = g_1(x_1) g_2(x_2) ... so that the
number of bins required is only Kd. This is equivalent to locating the
peaks of the function from the projections of the integrand onto the
coordinate axes. The efficiency of VEGAS depends on the validity of
this assumption. It is most efficient when the peaks of the integrand
are well-localized. If an integrand can be rewritten in a form which is
approximately separable this will increase the efficiency of integration
with VEGAS.
VEGAS incorporates a number of additional features, and combines both
stratified sampling and importance sampling. The integration region is
divided into a number of "boxes", with each box getting a fixed number
of points (the goal is 2). Each box can then have a fractional number
of bins, but if the ratio of bins-per-box is less than two, Vegas
switches to a kind variance reduction (rather than importance sampling).
-- Function: gsl_monte_vegas_state * gsl_monte_vegas_alloc (size_t DIM)
This function allocates and initializes a workspace for Monte Carlo
integration in DIM dimensions. The workspace is used to maintain
the state of the integration.
-- Function: int gsl_monte_vegas_init (gsl_monte_vegas_state* S)
This function initializes a previously allocated integration state.
This allows an existing workspace to be reused for different
integrations.
-- Function: int gsl_monte_vegas_integrate (gsl_monte_function * F,
double XL[], double XU[], size_t DIM, size_t CALLS, gsl_rng *
R, gsl_monte_vegas_state * S, double * RESULT, double *
ABSERR)
This routines uses the VEGAS Monte Carlo algorithm to integrate the
function F over the DIM-dimensional hypercubic region defined by
the lower and upper limits in the arrays XL and XU, each of size
DIM. The integration uses a fixed number of function calls CALLS,
and obtains random sampling points using the random number
generator R. A previously allocated workspace S must be supplied.
The result of the integration is returned in RESULT, with an
estimated absolute error ABSERR. The result and its error estimate
are based on a weighted average of independent samples. The
chi-squared per degree of freedom for the weighted average is
returned via the state struct component, S->CHISQ, and must be
consistent with 1 for the weighted average to be reliable.
-- Function: void gsl_monte_vegas_free (gsl_monte_vegas_state * S)
This function frees the memory associated with the integrator state
S.
The VEGAS algorithm computes a number of independent estimates of the
integral internally, according to the 'iterations' parameter described
below, and returns their weighted average. Random sampling of the
integrand can occasionally produce an estimate where the error is zero,
particularly if the function is constant in some regions. An estimate
with zero error causes the weighted average to break down and must be
handled separately. In the original Fortran implementations of VEGAS
the error estimate is made non-zero by substituting a small value
(typically '1e-30'). The implementation in GSL differs from this and
avoids the use of an arbitrary constant--it either assigns the value a
weight which is the average weight of the preceding estimates or
discards it according to the following procedure,
current estimate has zero error, weighted average has finite error
The current estimate is assigned a weight which is the average
weight of the preceding estimates.
current estimate has finite error, previous estimates had zero error
The previous estimates are discarded and the weighted averaging
procedure begins with the current estimate.
current estimate has zero error, previous estimates had zero error
The estimates are averaged using the arithmetic mean, but no error
is computed.
The convergence of the algorithm can be tested using the overall
chi-squared value of the results, which is available from the following
function:
-- Function: double gsl_monte_vegas_chisq (const gsl_monte_vegas_state
* S)
This function returns the chi-squared per degree of freedom for the
weighted estimate of the integral. The returned value should be
close to 1. A value which differs significantly from 1 indicates
that the values from different iterations are inconsistent. In
this case the weighted error will be under-estimated, and further
iterations of the algorithm are needed to obtain reliable results.
-- Function: void gsl_monte_vegas_runval (const gsl_monte_vegas_state *
S, double * RESULT, double * SIGMA)
This function returns the raw (unaveraged) values of the integral
RESULT and its error SIGMA from the most recent iteration of the
algorithm.
The VEGAS algorithm is highly configurable. Several parameters can
be changed using the following two functions.
-- Function: void gsl_monte_vegas_params_get (const
gsl_monte_vegas_state * S, gsl_monte_vegas_params * PARAMS)
This function copies the parameters of the integrator state into
the user-supplied PARAMS structure.
-- Function: void gsl_monte_vegas_params_set (gsl_monte_vegas_state *
S, const gsl_monte_vegas_params * PARAMS)
This function sets the integrator parameters based on values
provided in the PARAMS structure.
Typically the values of the parameters are first read using
'gsl_monte_vegas_params_get', the necessary changes are made to the
fields of the PARAMS structure, and the values are copied back into the
integrator state using 'gsl_monte_vegas_params_set'. The functions use
the 'gsl_monte_vegas_params' structure which contains the following
fields:
-- Variable: double alpha
The parameter 'alpha' controls the stiffness of the rebinning
algorithm. It is typically set between one and two. A value of
zero prevents rebinning of the grid. The default value is 1.5.
-- Variable: size_t iterations
The number of iterations to perform for each call to the routine.
The default value is 5 iterations.
-- Variable: int stage
Setting this determines the "stage" of the calculation. Normally,
'stage = 0' which begins with a new uniform grid and empty weighted
average. Calling VEGAS with 'stage = 1' retains the grid from the
previous run but discards the weighted average, so that one can
"tune" the grid using a relatively small number of points and then
do a large run with 'stage = 1' on the optimized grid. Setting
'stage = 2' keeps the grid and the weighted average from the
previous run, but may increase (or decrease) the number of
histogram bins in the grid depending on the number of calls
available. Choosing 'stage = 3' enters at the main loop, so that
nothing is changed, and is equivalent to performing additional
iterations in a previous call.
-- Variable: int mode
The possible choices are 'GSL_VEGAS_MODE_IMPORTANCE',
'GSL_VEGAS_MODE_STRATIFIED', 'GSL_VEGAS_MODE_IMPORTANCE_ONLY'.
This determines whether VEGAS will use importance sampling or
stratified sampling, or whether it can pick on its own. In low
dimensions VEGAS uses strict stratified sampling (more precisely,
stratified sampling is chosen if there are fewer than 2 bins per
box).
-- Variable: int verbose
-- Variable: FILE * ostream
These parameters set the level of information printed by VEGAS.
All information is written to the stream OSTREAM. The default
setting of VERBOSE is '-1', which turns off all output. A VERBOSE
value of '0' prints summary information about the weighted average
and final result, while a value of '1' also displays the grid
coordinates. A value of '2' prints information from the rebinning
procedure for each iteration.
The above fields and the CHISQ value can also be accessed directly in
the 'gsl_monte_vegas_state' but such use is deprecated.
25.5 Examples
=============
The example program below uses the Monte Carlo routines to estimate the
value of the following 3-dimensional integral from the theory of random
walks,
I = \int_{-pi}^{+pi} {dk_x/(2 pi)}
\int_{-pi}^{+pi} {dk_y/(2 pi)}
\int_{-pi}^{+pi} {dk_z/(2 pi)}
1 / (1 - cos(k_x)cos(k_y)cos(k_z)).
The analytic value of this integral can be shown to be I =
\Gamma(1/4)^4/(4 \pi^3) = 1.393203929685676859.... The integral gives
the mean time spent at the origin by a random walk on a body-centered
cubic lattice in three dimensions.
For simplicity we will compute the integral over the region (0,0,0)
to (\pi,\pi,\pi) and multiply by 8 to obtain the full result. The
integral is slowly varying in the middle of the region but has
integrable singularities at the corners (0,0,0), (0,\pi,\pi),
(\pi,0,\pi) and (\pi,\pi,0). The Monte Carlo routines only select
points which are strictly within the integration region and so no
special measures are needed to avoid these singularities.
#include
#include
#include
#include
#include
#include
/* Computation of the integral,
I = int (dx dy dz)/(2pi)^3 1/(1-cos(x)cos(y)cos(z))
over (-pi,-pi,-pi) to (+pi, +pi, +pi). The exact answer
is Gamma(1/4)^4/(4 pi^3). This example is taken from
C.Itzykson, J.M.Drouffe, "Statistical Field Theory -
Volume 1", Section 1.1, p21, which cites the original
paper M.L.Glasser, I.J.Zucker, Proc.Natl.Acad.Sci.USA 74
1800 (1977) */
/* For simplicity we compute the integral over the region
(0,0,0) -> (pi,pi,pi) and multiply by 8 */
double exact = 1.3932039296856768591842462603255;
double
g (double *k, size_t dim, void *params)
{
(void)(dim); /* avoid unused parameter warnings */
(void)(params);
double A = 1.0 / (M_PI * M_PI * M_PI);
return A / (1.0 - cos (k[0]) * cos (k[1]) * cos (k[2]));
}
void
display_results (char *title, double result, double error)
{
printf ("%s ==================\n", title);
printf ("result = % .6f\n", result);
printf ("sigma = % .6f\n", error);
printf ("exact = % .6f\n", exact);
printf ("error = % .6f = %.2g sigma\n", result - exact,
fabs (result - exact) / error);
}
int
main (void)
{
double res, err;
double xl[3] = { 0, 0, 0 };
double xu[3] = { M_PI, M_PI, M_PI };
const gsl_rng_type *T;
gsl_rng *r;
gsl_monte_function G = { &g, 3, 0 };
size_t calls = 500000;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
{
gsl_monte_plain_state *s = gsl_monte_plain_alloc (3);
gsl_monte_plain_integrate (&G, xl, xu, 3, calls, r, s,
&res, &err);
gsl_monte_plain_free (s);
display_results ("plain", res, err);
}
{
gsl_monte_miser_state *s = gsl_monte_miser_alloc (3);
gsl_monte_miser_integrate (&G, xl, xu, 3, calls, r, s,
&res, &err);
gsl_monte_miser_free (s);
display_results ("miser", res, err);
}
{
gsl_monte_vegas_state *s = gsl_monte_vegas_alloc (3);
gsl_monte_vegas_integrate (&G, xl, xu, 3, 10000, r, s,
&res, &err);
display_results ("vegas warm-up", res, err);
printf ("converging...\n");
do
{
gsl_monte_vegas_integrate (&G, xl, xu, 3, calls/5, r, s,
&res, &err);
printf ("result = % .6f sigma = % .6f "
"chisq/dof = %.1f\n", res, err, gsl_monte_vegas_chisq (s));
}
while (fabs (gsl_monte_vegas_chisq (s) - 1.0) > 0.5);
display_results ("vegas final", res, err);
gsl_monte_vegas_free (s);
}
gsl_rng_free (r);
return 0;
}
With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 1%. The estimated error 'sigma' is roughly
consistent with the actual error-the computed result differs from the
true result by about 1.4 standard deviations,
plain ==================
result = 1.412209
sigma = 0.013436
exact = 1.393204
error = 0.019005 = 1.4 sigma
The MISER algorithm reduces the error by a factor of four, and also
correctly estimates the error,
miser ==================
result = 1.391322
sigma = 0.003461
exact = 1.393204
error = -0.001882 = 0.54 sigma
In the case of the VEGAS algorithm the program uses an initial warm-up
run of 10,000 function calls to prepare, or "warm up", the grid. This
is followed by a main run with five iterations of 100,000 function
calls. The chi-squared per degree of freedom for the five iterations
are checked for consistency with 1, and the run is repeated if the
results have not converged. In this case the estimates are consistent
on the first pass.
vegas warm-up ==================
result = 1.392673
sigma = 0.003410
exact = 1.393204
error = -0.000531 = 0.16 sigma
converging...
result = 1.393281 sigma = 0.000362 chisq/dof = 1.5
vegas final ==================
result = 1.393281
sigma = 0.000362
exact = 1.393204
error = 0.000077 = 0.21 sigma
If the value of 'chisq' had differed significantly from 1 it would
indicate inconsistent results, with a correspondingly underestimated
error. The final estimate from VEGAS (using a similar number of
function calls) is significantly more accurate than the other two
algorithms.
25.6 References and Further Reading
===================================
The MISER algorithm is described in the following article by Press and
Farrar,
W.H. Press, G.R. Farrar, 'Recursive Stratified Sampling for
Multidimensional Monte Carlo Integration', Computers in Physics, v4
(1990), pp190-195.
The VEGAS algorithm is described in the following papers,
G.P. Lepage, 'A New Algorithm for Adaptive Multidimensional
Integration', Journal of Computational Physics 27, 192-203, (1978)
G.P. Lepage, 'VEGAS: An Adaptive Multi-dimensional Integration
Program', Cornell preprint CLNS 80-447, March 1980
26 Simulated Annealing
**********************
Stochastic search techniques are used when the structure of a space is
not well understood or is not smooth, so that techniques like Newton's
method (which requires calculating Jacobian derivative matrices) cannot
be used. In particular, these techniques are frequently used to solve
combinatorial optimization problems, such as the traveling salesman
problem.
The goal is to find a point in the space at which a real valued
"energy function" (or "cost function") is minimized. Simulated
annealing is a minimization technique which has given good results in
avoiding local minima; it is based on the idea of taking a random walk
through the space at successively lower temperatures, where the
probability of taking a step is given by a Boltzmann distribution.
The functions described in this chapter are declared in the header
file 'gsl_siman.h'.
26.1 Simulated Annealing algorithm
==================================
The simulated annealing algorithm takes random walks through the problem
space, looking for points with low energies; in these random walks, the
probability of taking a step is determined by the Boltzmann
distribution,
p = e^{-(E_{i+1} - E_i)/(kT)}
if E_{i+1} > E_i, and p = 1 when E_{i+1} <= E_i.
In other words, a step will occur if the new energy is lower. If the
new energy is higher, the transition can still occur, and its likelihood
is proportional to the temperature T and inversely proportional to the
energy difference E_{i+1} - E_i.
The temperature T is initially set to a high value, and a random walk
is carried out at that temperature. Then the temperature is lowered
very slightly according to a "cooling schedule", for example: T ->
T/mu_T where \mu_T is slightly greater than 1.
The slight probability of taking a step that gives higher energy is
what allows simulated annealing to frequently get out of local minima.
26.2 Simulated Annealing functions
==================================
-- Function: void gsl_siman_solve (const gsl_rng * R, void * X0_P,
gsl_siman_Efunc_t EF, gsl_siman_step_t TAKE_STEP,
gsl_siman_metric_t DISTANCE, gsl_siman_print_t PRINT_POSITION,
gsl_siman_copy_t COPYFUNC, gsl_siman_copy_construct_t
COPY_CONSTRUCTOR, gsl_siman_destroy_t DESTRUCTOR, size_t
ELEMENT_SIZE, gsl_siman_params_t PARAMS)
This function performs a simulated annealing search through a given
space. The space is specified by providing the functions EF and
DISTANCE. The simulated annealing steps are generated using the
random number generator R and the function TAKE_STEP.
The starting configuration of the system should be given by X0_P.
The routine offers two modes for updating configurations, a
fixed-size mode and a variable-size mode. In the fixed-size mode
the configuration is stored as a single block of memory of size
ELEMENT_SIZE. Copies of this configuration are created, copied and
destroyed internally using the standard library functions 'malloc',
'memcpy' and 'free'. The function pointers COPYFUNC,
COPY_CONSTRUCTOR and DESTRUCTOR should be null pointers in
fixed-size mode. In the variable-size mode the functions COPYFUNC,
COPY_CONSTRUCTOR and DESTRUCTOR are used to create, copy and
destroy configurations internally. The variable ELEMENT_SIZE
should be zero in the variable-size mode.
The PARAMS structure (described below) controls the run by
providing the temperature schedule and other tunable parameters to
the algorithm.
On exit the best result achieved during the search is placed in
'*X0_P'. If the annealing process has been successful this should
be a good approximation to the optimal point in the space.
If the function pointer PRINT_POSITION is not null, a debugging log
will be printed to 'stdout' with the following columns:
#-iter #-evals temperature position energy best_energy
and the output of the function PRINT_POSITION itself. If
PRINT_POSITION is null then no information is printed.
The simulated annealing routines require several user-specified
functions to define the configuration space and energy function. The
prototypes for these functions are given below.
-- Data Type: gsl_siman_Efunc_t
This function type should return the energy of a configuration XP.
double (*gsl_siman_Efunc_t) (void *xp)
-- Data Type: gsl_siman_step_t
This function type should modify the configuration XP using a
random step taken from the generator R, up to a maximum distance of
STEP_SIZE.
void (*gsl_siman_step_t) (const gsl_rng *r, void *xp,
double step_size)
-- Data Type: gsl_siman_metric_t
This function type should return the distance between two
configurations XP and YP.
double (*gsl_siman_metric_t) (void *xp, void *yp)
-- Data Type: gsl_siman_print_t
This function type should print the contents of the configuration
XP.
void (*gsl_siman_print_t) (void *xp)
-- Data Type: gsl_siman_copy_t
This function type should copy the configuration SOURCE into DEST.
void (*gsl_siman_copy_t) (void *source, void *dest)
-- Data Type: gsl_siman_copy_construct_t
This function type should create a new copy of the configuration
XP.
void * (*gsl_siman_copy_construct_t) (void *xp)
-- Data Type: gsl_siman_destroy_t
This function type should destroy the configuration XP, freeing its
memory.
void (*gsl_siman_destroy_t) (void *xp)
-- Data Type: gsl_siman_params_t
These are the parameters that control a run of 'gsl_siman_solve'.
This structure contains all the information needed to control the
search, beyond the energy function, the step function and the
initial guess.
'int n_tries'
The number of points to try for each step.
'int iters_fixed_T'
The number of iterations at each temperature.
'double step_size'
The maximum step size in the random walk.
'double k, t_initial, mu_t, t_min'
The parameters of the Boltzmann distribution and cooling
schedule.
26.3 Examples
=============
The simulated annealing package is clumsy, and it has to be because it
is written in C, for C callers, and tries to be polymorphic at the same
time. But here we provide some examples which can be pasted into your
application with little change and should make things easier.
26.3.1 Trivial example
----------------------
The first example, in one dimensional Cartesian space, sets up an energy
function which is a damped sine wave; this has many local minima, but
only one global minimum, somewhere between 1.0 and 1.5. The initial
guess given is 15.5, which is several local minima away from the global
minimum.
#include
#include
#include
#include
/* set up parameters for this simulated annealing run */
/* how many points do we try before stepping */
#define N_TRIES 200
/* how many iterations for each T? */
#define ITERS_FIXED_T 1000
/* max step size in random walk */
#define STEP_SIZE 1.0
/* Boltzmann constant */
#define K 1.0
/* initial temperature */
#define T_INITIAL 0.008
/* damping factor for temperature */
#define MU_T 1.003
#define T_MIN 2.0e-6
gsl_siman_params_t params
= {N_TRIES, ITERS_FIXED_T, STEP_SIZE,
K, T_INITIAL, MU_T, T_MIN};
/* now some functions to test in one dimension */
double E1(void *xp)
{
double x = * ((double *) xp);
return exp(-pow((x-1.0),2.0))*sin(8*x);
}
double M1(void *xp, void *yp)
{
double x = *((double *) xp);
double y = *((double *) yp);
return fabs(x - y);
}
void S1(const gsl_rng * r, void *xp, double step_size)
{
double old_x = *((double *) xp);
double new_x;
double u = gsl_rng_uniform(r);
new_x = u * 2 * step_size - step_size + old_x;
memcpy(xp, &new_x, sizeof(new_x));
}
void P1(void *xp)
{
printf ("%12g", *((double *) xp));
}
int
main(void)
{
const gsl_rng_type * T;
gsl_rng * r;
double x_initial = 15.5;
gsl_rng_env_setup();
T = gsl_rng_default;
r = gsl_rng_alloc(T);
gsl_siman_solve(r, &x_initial, E1, S1, M1, P1,
NULL, NULL, NULL,
sizeof(double), params);
gsl_rng_free (r);
return 0;
}
Here are a couple of plots that are generated by running 'siman_test'
in the following way:
$ ./siman_test | awk '!/^#/ {print $1, $4}'
| graph -y 1.34 1.4 -W0 -X generation -Y position
| plot -Tps > siman-test.eps
$ ./siman_test | awk '!/^#/ {print $1, $5}'
| graph -y -0.88 -0.83 -W0 -X generation -Y energy
| plot -Tps > siman-energy.eps
26.3.2 Traveling Salesman Problem
---------------------------------
The TSP ("Traveling Salesman Problem") is the classic combinatorial
optimization problem. I have provided a very simple version of it,
based on the coordinates of twelve cities in the southwestern United
States. This should maybe be called the "Flying Salesman Problem",
since I am using the great-circle distance between cities, rather than
the driving distance. Also: I assume the earth is a sphere, so I don't
use geoid distances.
The 'gsl_siman_solve' routine finds a route which is 3490.62
Kilometers long; this is confirmed by an exhaustive search of all
possible routes with the same initial city.
The full code can be found in 'siman/siman_tsp.c', but I include here
some plots generated in the following way:
$ ./siman_tsp > tsp.output
$ grep -v "^#" tsp.output
| awk '{print $1, $NF}'
| graph -y 3300 6500 -W0 -X generation -Y distance
-L "TSP - 12 southwest cities"
| plot -Tps > 12-cities.eps
$ grep initial_city_coord tsp.output
| awk '{print $2, $3}'
| graph -X "longitude (- means west)" -Y "latitude"
-L "TSP - initial-order" -f 0.03 -S 1 0.1
| plot -Tps > initial-route.eps
$ grep final_city_coord tsp.output
| awk '{print $2, $3}'
| graph -X "longitude (- means west)" -Y "latitude"
-L "TSP - final-order" -f 0.03 -S 1 0.1
| plot -Tps > final-route.eps
This is the output showing the initial order of the cities; longitude is
negative, since it is west and I want the plot to look like a map.
# initial coordinates of cities (longitude and latitude)
###initial_city_coord: -105.95 35.68 Santa Fe
###initial_city_coord: -112.07 33.54 Phoenix
###initial_city_coord: -106.62 35.12 Albuquerque
###initial_city_coord: -103.2 34.41 Clovis
###initial_city_coord: -107.87 37.29 Durango
###initial_city_coord: -96.77 32.79 Dallas
###initial_city_coord: -105.92 35.77 Tesuque
###initial_city_coord: -107.84 35.15 Grants
###initial_city_coord: -106.28 35.89 Los Alamos
###initial_city_coord: -106.76 32.34 Las Cruces
###initial_city_coord: -108.58 37.35 Cortez
###initial_city_coord: -108.74 35.52 Gallup
###initial_city_coord: -105.95 35.68 Santa Fe
The optimal route turns out to be:
# final coordinates of cities (longitude and latitude)
###final_city_coord: -105.95 35.68 Santa Fe
###final_city_coord: -103.2 34.41 Clovis
###final_city_coord: -96.77 32.79 Dallas
###final_city_coord: -106.76 32.34 Las Cruces
###final_city_coord: -112.07 33.54 Phoenix
###final_city_coord: -108.74 35.52 Gallup
###final_city_coord: -108.58 37.35 Cortez
###final_city_coord: -107.87 37.29 Durango
###final_city_coord: -107.84 35.15 Grants
###final_city_coord: -106.62 35.12 Albuquerque
###final_city_coord: -106.28 35.89 Los Alamos
###final_city_coord: -105.92 35.77 Tesuque
###final_city_coord: -105.95 35.68 Santa Fe
Here's a plot of the cost function (energy) versus generation (point in
the calculation at which a new temperature is set) for this problem:
26.4 References and Further Reading
===================================
Further information is available in the following book,
'Modern Heuristic Techniques for Combinatorial Problems', Colin R.
Reeves (ed.), McGraw-Hill, 1995 (ISBN 0-07-709239-2).
27 Ordinary Differential Equations
**********************************
This chapter describes functions for solving ordinary differential
equation (ODE) initial value problems. The library provides a variety
of low-level methods, such as Runge-Kutta and Bulirsch-Stoer routines,
and higher-level components for adaptive step-size control. The
components can be combined by the user to achieve the desired solution,
with full access to any intermediate steps. A driver object can be used
as a high level wrapper for easy use of low level functions.
These functions are declared in the header file 'gsl_odeiv2.h'. This
is a new interface in version 1.15 and uses the prefix 'gsl_odeiv2' for
all functions. It is recommended over the previous 'gsl_odeiv'
implementation defined in 'gsl_odeiv.h' The old interface has been
retained under the original name for backwards compatibility.
27.1 Defining the ODE System
============================
The routines solve the general n-dimensional first-order system,
dy_i(t)/dt = f_i(t, y_1(t), ..., y_n(t))
for i = 1, \dots, n. The stepping functions rely on the vector of
derivatives f_i and the Jacobian matrix, J_{ij} = df_i(t,y(t)) / dy_j.
A system of equations is defined using the 'gsl_odeiv2_system' datatype.
-- Data Type: gsl_odeiv2_system
This data type defines a general ODE system with arbitrary
parameters.
'int (* function) (double t, const double y[], double dydt[], void * params)'
This function should store the vector elements f_i(t,y,params)
in the array DYDT, for arguments (T,Y) and parameters PARAMS.
The function should return 'GSL_SUCCESS' if the calculation
was completed successfully. Any other return value indicates
an error. A special return value 'GSL_EBADFUNC' causes
'gsl_odeiv2' routines to immediately stop and return. If
'function' is modified (for example contents of PARAMS), the
user must call an appropriate reset function
('gsl_odeiv2_driver_reset', 'gsl_odeiv2_evolve_reset' or
'gsl_odeiv2_step_reset') before continuing. Use return values
distinct from standard GSL error codes to distinguish your
function as the source of the error.
'int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);'
This function should store the vector of derivative elements
in the array DFDT and the Jacobian matrix J_{ij} in the array
DFDY, regarded as a row-ordered matrix 'J(i,j) = dfdy[i *
dimension + j]' where 'dimension' is the dimension of the
system.
Not all of the stepper algorithms of 'gsl_odeiv2' make use of
the Jacobian matrix, so it may not be necessary to provide
this function (the 'jacobian' element of the struct can be
replaced by a null pointer for those algorithms).
The function should return 'GSL_SUCCESS' if the calculation
was completed successfully. Any other return value indicates
an error. A special return value 'GSL_EBADFUNC' causes
'gsl_odeiv2' routines to immediately stop and return. If
'jacobian' is modified (for example contents of PARAMS), the
user must call an appropriate reset function
('gsl_odeiv2_driver_reset', 'gsl_odeiv2_evolve_reset' or
'gsl_odeiv2_step_reset') before continuing. Use return values
distinct from standard GSL error codes to distinguish your
function as the source of the error.
'size_t dimension;'
This is the dimension of the system of equations.
'void * params'
This is a pointer to the arbitrary parameters of the system.
27.2 Stepping Functions
=======================
The lowest level components are the "stepping functions" which advance a
solution from time t to t+h for a fixed step-size h and estimate the
resulting local error.
-- Function: gsl_odeiv2_step * gsl_odeiv2_step_alloc (const
gsl_odeiv2_step_type * T, size_t DIM)
This function returns a pointer to a newly allocated instance of a
stepping function of type T for a system of DIM dimensions. Please
note that if you use a stepper method that requires access to a
driver object, it is advisable to use a driver allocation method,
which automatically allocates a stepper, too.
-- Function: int gsl_odeiv2_step_reset (gsl_odeiv2_step * S)
This function resets the stepping function S. It should be used
whenever the next use of S will not be a continuation of a previous
step.
-- Function: void gsl_odeiv2_step_free (gsl_odeiv2_step * S)
This function frees all the memory associated with the stepping
function S.
-- Function: const char * gsl_odeiv2_step_name (const gsl_odeiv2_step *
S)
This function returns a pointer to the name of the stepping
function. For example,
printf ("step method is '%s'\n",
gsl_odeiv2_step_name (s));
would print something like 'step method is 'rkf45''.
-- Function: unsigned int gsl_odeiv2_step_order (const gsl_odeiv2_step
* S)
This function returns the order of the stepping function on the
previous step. The order can vary if the stepping function itself
is adaptive.
-- Function: int gsl_odeiv2_step_set_driver (gsl_odeiv2_step * S, const
gsl_odeiv2_driver * D)
This function sets a pointer of the driver object D for stepper S,
to allow the stepper to access control (and evolve) object through
the driver object. This is a requirement for some steppers, to get
the desired error level for internal iteration of stepper.
Allocation of a driver object calls this function automatically.
-- Function: int gsl_odeiv2_step_apply (gsl_odeiv2_step * S, double T,
double H, double Y[], double YERR[], const double DYDT_IN[],
double DYDT_OUT[], const gsl_odeiv2_system * SYS)
This function applies the stepping function S to the system of
equations defined by SYS, using the step-size H to advance the
system from time T and state Y to time T+H. The new state of the
system is stored in Y on output, with an estimate of the absolute
error in each component stored in YERR. If the argument DYDT_IN is
not null it should point an array containing the derivatives for
the system at time T on input. This is optional as the derivatives
will be computed internally if they are not provided, but allows
the reuse of existing derivative information. On output the new
derivatives of the system at time T+H will be stored in DYDT_OUT if
it is not null.
The stepping function returns 'GSL_FAILURE' if it is unable to
compute the requested step. Also, if the user-supplied functions
defined in the system SYS return a status other than 'GSL_SUCCESS'
the step will be aborted. In that case, the elements of Y will be
restored to their pre-step values and the error code from the
user-supplied function will be returned. Failure may be due to a
singularity in the system or too large step-size H. In that case
the step should be attempted again with a smaller step-size, e.g.
H/2.
If the driver object is not appropriately set via
'gsl_odeiv2_step_set_driver' for those steppers that need it, the
stepping function returns 'GSL_EFAULT'. If the user-supplied
functions defined in the system SYS returns 'GSL_EBADFUNC', the
function returns immediately with the same return code. In this
case the user must call 'gsl_odeiv2_step_reset' before calling this
function again.
The following algorithms are available,
-- Step Type: gsl_odeiv2_step_rk2
Explicit embedded Runge-Kutta (2, 3) method.
-- Step Type: gsl_odeiv2_step_rk4
Explicit 4th order (classical) Runge-Kutta. Error estimation is
carried out by the step doubling method. For more efficient
estimate of the error, use the embedded methods described below.
-- Step Type: gsl_odeiv2_step_rkf45
Explicit embedded Runge-Kutta-Fehlberg (4, 5) method. This method
is a good general-purpose integrator.
-- Step Type: gsl_odeiv2_step_rkck
Explicit embedded Runge-Kutta Cash-Karp (4, 5) method.
-- Step Type: gsl_odeiv2_step_rk8pd
Explicit embedded Runge-Kutta Prince-Dormand (8, 9) method.
-- Step Type: gsl_odeiv2_step_rk1imp
Implicit Gaussian first order Runge-Kutta. Also known as implicit
Euler or backward Euler method. Error estimation is carried out by
the step doubling method. This algorithm requires the Jacobian and
access to the driver object via 'gsl_odeiv2_step_set_driver'.
-- Step Type: gsl_odeiv2_step_rk2imp
Implicit Gaussian second order Runge-Kutta. Also known as implicit
mid-point rule. Error estimation is carried out by the step
doubling method. This stepper requires the Jacobian and access to
the driver object via 'gsl_odeiv2_step_set_driver'.
-- Step Type: gsl_odeiv2_step_rk4imp
Implicit Gaussian 4th order Runge-Kutta. Error estimation is
carried out by the step doubling method. This algorithm requires
the Jacobian and access to the driver object via
'gsl_odeiv2_step_set_driver'.
-- Step Type: gsl_odeiv2_step_bsimp
Implicit Bulirsch-Stoer method of Bader and Deuflhard. The method
is generally suitable for stiff problems. This stepper requires
the Jacobian.
-- Step Type: gsl_odeiv2_step_msadams
A variable-coefficient linear multistep Adams method in Nordsieck
form. This stepper uses explicit Adams-Bashforth (predictor) and
implicit Adams-Moulton (corrector) methods in P(EC)^m functional
iteration mode. Method order varies dynamically between 1 and 12.
This stepper requires the access to the driver object via
'gsl_odeiv2_step_set_driver'.
-- Step Type: gsl_odeiv2_step_msbdf
A variable-coefficient linear multistep backward differentiation
formula (BDF) method in Nordsieck form. This stepper uses the
explicit BDF formula as predictor and implicit BDF formula as
corrector. A modified Newton iteration method is used to solve the
system of non-linear equations. Method order varies dynamically
between 1 and 5. The method is generally suitable for stiff
problems. This stepper requires the Jacobian and the access to the
driver object via 'gsl_odeiv2_step_set_driver'.
27.3 Adaptive Step-size Control
===============================
The control function examines the proposed change to the solution
produced by a stepping function and attempts to determine the optimal
step-size for a user-specified level of error.
-- Function: gsl_odeiv2_control * gsl_odeiv2_control_standard_new
(double EPS_ABS, double EPS_REL, double A_Y, double A_DYDT)
The standard control object is a four parameter heuristic based on
absolute and relative errors EPS_ABS and EPS_REL, and scaling
factors A_Y and A_DYDT for the system state y(t) and derivatives
y'(t) respectively.
The step-size adjustment procedure for this method begins by
computing the desired error level D_i for each component,
D_i = eps_abs + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)
and comparing it with the observed error E_i = |yerr_i|. If the
observed error E exceeds the desired error level D by more than 10%
for any component then the method reduces the step-size by an
appropriate factor,
h_new = h_old * S * (E/D)^(-1/q)
where q is the consistency order of the method (e.g. q=4 for 4(5)
embedded RK), and S is a safety factor of 0.9. The ratio E/D is
taken to be the maximum of the ratios E_i/D_i.
If the observed error E is less than 50% of the desired error level
D for the maximum ratio E_i/D_i then the algorithm takes the
opportunity to increase the step-size to bring the error in line
with the desired level,
h_new = h_old * S * (E/D)^(-1/(q+1))
This encompasses all the standard error scaling methods. To avoid
uncontrolled changes in the stepsize, the overall scaling factor is
limited to the range 1/5 to 5.
-- Function: gsl_odeiv2_control * gsl_odeiv2_control_y_new (double
EPS_ABS, double EPS_REL)
This function creates a new control object which will keep the
local error on each step within an absolute error of EPS_ABS and
relative error of EPS_REL with respect to the solution y_i(t).
This is equivalent to the standard control object with A_Y=1 and
A_DYDT=0.
-- Function: gsl_odeiv2_control * gsl_odeiv2_control_yp_new (double
EPS_ABS, double EPS_REL)
This function creates a new control object which will keep the
local error on each step within an absolute error of EPS_ABS and
relative error of EPS_REL with respect to the derivatives of the
solution y'_i(t). This is equivalent to the standard control
object with A_Y=0 and A_DYDT=1.
-- Function: gsl_odeiv2_control * gsl_odeiv2_control_scaled_new (double
EPS_ABS, double EPS_REL, double A_Y, double A_DYDT, const
double SCALE_ABS[], size_t DIM)
This function creates a new control object which uses the same
algorithm as 'gsl_odeiv2_control_standard_new' but with an absolute
error which is scaled for each component by the array SCALE_ABS.
The formula for D_i for this control object is,
D_i = eps_abs * s_i + eps_rel * (a_y |y_i| + a_dydt h |y\prime_i|)
where s_i is the i-th component of the array SCALE_ABS. The same
error control heuristic is used by the Matlab ODE suite.
-- Function: gsl_odeiv2_control * gsl_odeiv2_control_alloc (const
gsl_odeiv2_control_type * T)
This function returns a pointer to a newly allocated instance of a
control function of type T. This function is only needed for
defining new types of control functions. For most purposes the
standard control functions described above should be sufficient.
-- Function: int gsl_odeiv2_control_init (gsl_odeiv2_control * C,
double EPS_ABS, double EPS_REL, double A_Y, double A_DYDT)
This function initializes the control function C with the
parameters EPS_ABS (absolute error), EPS_REL (relative error), A_Y
(scaling factor for y) and A_DYDT (scaling factor for derivatives).
-- Function: void gsl_odeiv2_control_free (gsl_odeiv2_control * C)
This function frees all the memory associated with the control
function C.
-- Function: int gsl_odeiv2_control_hadjust (gsl_odeiv2_control * C,
gsl_odeiv2_step * S, const double Y[], const double YERR[],
const double DYDT[], double * H)
This function adjusts the step-size H using the control function C,
and the current values of Y, YERR and DYDT. The stepping function
STEP is also needed to determine the order of the method. If the
error in the y-values YERR is found to be too large then the
step-size H is reduced and the function returns
'GSL_ODEIV_HADJ_DEC'. If the error is sufficiently small then H
may be increased and 'GSL_ODEIV_HADJ_INC' is returned. The
function returns 'GSL_ODEIV_HADJ_NIL' if the step-size is
unchanged. The goal of the function is to estimate the largest
step-size which satisfies the user-specified accuracy requirements
for the current point.
-- Function: const char * gsl_odeiv2_control_name (const
gsl_odeiv2_control * C)
This function returns a pointer to the name of the control
function. For example,
printf ("control method is '%s'\n",
gsl_odeiv2_control_name (c));
would print something like 'control method is 'standard''
-- Function: int gsl_odeiv2_control_errlevel (gsl_odeiv2_control * C,
const double Y, const double DYDT, const double H, const
size_t IND, double * ERRLEV)
This function calculates the desired error level of the IND-th
component to ERRLEV. It requires the value (Y) and value of the
derivative (DYDT) of the component, and the current step size H.
-- Function: int gsl_odeiv2_control_set_driver (gsl_odeiv2_control * C,
const gsl_odeiv2_driver * D)
This function sets a pointer of the driver object D for control
object C.
27.4 Evolution
==============
The evolution function combines the results of a stepping function and
control function to reliably advance the solution forward one step using
an acceptable step-size.
-- Function: gsl_odeiv2_evolve * gsl_odeiv2_evolve_alloc (size_t DIM)
This function returns a pointer to a newly allocated instance of an
evolution function for a system of DIM dimensions.
-- Function: int gsl_odeiv2_evolve_apply (gsl_odeiv2_evolve * E,
gsl_odeiv2_control * CON, gsl_odeiv2_step * STEP, const
gsl_odeiv2_system * SYS, double * T, double T1, double * H,
double Y[])
This function advances the system (E, SYS) from time T and position
Y using the stepping function STEP. The new time and position are
stored in T and Y on output.
The initial step-size is taken as H. The control function CON is
applied to check whether the local error estimated by the stepping
function STEP using step-size H exceeds the required error
tolerance. If the error is too high, the step is retried by
calling STEP with a decreased step-size. This process is continued
until an acceptable step-size is found. An estimate of the local
error for the step can be obtained from the components of the array
'E->yerr[]'.
If the user-supplied functions defined in the system SYS returns
'GSL_EBADFUNC', the function returns immediately with the same
return code. In this case the user must call
'gsl_odeiv2_step_reset' and 'gsl_odeiv2_evolve_reset' before
calling this function again.
Otherwise, if the user-supplied functions defined in the system SYS
or the stepping function STEP return a status other than
'GSL_SUCCESS', the step is retried with a decreased step-size. If
the step-size decreases below machine precision, a status of
'GSL_FAILURE' is returned if the user functions returned
'GSL_SUCCESS'. Otherwise the value returned by user function is
returned. If no acceptable step can be made, T and Y will be
restored to their pre-step values and H contains the final
attempted step-size.
If the step is successful the function returns a suggested
step-size for the next step in H. The maximum time T1 is
guaranteed not to be exceeded by the time-step. On the final
time-step the value of T will be set to T1 exactly.
-- Function: int gsl_odeiv2_evolve_apply_fixed_step (gsl_odeiv2_evolve
* E, gsl_odeiv2_control * CON, gsl_odeiv2_step * STEP, const
gsl_odeiv2_system * SYS, double * T, const double H, double
Y[])
This function advances the ODE-system (E, SYS, CON) from time T and
position Y using the stepping function STEP by a specified step
size H. If the local error estimated by the stepping function
exceeds the desired error level, the step is not taken and the
function returns 'GSL_FAILURE'. Otherwise the value returned by
user function is returned.
-- Function: int gsl_odeiv2_evolve_reset (gsl_odeiv2_evolve * E)
This function resets the evolution function E. It should be used
whenever the next use of E will not be a continuation of a previous
step.
-- Function: void gsl_odeiv2_evolve_free (gsl_odeiv2_evolve * E)
This function frees all the memory associated with the evolution
function E.
-- Function: int gsl_odeiv2_evolve_set_driver (gsl_odeiv2_evolve * E,
const gsl_odeiv2_driver * D)
This function sets a pointer of the driver object D for evolve
object E.
If a system has discontinuous changes in the derivatives at known
points, it is advisable to evolve the system between each discontinuity
in sequence. For example, if a step-change in an external driving force
occurs at times t_a, t_b and t_c then evolution should be carried out
over the ranges (t_0,t_a), (t_a,t_b), (t_b,t_c), and (t_c,t_1)
separately and not directly over the range (t_0,t_1).
27.5 Driver
===========
The driver object is a high level wrapper that combines the evolution,
control and stepper objects for easy use.
-- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_y_new (const
gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type * T, const
double HSTART, const double EPSABS, const double EPSREL)
-- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_yp_new (const
gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type * T, const
double HSTART, const double EPSABS, const double EPSREL)
-- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_standard_new
(const gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type *
T, const double HSTART, const double EPSABS, const double
EPSREL, const double A_Y, const double A_DYDT)
-- Function: gsl_odeiv2_driver * gsl_odeiv2_driver_alloc_scaled_new
(const gsl_odeiv2_system * SYS, const gsl_odeiv2_step_type *
T, const double HSTART, const double EPSABS, const double
EPSREL, const double A_Y, const double A_DYDT, const double
SCALE_ABS[])
These functions return a pointer to a newly allocated instance of a
driver object. The functions automatically allocate and initialise
the evolve, control and stepper objects for ODE system SYS using
stepper type T. The initial step size is given in HSTART. The
rest of the arguments follow the syntax and semantics of the
control functions with same name ('gsl_odeiv2_control_*_new').
-- Function: int gsl_odeiv2_driver_set_hmin (gsl_odeiv2_driver * D,
const double HMIN)
The function sets a minimum for allowed step size HMIN for driver
D. Default value is 0.
-- Function: int gsl_odeiv2_driver_set_hmax (gsl_odeiv2_driver * D,
const double HMAX)
The function sets a maximum for allowed step size HMAX for driver
D. Default value is 'GSL_DBL_MAX'.
-- Function: int gsl_odeiv2_driver_set_nmax (gsl_odeiv2_driver * D,
const unsigned long int NMAX)
The function sets a maximum for allowed number of steps NMAX for
driver D. Default value of 0 sets no limit for steps.
-- Function: int gsl_odeiv2_driver_apply (gsl_odeiv2_driver * D, double
* T, const double T1, double Y[])
This function evolves the driver system D from T to T1. Initially
vector Y should contain the values of dependent variables at point
T. If the function is unable to complete the calculation, an error
code from 'gsl_odeiv2_evolve_apply' is returned, and T and Y
contain the values from last successful step.
If maximum number of steps is reached, a value of 'GSL_EMAXITER' is
returned. If the step size drops below minimum value, the function
returns with 'GSL_ENOPROG'. If the user-supplied functions defined
in the system SYS returns 'GSL_EBADFUNC', the function returns
immediately with the same return code. In this case the user must
call 'gsl_odeiv2_driver_reset' before calling this function again.
-- Function: int gsl_odeiv2_driver_apply_fixed_step (gsl_odeiv2_driver
* D, double * T, const double H, const unsigned long int N,
double Y[])
This function evolves the driver system D from T with N steps of
size H. If the function is unable to complete the calculation, an
error code from 'gsl_odeiv2_evolve_apply_fixed_step' is returned,
and T and Y contain the values from last successful step.
-- Function: int gsl_odeiv2_driver_reset (gsl_odeiv2_driver * D)
This function resets the evolution and stepper objects.
-- Function: int gsl_odeiv2_driver_reset_hstart (gsl_odeiv2_driver * D,
const double HSTART)
The routine resets the evolution and stepper objects and sets new
initial step size to HSTART. This function can be used e.g. to
change the direction of integration.
-- Function: int gsl_odeiv2_driver_free (gsl_odeiv2_driver * D)
This function frees the driver object, and the related evolution,
stepper and control objects.
27.6 Examples
=============
The following program solves the second-order nonlinear Van der Pol
oscillator equation,
u''(t) + \mu u'(t) (u(t)^2 - 1) + u(t) = 0
This can be converted into a first order system suitable for use with
the routines described in this chapter by introducing a separate
variable for the velocity, v = u'(t),
u' = v
v' = -u + \mu v (1-u^2)
The program begins by defining functions for these derivatives and their
Jacobian. The main function uses driver level functions to solve the
problem. The program evolves the solution from (u, v) = (1, 0) at t=0
to t=100. The step-size h is automatically adjusted by the controller
to maintain an absolute accuracy of 10^{-6} in the function values (u,
v). The loop in the example prints the solution at the points t_i = 1,
2, \dots, 100.
#include
#include
#include
#include
int
func (double t, const double y[], double f[],
void *params)
{
(void)(t); /* avoid unused parameter warning */
double mu = *(double *)params;
f[0] = y[1];
f[1] = -y[0] - mu*y[1]*(y[0]*y[0] - 1);
return GSL_SUCCESS;
}
int
jac (double t, const double y[], double *dfdy,
double dfdt[], void *params)
{
(void)(t); /* avoid unused parameter warning */
double mu = *(double *)params;
gsl_matrix_view dfdy_mat
= gsl_matrix_view_array (dfdy, 2, 2);
gsl_matrix * m = &dfdy_mat.matrix;
gsl_matrix_set (m, 0, 0, 0.0);
gsl_matrix_set (m, 0, 1, 1.0);
gsl_matrix_set (m, 1, 0, -2.0*mu*y[0]*y[1] - 1.0);
gsl_matrix_set (m, 1, 1, -mu*(y[0]*y[0] - 1.0));
dfdt[0] = 0.0;
dfdt[1] = 0.0;
return GSL_SUCCESS;
}
int
main (void)
{
double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};
gsl_odeiv2_driver * d =
gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk8pd,
1e-6, 1e-6, 0.0);
int i;
double t = 0.0, t1 = 100.0;
double y[2] = { 1.0, 0.0 };
for (i = 1; i <= 100; i++)
{
double ti = i * t1 / 100.0;
int status = gsl_odeiv2_driver_apply (d, &t, ti, y);
if (status != GSL_SUCCESS)
{
printf ("error, return value=%d\n", status);
break;
}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv2_driver_free (d);
return 0;
}
The user can work with the lower level functions directly, as in the
following example. In this case an intermediate result is printed after
each successful step instead of equidistant time points.
int
main (void)
{
const gsl_odeiv2_step_type * T
= gsl_odeiv2_step_rk8pd;
gsl_odeiv2_step * s
= gsl_odeiv2_step_alloc (T, 2);
gsl_odeiv2_control * c
= gsl_odeiv2_control_y_new (1e-6, 0.0);
gsl_odeiv2_evolve * e
= gsl_odeiv2_evolve_alloc (2);
double mu = 10;
gsl_odeiv2_system sys = {func, jac, 2, &mu};
double t = 0.0, t1 = 100.0;
double h = 1e-6;
double y[2] = { 1.0, 0.0 };
while (t < t1)
{
int status = gsl_odeiv2_evolve_apply (e, c, s,
&sys,
&t, t1,
&h, y);
if (status != GSL_SUCCESS)
break;
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv2_evolve_free (e);
gsl_odeiv2_control_free (c);
gsl_odeiv2_step_free (s);
return 0;
}
For functions with multiple parameters, the appropriate information can
be passed in through the PARAMS argument in 'gsl_odeiv2_system'
definition (MU in this example) by using a pointer to a struct.
It is also possible to work with a non-adaptive integrator, using only
the stepping function itself, 'gsl_odeiv2_driver_apply_fixed_step' or
'gsl_odeiv2_evolve_apply_fixed_step'. The following program uses the
driver level function, with fourth-order Runge-Kutta stepping function
with a fixed stepsize of 0.001.
int
main (void)
{
double mu = 10;
gsl_odeiv2_system sys = { func, jac, 2, &mu };
gsl_odeiv2_driver *d =
gsl_odeiv2_driver_alloc_y_new (&sys, gsl_odeiv2_step_rk4,
1e-3, 1e-8, 1e-8);
double t = 0.0;
double y[2] = { 1.0, 0.0 };
int i, s;
for (i = 0; i < 100; i++)
{
s = gsl_odeiv2_driver_apply_fixed_step (d, &t, 1e-3, 1000, y);
if (s != GSL_SUCCESS)
{
printf ("error: driver returned %d\n", s);
break;
}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv2_driver_free (d);
return s;
}
27.7 References and Further Reading
===================================
Ascher, U.M., Petzold, L.R., 'Computer Methods for Ordinary
Differential and Differential-Algebraic Equations', SIAM,
Philadelphia, 1998.
Hairer, E., Norsett, S. P., Wanner, G., 'Solving Ordinary
Differential Equations I: Nonstiff Problems', Springer, Berlin,
1993.
Hairer, E., Wanner, G., 'Solving Ordinary Differential Equations
II: Stiff and Differential-Algebraic Problems', Springer, Berlin,
1996.
Many of the basic Runge-Kutta formulas can be found in the Handbook
of Mathematical Functions,
Abramowitz & Stegun (eds.), 'Handbook of Mathematical Functions',
Section 25.5.
The implicit Bulirsch-Stoer algorithm 'bsimp' is described in the
following paper,
G. Bader and P. Deuflhard, "A Semi-Implicit Mid-Point Rule for
Stiff Systems of Ordinary Differential Equations.", Numer. Math.
41, 373-398, 1983.
The Adams and BDF multistep methods 'msadams' and 'msbdf' are based on
the following articles,
G. D. Byrne and A. C. Hindmarsh, "A Polyalgorithm for the Numerical
Solution of Ordinary Differential Equations.", ACM Trans. Math.
Software, 1, 71-96, 1975.
P. N. Brown, G. D. Byrne and A. C. Hindmarsh, "VODE: A
Variable-coefficient ODE Solver.", SIAM J. Sci. Stat. Comput.
10, 1038-1051, 1989.
A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D.
E. Shumaker and C. S. Woodward, "SUNDIALS: Suite of Nonlinear and
Differential/Algebraic Equation Solvers.", ACM Trans. Math.
Software 31, 363-396, 2005.
28 Interpolation
****************
This chapter describes functions for performing interpolation. The
library provides a variety of interpolation methods, including Cubic,
Akima, and Steffen splines. The interpolation types are
interchangeable, allowing different methods to be used without
recompiling. Interpolations can be defined for both normal and periodic
boundary conditions. Additional functions are available for computing
derivatives and integrals of interpolating functions. Routines are
provided for interpolating both one and two dimensional datasets.
These interpolation methods produce curves that pass through each
datapoint. To interpolate noisy data with a smoothing curve see *note
Basis Splines::.
The functions described in this section are declared in the header
files 'gsl_interp.h' and 'gsl_spline.h'.
28.1 Introduction to 1D Interpolation
=====================================
Given a set of data points (x_1, y_1) \dots (x_n, y_n) the routines
described in this section compute a continuous interpolating function
y(x) such that y(x_i) = y_i. The interpolation is piecewise smooth, and
its behavior at the end-points is determined by the type of
interpolation used.
28.2 1D Interpolation Functions
===============================
The interpolation function for a given dataset is stored in a
'gsl_interp' object. These are created by the following functions.
-- Function: gsl_interp * gsl_interp_alloc (const gsl_interp_type * T,
size_t SIZE)
This function returns a pointer to a newly allocated interpolation
object of type T for SIZE data-points.
-- Function: int gsl_interp_init (gsl_interp * INTERP, const double
XA[], const double YA[], size_t SIZE)
This function initializes the interpolation object INTERP for the
data (XA,YA) where XA and YA are arrays of size SIZE. The
interpolation object ('gsl_interp') does not save the data arrays
XA and YA and only stores the static state computed from the data.
The XA data array is always assumed to be strictly ordered, with
increasing x values; the behavior for other arrangements is not
defined.
-- Function: void gsl_interp_free (gsl_interp * INTERP)
This function frees the interpolation object INTERP.
28.3 1D Interpolation Types
===========================
The interpolation library provides the following interpolation types:
-- Interpolation Type: gsl_interp_linear
Linear interpolation. This interpolation method does not require
any additional memory.
-- Interpolation Type: gsl_interp_polynomial
Polynomial interpolation. This method should only be used for
interpolating small numbers of points because polynomial
interpolation introduces large oscillations, even for well-behaved
datasets. The number of terms in the interpolating polynomial is
equal to the number of points.
-- Interpolation Type: gsl_interp_cspline
Cubic spline with natural boundary conditions. The resulting curve
is piecewise cubic on each interval, with matching first and second
derivatives at the supplied data-points. The second derivative is
chosen to be zero at the first point and last point.
-- Interpolation Type: gsl_interp_cspline_periodic
Cubic spline with periodic boundary conditions. The resulting
curve is piecewise cubic on each interval, with matching first and
second derivatives at the supplied data-points. The derivatives at
the first and last points are also matched. Note that the last
point in the data must have the same y-value as the first point,
otherwise the resulting periodic interpolation will have a
discontinuity at the boundary.
-- Interpolation Type: gsl_interp_akima
Non-rounded Akima spline with natural boundary conditions. This
method uses the non-rounded corner algorithm of Wodicka.
-- Interpolation Type: gsl_interp_akima_periodic
Non-rounded Akima spline with periodic boundary conditions. This
method uses the non-rounded corner algorithm of Wodicka.
-- Interpolation Type: gsl_interp_steffen
Steffen's method guarantees the monotonicity of the interpolating
function between the given data points. Therefore, minima and
maxima can only occur exactly at the data points, and there can
never be spurious oscillations between data points. The
interpolated function is piecewise cubic in each interval. The
resulting curve and its first derivative are guaranteed to be
continuous, but the second derivative may be discontinuous.
The following related functions are available:
-- Function: const char * gsl_interp_name (const gsl_interp * INTERP)
This function returns the name of the interpolation type used by
INTERP. For example,
printf ("interp uses '%s' interpolation.\n",
gsl_interp_name (interp));
would print something like,
interp uses 'cspline' interpolation.
-- Function: unsigned int gsl_interp_min_size (const gsl_interp *
INTERP)
-- Function: unsigned int gsl_interp_type_min_size (const
gsl_interp_type * T)
These functions return the minimum number of points required by the
interpolation object INTERP or interpolation type T. For example,
Akima spline interpolation requires a minimum of 5 points.
28.4 1D Index Look-up and Acceleration
======================================
The state of searches can be stored in a 'gsl_interp_accel' object,
which is a kind of iterator for interpolation lookups. It caches the
previous value of an index lookup. When the subsequent interpolation
point falls in the same interval its index value can be returned
immediately.
-- Function: size_t gsl_interp_bsearch (const double X_ARRAY[], double
X, size_t INDEX_LO, size_t INDEX_HI)
This function returns the index i of the array X_ARRAY such that
'x_array[i] <= x < x_array[i+1]'. The index is searched for in the
range [INDEX_LO,INDEX_HI]. An inline version of this function is
used when 'HAVE_INLINE' is defined.
-- Function: gsl_interp_accel * gsl_interp_accel_alloc (void)
This function returns a pointer to an accelerator object, which is
a kind of iterator for interpolation lookups. It tracks the state
of lookups, thus allowing for application of various acceleration
strategies.
-- Function: size_t gsl_interp_accel_find (gsl_interp_accel * A, const
double X_ARRAY[], size_t SIZE, double X)
This function performs a lookup action on the data array X_ARRAY of
size SIZE, using the given accelerator A. This is how lookups are
performed during evaluation of an interpolation. The function
returns an index i such that 'x_array[i] <= x < x_array[i+1]'. An
inline version of this function is used when 'HAVE_INLINE' is
defined.
-- Function: int gsl_interp_accel_reset (gsl_interp_accel * ACC);
This function reinitializes the accelerator object ACC. It should
be used when the cached information is no longer applicable--for
example, when switching to a new dataset.
-- Function: void gsl_interp_accel_free (gsl_interp_accel* ACC)
This function frees the accelerator object ACC.
28.5 1D Evaluation of Interpolating Functions
=============================================
-- Function: double gsl_interp_eval (const gsl_interp * INTERP, const
double XA[], const double YA[], double X, gsl_interp_accel *
ACC)
-- Function: int gsl_interp_eval_e (const gsl_interp * INTERP, const
double XA[], const double YA[], double X, gsl_interp_accel *
ACC, double * Y)
These functions return the interpolated value of Y for a given
point X, using the interpolation object INTERP, data arrays XA and
YA and the accelerator ACC. When X is outside the range of XA, the
error code 'GSL_EDOM' is returned with a value of 'GSL_NAN' for Y.
-- Function: double gsl_interp_eval_deriv (const gsl_interp * INTERP,
const double XA[], const double YA[], double X,
gsl_interp_accel * ACC)
-- Function: int gsl_interp_eval_deriv_e (const gsl_interp * INTERP,
const double XA[], const double YA[], double X,
gsl_interp_accel * ACC, double * D)
These functions return the derivative D of an interpolated function
for a given point X, using the interpolation object INTERP, data
arrays XA and YA and the accelerator ACC.
-- Function: double gsl_interp_eval_deriv2 (const gsl_interp * INTERP,
const double XA[], const double YA[], double X,
gsl_interp_accel * ACC)
-- Function: int gsl_interp_eval_deriv2_e (const gsl_interp * INTERP,
const double XA[], const double YA[], double X,
gsl_interp_accel * ACC, double * D2)
These functions return the second derivative D2 of an interpolated
function for a given point X, using the interpolation object
INTERP, data arrays XA and YA and the accelerator ACC.
-- Function: double gsl_interp_eval_integ (const gsl_interp * INTERP,
const double XA[], const double YA[], double A, double B,
gsl_interp_accel * ACC)
-- Function: int gsl_interp_eval_integ_e (const gsl_interp * INTERP,
const double XA[], const double YA[], double A, double B,
gsl_interp_accel * ACC, double * RESULT)
These functions return the numerical integral RESULT of an
interpolated function over the range [A, B], using the
interpolation object INTERP, data arrays XA and YA and the
accelerator ACC.
28.6 1D Higher-level Interface
==============================
The functions described in the previous sections required the user to
supply pointers to the x and y arrays on each call. The following
functions are equivalent to the corresponding 'gsl_interp' functions but
maintain a copy of this data in the 'gsl_spline' object. This removes
the need to pass both XA and YA as arguments on each evaluation. These
functions are defined in the header file 'gsl_spline.h'.
-- Function: gsl_spline * gsl_spline_alloc (const gsl_interp_type * T,
size_t SIZE)
-- Function: int gsl_spline_init (gsl_spline * SPLINE, const double
XA[], const double YA[], size_t SIZE)
-- Function: void gsl_spline_free (gsl_spline * SPLINE)
-- Function: const char * gsl_spline_name (const gsl_spline * SPLINE)
-- Function: unsigned int gsl_spline_min_size (const gsl_spline *
SPLINE)
-- Function: double gsl_spline_eval (const gsl_spline * SPLINE, double
X, gsl_interp_accel * ACC)
-- Function: int gsl_spline_eval_e (const gsl_spline * SPLINE, double
X, gsl_interp_accel * ACC, double * Y)
-- Function: double gsl_spline_eval_deriv (const gsl_spline * SPLINE,
double X, gsl_interp_accel * ACC)
-- Function: int gsl_spline_eval_deriv_e (const gsl_spline * SPLINE,
double X, gsl_interp_accel * ACC, double * D)
-- Function: double gsl_spline_eval_deriv2 (const gsl_spline * SPLINE,
double X, gsl_interp_accel * ACC)
-- Function: int gsl_spline_eval_deriv2_e (const gsl_spline * SPLINE,
double X, gsl_interp_accel * ACC, double * D2)
-- Function: double gsl_spline_eval_integ (const gsl_spline * SPLINE,
double A, double B, gsl_interp_accel * ACC)
-- Function: int gsl_spline_eval_integ_e (const gsl_spline * SPLINE,
double A, double B, gsl_interp_accel * ACC, double * RESULT)
28.7 Examples of 1D Interpolation
=================================
The following program demonstrates the use of the interpolation and
spline functions. It computes a cubic spline interpolation of the
10-point dataset (x_i, y_i) where x_i = i + \sin(i)/2 and y_i = i +
\cos(i^2) for i = 0 \dots 9.
#include
#include
#include
#include
#include
int
main (void)
{
int i;
double xi, yi, x[10], y[10];
printf ("#m=0,S=2\n");
for (i = 0; i < 10; i++)
{
x[i] = i + 0.5 * sin (i);
y[i] = i + cos (i * i);
printf ("%g %g\n", x[i], y[i]);
}
printf ("#m=1,S=0\n");
{
gsl_interp_accel *acc
= gsl_interp_accel_alloc ();
gsl_spline *spline
= gsl_spline_alloc (gsl_interp_cspline, 10);
gsl_spline_init (spline, x, y, 10);
for (xi = x[0]; xi < x[9]; xi += 0.01)
{
yi = gsl_spline_eval (spline, xi, acc);
printf ("%g %g\n", xi, yi);
}
gsl_spline_free (spline);
gsl_interp_accel_free (acc);
}
return 0;
}
The output is designed to be used with the GNU plotutils 'graph'
program,
$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps
The result shows a smooth interpolation of the original points. The
interpolation method can be changed simply by varying the first argument
of 'gsl_spline_alloc'.
The next program demonstrates a periodic cubic spline with 4 data
points. Note that the first and last points must be supplied with the
same y-value for a periodic spline.
#include
#include
#include
#include
#include
int
main (void)
{
int N = 4;
double x[4] = {0.00, 0.10, 0.27, 0.30};
double y[4] = {0.15, 0.70, -0.10, 0.15};
/* Note: y[0] == y[3] for periodic data */
gsl_interp_accel *acc = gsl_interp_accel_alloc ();
const gsl_interp_type *t = gsl_interp_cspline_periodic;
gsl_spline *spline = gsl_spline_alloc (t, N);
int i; double xi, yi;
printf ("#m=0,S=5\n");
for (i = 0; i < N; i++)
{
printf ("%g %g\n", x[i], y[i]);
}
printf ("#m=1,S=0\n");
gsl_spline_init (spline, x, y, N);
for (i = 0; i <= 100; i++)
{
xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
yi = gsl_spline_eval (spline, xi, acc);
printf ("%g %g\n", xi, yi);
}
gsl_spline_free (spline);
gsl_interp_accel_free (acc);
return 0;
}
The output can be plotted with GNU 'graph'.
$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps
The result shows a periodic interpolation of the original points. The
slope of the fitted curve is the same at the beginning and end of the
data, and the second derivative is also.
The next program illustrates the difference between the cubic spline,
Akima, and Steffen interpolation types on a difficult dataset.
#include
#include
#include
#include
#include
int
main(void)
{
size_t i;
const size_t N = 9;
/* this dataset is taken from
* J. M. Hyman, Accurate Monotonicity preserving cubic interpolation,
* SIAM J. Sci. Stat. Comput. 4, 4, 1983. */
const double x[] = { 7.99, 8.09, 8.19, 8.7, 9.2,
10.0, 12.0, 15.0, 20.0 };
const double y[] = { 0.0, 2.76429e-5, 4.37498e-2,
0.169183, 0.469428, 0.943740,
0.998636, 0.999919, 0.999994 };
gsl_interp_accel *acc = gsl_interp_accel_alloc();
gsl_spline *spline_cubic = gsl_spline_alloc(gsl_interp_cspline, N);
gsl_spline *spline_akima = gsl_spline_alloc(gsl_interp_akima, N);
gsl_spline *spline_steffen = gsl_spline_alloc(gsl_interp_steffen, N);
gsl_spline_init(spline_cubic, x, y, N);
gsl_spline_init(spline_akima, x, y, N);
gsl_spline_init(spline_steffen, x, y, N);
for (i = 0; i < N; ++i)
printf("%g %g\n", x[i], y[i]);
printf("\n\n");
for (i = 0; i <= 100; ++i)
{
double xi = (1 - i / 100.0) * x[0] + (i / 100.0) * x[N-1];
double yi_cubic = gsl_spline_eval(spline_cubic, xi, acc);
double yi_akima = gsl_spline_eval(spline_akima, xi, acc);
double yi_steffen = gsl_spline_eval(spline_steffen, xi, acc);
printf("%g %g %g %g\n", xi, yi_cubic, yi_akima, yi_steffen);
}
gsl_spline_free(spline_cubic);
gsl_spline_free(spline_akima);
gsl_spline_free(spline_steffen);
gsl_interp_accel_free(acc);
return 0;
}
The cubic method exhibits a local maxima between the 6th and 7th data
points and continues oscillating for the rest of the data. Akima also
shows a local maxima but recovers and follows the data well after the
7th grid point. Steffen preserves monotonicity in all intervals and
does not exhibit oscillations, at the expense of having a discontinuous
second derivative.
28.8 References and Further Reading
===================================
Descriptions of the interpolation algorithms and further references can
be found in the following publications:
C.W. Ueberhuber, 'Numerical Computation (Volume 1), Chapter 9
"Interpolation"', Springer (1997), ISBN 3-540-62058-3.
D.M. Young, R.T. Gregory, 'A Survey of Numerical Mathematics
(Volume 1), Chapter 6.8', Dover (1988), ISBN 0-486-65691-8.
M. Steffen, 'A simple method for monotonic interpolation in one
dimension', Astron. Astrophys. 239, 443-450, 1990.
28.9 Introduction to 2D Interpolation
=====================================
Given a set of x coordinates x_1,...,x_m and a set of y coordinates
y_1,...,y_n, each in increasing order, plus a set of function values
z_{ij} for each grid point (x_i,y_j), the routines described in this
section compute a continuous interpolation function z(x,y) such that
z(x_i,y_j) = z_{ij}.
28.10 2D Interpolation Functions
================================
The interpolation function for a given dataset is stored in a
'gsl_interp2d' object. These are created by the following functions.
-- Function: gsl_interp2d * gsl_interp2d_alloc (const gsl_interp2d_type
* T, const size_t XSIZE, const size_t YSIZE)
This function returns a pointer to a newly allocated interpolation
object of type T for XSIZE grid points in the x direction and YSIZE
grid points in the y direction.
-- Function: int gsl_interp2d_init (gsl_interp2d * INTERP, const double
XA[], const double YA[], const double ZA[], const size_t
XSIZE, const size_t YSIZE)
This function initializes the interpolation object INTERP for the
data (XA,YA,ZA) where XA and YA are arrays of the x and y grid
points of size XSIZE and YSIZE respectively, and ZA is an array of
function values of size XSIZE*YSIZE. The interpolation object
('gsl_interp2d') does not save the data arrays XA, YA, and ZA and
only stores the static state computed from the data. The XA and YA
data arrays are always assumed to be strictly ordered, with
increasing x,y values; the behavior for other arrangements is not
defined.
-- Function: void gsl_interp2d_free (gsl_interp2d * INTERP)
This function frees the interpolation object INTERP.
28.11 2D Interpolation Grids
============================
The 2D interpolation routines access the function values z_{ij} with the
following ordering:
z_ij = za[j*xsize + i]
with i = 0,...,xsize-1 and j = 0,...,ysize-1. However, for ease of
use, the following functions are provided to add and retrieve elements
from the function grid without requiring knowledge of the internal
ordering.
-- Function: int gsl_interp2d_set (const gsl_interp2d * INTERP, double
ZA[], const size_t I, const size_t J, const double Z)
This function sets the value z_{ij} for grid point (I,J) of the
array ZA to Z.
-- Function: double gsl_interp2d_get (const gsl_interp2d * INTERP,
const double ZA[], const size_t I, const size_t J)
This function returns the value z_{ij} for grid point (I,J) stored
in the array ZA.
-- Function: size_t gsl_interp2d_idx (const gsl_interp2d * INTERP,
const size_t I, const size_t J)
This function returns the index corresponding to the grid point
(I,J). The index is given by j*xsize + i.
28.12 2D Interpolation Types
============================
The interpolation library provides the following 2D interpolation types:
-- Interpolation Type: gsl_interp2d_bilinear
Bilinear interpolation. This interpolation method does not require
any additional memory.
-- Interpolation Type: gsl_interp2d_bicubic
Bicubic interpolation.
-- Function: const char * gsl_interp2d_name (const gsl_interp2d *
INTERP)
This function returns the name of the interpolation type used by
INTERP. For example,
printf ("interp uses '%s' interpolation.\n",
gsl_interp2d_name (interp));
would print something like,
interp uses 'bilinear' interpolation.
-- Function: unsigned int gsl_interp2d_min_size (const gsl_interp2d *
INTERP)
-- Function: unsigned int gsl_interp2d_type_min_size (const
gsl_interp2d_type * T)
These functions return the minimum number of points required by the
interpolation object INTERP or interpolation type T. For example,
bicubic interpolation requires a minimum of 4 points.
28.13 2D Evaluation of Interpolating Functions
==============================================
-- Function: double gsl_interp2d_eval (const gsl_interp2d * INTERP,
const double XA[], const double YA[], const double ZA[], const
double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_e (const gsl_interp2d * INTERP,
const double XA[], const double YA[], const double ZA[], const
double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * Z)
These functions return the interpolated value of Z for a given
point (X,Y), using the interpolation object INTERP, data arrays XA,
YA, and ZA and the accelerators XACC and YACC. When X is outside
the range of XA or Y is outside the range of YA, the error code
'GSL_EDOM' is returned.
-- Function: double gsl_interp2d_eval_extrap (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_extrap_e (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * Z)
These functions return the interpolated value of Z for a given
point (X,Y), using the interpolation object INTERP, data arrays XA,
YA, and ZA and the accelerators XACC and YACC. The functions
perform no bounds checking, so when X is outside the range of XA or
Y is outside the range of YA, extrapolation is performed.
-- Function: double gsl_interp2d_eval_deriv_x (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_deriv_x_e (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * D)
These functions return the interpolated value D = \partial z /
\partial x for a given point (X,Y), using the interpolation object
INTERP, data arrays XA, YA, and ZA and the accelerators XACC and
YACC. When X is outside the range of XA or Y is outside the range
of YA, the error code 'GSL_EDOM' is returned.
-- Function: double gsl_interp2d_eval_deriv_y (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_deriv_y_e (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * D)
These functions return the interpolated value D = \partial z /
\partial y for a given point (X,Y), using the interpolation object
INTERP, data arrays XA, YA, and ZA and the accelerators XACC and
YACC. When X is outside the range of XA or Y is outside the range
of YA, the error code 'GSL_EDOM' is returned.
-- Function: double gsl_interp2d_eval_deriv_xx (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_deriv_xx_e (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * D)
These functions return the interpolated value D = \partial^2 z /
\partial x^2 for a given point (X,Y), using the interpolation
object INTERP, data arrays XA, YA, and ZA and the accelerators XACC
and YACC. When X is outside the range of XA or Y is outside the
range of YA, the error code 'GSL_EDOM' is returned.
-- Function: double gsl_interp2d_eval_deriv_yy (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_deriv_yy_e (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * D)
These functions return the interpolated value D = \partial^2 z /
\partial y^2 for a given point (X,Y), using the interpolation
object INTERP, data arrays XA, YA, and ZA and the accelerators XACC
and YACC. When X is outside the range of XA or Y is outside the
range of YA, the error code 'GSL_EDOM' is returned.
-- Function: double gsl_interp2d_eval_deriv_xy (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_interp2d_eval_deriv_xy_e (const gsl_interp2d *
INTERP, const double XA[], const double YA[], const double
ZA[], const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * D)
These functions return the interpolated value D = \partial^2 z /
\partial x \partial y for a given point (X,Y), using the
interpolation object INTERP, data arrays XA, YA, and ZA and the
accelerators XACC and YACC. When X is outside the range of XA or Y
is outside the range of YA, the error code 'GSL_EDOM' is returned.
28.14 2D Higher-level Interface
===============================
The functions described in the previous sections required the user to
supply pointers to the x, y, and z arrays on each call. The following
functions are equivalent to the corresponding 'gsl_interp2d' functions
but maintain a copy of this data in the 'gsl_spline2d' object. This
removes the need to pass XA, YA, and ZA as arguments on each evaluation.
These functions are defined in the header file 'gsl_spline2d.h'.
-- Function: gsl_spline2d * gsl_spline2d_alloc (const gsl_interp2d_type
* T, size_t XSIZE, size_t YSIZE)
-- Function: int gsl_spline2d_init (gsl_spline2d * SPLINE, const double
XA[], const double YA[], const double ZA[], size_t XSIZE,
size_t YSIZE)
-- Function: void gsl_spline2d_free (gsl_spline2d * SPLINE)
-- Function: const char * gsl_spline2d_name (const gsl_spline2d *
SPLINE)
-- Function: unsigned int gsl_spline2d_min_size (const gsl_spline2d *
SPLINE)
-- Function: double gsl_spline2d_eval (const gsl_spline2d * SPLINE,
const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC)
-- Function: int gsl_spline2d_eval_e (const gsl_spline2d * SPLINE,
const double X, const double Y, gsl_interp_accel * XACC,
gsl_interp_accel * YACC, double * Z)
-- Function: double gsl_spline2d_eval_deriv_x (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC)
-- Function: int gsl_spline2d_eval_deriv_x_e (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC, double * D)
-- Function: double gsl_spline2d_eval_deriv_y (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC)
-- Function: int gsl_spline2d_eval_deriv_y_e (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC, double * D)
-- Function: double gsl_spline2d_eval_deriv_xx (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC)
-- Function: int gsl_spline2d_eval_deriv_xx_e (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC, double * D)
-- Function: double gsl_spline2d_eval_deriv_yy (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC)
-- Function: int gsl_spline2d_eval_deriv_yy_e (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC, double * D)
-- Function: double gsl_spline2d_eval_deriv_xy (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC)
-- Function: int gsl_spline2d_eval_deriv_xy_e (const gsl_spline2d *
SPLINE, const double X, const double Y, gsl_interp_accel *
XACC, gsl_interp_accel * YACC, double * D)
-- Function: int gsl_spline2d_set (const gsl_spline2d * SPLINE, double
ZA[], const size_t I, const size_t J, const double Z)
-- Function: double gsl_spline2d_get (const gsl_spline2d * SPLINE,
const double ZA[], const size_t I, const size_t J)
This function returns the value z_{ij} for grid point (I,J) stored
in the array ZA.
28.15 2D Interpolation Example programs
=======================================
The following example performs bilinear interpolation on the unit
square, using z values of (0,1,0.5,1) going clockwise around the square.
#include
#include
#include
#include
#include
int
main()
{
const gsl_interp2d_type *T = gsl_interp2d_bilinear;
const size_t N = 100; /* number of points to interpolate */
const double xa[] = { 0.0, 1.0 }; /* define unit square */
const double ya[] = { 0.0, 1.0 };
const size_t nx = sizeof(xa) / sizeof(double); /* x grid points */
const size_t ny = sizeof(ya) / sizeof(double); /* y grid points */
double *za = malloc(nx * ny * sizeof(double));
gsl_spline2d *spline = gsl_spline2d_alloc(T, nx, ny);
gsl_interp_accel *xacc = gsl_interp_accel_alloc();
gsl_interp_accel *yacc = gsl_interp_accel_alloc();
size_t i, j;
/* set z grid values */
gsl_spline2d_set(spline, za, 0, 0, 0.0);
gsl_spline2d_set(spline, za, 0, 1, 1.0);
gsl_spline2d_set(spline, za, 1, 1, 0.5);
gsl_spline2d_set(spline, za, 1, 0, 1.0);
/* initialize interpolation */
gsl_spline2d_init(spline, xa, ya, za, nx, ny);
/* interpolate N values in x and y and print out grid for plotting */
for (i = 0; i < N; ++i)
{
double xi = i / (N - 1.0);
for (j = 0; j < N; ++j)
{
double yj = j / (N - 1.0);
double zij = gsl_spline2d_eval(spline, xi, yj, xacc, yacc);
printf("%f %f %f\n", xi, yj, zij);
}
printf("\n");
}
gsl_spline2d_free(spline);
gsl_interp_accel_free(xacc);
gsl_interp_accel_free(yacc);
free(za);
return 0;
}
The results of the interpolation are shown in the following plot, where
the corners are labeled with their fixed z values.
29 Numerical Differentiation
****************************
The functions described in this chapter compute numerical derivatives by
finite differencing. An adaptive algorithm is used to find the best
choice of finite difference and to estimate the error in the derivative.
These functions are declared in the header file 'gsl_deriv.h'.
29.1 Functions
==============
-- Function: int gsl_deriv_central (const gsl_function * F, double X,
double H, double * RESULT, double * ABSERR)
This function computes the numerical derivative of the function F
at the point X using an adaptive central difference algorithm with
a step-size of H. The derivative is returned in RESULT and an
estimate of its absolute error is returned in ABSERR.
The initial value of H is used to estimate an optimal step-size,
based on the scaling of the truncation error and round-off error in
the derivative calculation. The derivative is computed using a
5-point rule for equally spaced abscissae at x-h, x-h/2, x, x+h/2,
x+h, with an error estimate taken from the difference between the
5-point rule and the corresponding 3-point rule x-h, x, x+h. Note
that the value of the function at x does not contribute to the
derivative calculation, so only 4-points are actually used.
-- Function: int gsl_deriv_forward (const gsl_function * F, double X,
double H, double * RESULT, double * ABSERR)
This function computes the numerical derivative of the function F
at the point X using an adaptive forward difference algorithm with
a step-size of H. The function is evaluated only at points greater
than X, and never at X itself. The derivative is returned in
RESULT and an estimate of its absolute error is returned in ABSERR.
This function should be used if f(x) has a discontinuity at X, or
is undefined for values less than X.
The initial value of H is used to estimate an optimal step-size,
based on the scaling of the truncation error and round-off error in
the derivative calculation. The derivative at x is computed using
an "open" 4-point rule for equally spaced abscissae at x+h/4,
x+h/2, x+3h/4, x+h, with an error estimate taken from the
difference between the 4-point rule and the corresponding 2-point
rule x+h/2, x+h.
-- Function: int gsl_deriv_backward (const gsl_function * F, double X,
double H, double * RESULT, double * ABSERR)
This function computes the numerical derivative of the function F
at the point X using an adaptive backward difference algorithm with
a step-size of H. The function is evaluated only at points less
than X, and never at X itself. The derivative is returned in
RESULT and an estimate of its absolute error is returned in ABSERR.
This function should be used if f(x) has a discontinuity at X, or
is undefined for values greater than X.
This function is equivalent to calling 'gsl_deriv_forward' with a
negative step-size.
29.2 Examples
=============
The following code estimates the derivative of the function f(x) =
x^{3/2} at x=2 and at x=0. The function f(x) is undefined for x<0 so
the derivative at x=0 is computed using 'gsl_deriv_forward'.
#include
#include
#include
double f (double x, void * params)
{
(void)(params); /* avoid unused parameter warning */
return pow (x, 1.5);
}
int
main (void)
{
gsl_function F;
double result, abserr;
F.function = &f;
F.params = 0;
printf ("f(x) = x^(3/2)\n");
gsl_deriv_central (&F, 2.0, 1e-8, &result, &abserr);
printf ("x = 2.0\n");
printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
printf ("exact = %.10f\n\n", 1.5 * sqrt(2.0));
gsl_deriv_forward (&F, 0.0, 1e-8, &result, &abserr);
printf ("x = 0.0\n");
printf ("f'(x) = %.10f +/- %.10f\n", result, abserr);
printf ("exact = %.10f\n", 0.0);
return 0;
}
Here is the output of the program,
$ ./a.out
f(x) = x^(3/2)
x = 2.0
f'(x) = 2.1213203120 +/- 0.0000005006
exact = 2.1213203436
x = 0.0
f'(x) = 0.0000000160 +/- 0.0000000339
exact = 0.0000000000
29.3 References and Further Reading
===================================
The algorithms used by these functions are described in the following
sources:
Abramowitz and Stegun, 'Handbook of Mathematical Functions',
Section 25.3.4, and Table 25.5 (Coefficients for Differentiation).
S.D. Conte and Carl de Boor, 'Elementary Numerical Analysis: An
Algorithmic Approach', McGraw-Hill, 1972.
30 Chebyshev Approximations
***************************
This chapter describes routines for computing Chebyshev approximations
to univariate functions. A Chebyshev approximation is a truncation of
the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials
T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on
the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}. The
first few Chebyshev polynomials are, T_0(x) = 1, T_1(x) = x, T_2(x) = 2
x^2 - 1. For further information see Abramowitz & Stegun, Chapter 22.
The functions described in this chapter are declared in the header
file 'gsl_chebyshev.h'.
30.1 Definitions
================
A Chebyshev series is stored using the following structure,
typedef struct
{
double * c; /* coefficients c[0] .. c[order] */
int order; /* order of expansion */
double a; /* lower interval point */
double b; /* upper interval point */
...
} gsl_cheb_series
The approximation is made over the range [a,b] using ORDER+1 terms,
including the coefficient c[0]. The series is computed using the
following convention,
f(x) = (c_0 / 2) + \sum_{n=1} c_n T_n(x)
which is needed when accessing the coefficients directly.
30.2 Creation and Calculation of Chebyshev Series
=================================================
-- Function: gsl_cheb_series * gsl_cheb_alloc (const size_t N)
This function allocates space for a Chebyshev series of order N and
returns a pointer to a new 'gsl_cheb_series' struct.
-- Function: void gsl_cheb_free (gsl_cheb_series * CS)
This function frees a previously allocated Chebyshev series CS.
-- Function: int gsl_cheb_init (gsl_cheb_series * CS, const
gsl_function * F, const double A, const double B)
This function computes the Chebyshev approximation CS for the
function F over the range (a,b) to the previously specified order.
The computation of the Chebyshev approximation is an O(n^2)
process, and requires n function evaluations.
30.3 Auxiliary Functions
========================
The following functions provide information about an existing Chebyshev
series.
-- Function: size_t gsl_cheb_order (const gsl_cheb_series * CS)
This function returns the order of Chebyshev series CS.
-- Function: size_t gsl_cheb_size (const gsl_cheb_series * CS)
-- Function: double * gsl_cheb_coeffs (const gsl_cheb_series * CS)
These functions return the size of the Chebyshev coefficient array
'c[]' and a pointer to its location in memory for the Chebyshev
series CS.
30.4 Chebyshev Series Evaluation
================================
-- Function: double gsl_cheb_eval (const gsl_cheb_series * CS, double
X)
This function evaluates the Chebyshev series CS at a given point X.
-- Function: int gsl_cheb_eval_err (const gsl_cheb_series * CS, const
double X, double * RESULT, double * ABSERR)
This function computes the Chebyshev series CS at a given point X,
estimating both the series RESULT and its absolute error ABSERR.
The error estimate is made from the first neglected term in the
series.
-- Function: double gsl_cheb_eval_n (const gsl_cheb_series * CS, size_t
ORDER, double X)
This function evaluates the Chebyshev series CS at a given point X,
to (at most) the given order ORDER.
-- Function: int gsl_cheb_eval_n_err (const gsl_cheb_series * CS, const
size_t ORDER, const double X, double * RESULT, double *
ABSERR)
This function evaluates a Chebyshev series CS at a given point X,
estimating both the series RESULT and its absolute error ABSERR, to
(at most) the given order ORDER. The error estimate is made from
the first neglected term in the series.
30.5 Derivatives and Integrals
==============================
The following functions allow a Chebyshev series to be differentiated or
integrated, producing a new Chebyshev series. Note that the error
estimate produced by evaluating the derivative series will be
underestimated due to the contribution of higher order terms being
neglected.
-- Function: int gsl_cheb_calc_deriv (gsl_cheb_series * DERIV, const
gsl_cheb_series * CS)
This function computes the derivative of the series CS, storing the
derivative coefficients in the previously allocated DERIV. The two
series CS and DERIV must have been allocated with the same order.
-- Function: int gsl_cheb_calc_integ (gsl_cheb_series * INTEG, const
gsl_cheb_series * CS)
This function computes the integral of the series CS, storing the
integral coefficients in the previously allocated INTEG. The two
series CS and INTEG must have been allocated with the same order.
The lower limit of the integration is taken to be the left hand end
of the range A.
30.6 Examples
=============
The following example program computes Chebyshev approximations to a
step function. This is an extremely difficult approximation to make,
due to the discontinuity, and was chosen as an example where
approximation error is visible. For smooth functions the Chebyshev
approximation converges extremely rapidly and errors would not be
visible.
#include
#include
#include
double
f (double x, void *p)
{
(void)(p); /* avoid unused parameter warning */
if (x < 0.5)
return 0.25;
else
return 0.75;
}
int
main (void)
{
int i, n = 10000;
gsl_cheb_series *cs = gsl_cheb_alloc (40);
gsl_function F;
F.function = f;
F.params = 0;
gsl_cheb_init (cs, &F, 0.0, 1.0);
for (i = 0; i < n; i++)
{
double x = i / (double)n;
double r10 = gsl_cheb_eval_n (cs, 10, x);
double r40 = gsl_cheb_eval (cs, x);
printf ("%g %g %g %g\n",
x, GSL_FN_EVAL (&F, x), r10, r40);
}
gsl_cheb_free (cs);
return 0;
}
The output from the program gives the original function, 10-th order
approximation and 40-th order approximation, all sampled at intervals of
0.001 in x.
30.7 References and Further Reading
===================================
The following paper describes the use of Chebyshev series,
R. Broucke, "Ten Subroutines for the Manipulation of Chebyshev
Series [C1] (Algorithm 446)". 'Communications of the ACM' 16(4),
254-256 (1973)
31 Series Acceleration
**********************
The functions described in this chapter accelerate the convergence of a
series using the Levin u-transform. This method takes a small number of
terms from the start of a series and uses a systematic approximation to
compute an extrapolated value and an estimate of its error. The
u-transform works for both convergent and divergent series, including
asymptotic series.
These functions are declared in the header file 'gsl_sum.h'.
31.1 Acceleration functions
===========================
The following functions compute the full Levin u-transform of a series
with its error estimate. The error estimate is computed by propagating
rounding errors from each term through to the final extrapolation.
These functions are intended for summing analytic series where each
term is known to high accuracy, and the rounding errors are assumed to
originate from finite precision. They are taken to be relative errors
of order 'GSL_DBL_EPSILON' for each term.
The calculation of the error in the extrapolated value is an O(N^2)
process, which is expensive in time and memory. A faster but less
reliable method which estimates the error from the convergence of the
extrapolated value is described in the next section. For the method
described here a full table of intermediate values and derivatives
through to O(N) must be computed and stored, but this does give a
reliable error estimate.
-- Function: gsl_sum_levin_u_workspace * gsl_sum_levin_u_alloc (size_t
N)
This function allocates a workspace for a Levin u-transform of N
terms. The size of the workspace is O(2n^2 + 3n).
-- Function: void gsl_sum_levin_u_free (gsl_sum_levin_u_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_sum_levin_u_accel (const double * ARRAY, size_t
ARRAY_SIZE, gsl_sum_levin_u_workspace * W, double * SUM_ACCEL,
double * ABSERR)
This function takes the terms of a series in ARRAY of size
ARRAY_SIZE and computes the extrapolated limit of the series using
a Levin u-transform. Additional working space must be provided in
W. The extrapolated sum is stored in SUM_ACCEL, with an estimate
of the absolute error stored in ABSERR. The actual term-by-term
sum is returned in 'w->sum_plain'. The algorithm calculates the
truncation error (the difference between two successive
extrapolations) and round-off error (propagated from the individual
terms) to choose an optimal number of terms for the extrapolation.
All the terms of the series passed in through ARRAY should be
non-zero.
31.2 Acceleration functions without error estimation
====================================================
The functions described in this section compute the Levin u-transform of
series and attempt to estimate the error from the "truncation error" in
the extrapolation, the difference between the final two approximations.
Using this method avoids the need to compute an intermediate table of
derivatives because the error is estimated from the behavior of the
extrapolated value itself. Consequently this algorithm is an O(N)
process and only requires O(N) terms of storage. If the series
converges sufficiently fast then this procedure can be acceptable. It
is appropriate to use this method when there is a need to compute many
extrapolations of series with similar convergence properties at
high-speed. For example, when numerically integrating a function
defined by a parameterized series where the parameter varies only
slightly. A reliable error estimate should be computed first using the
full algorithm described above in order to verify the consistency of the
results.
-- Function: gsl_sum_levin_utrunc_workspace *
gsl_sum_levin_utrunc_alloc (size_t N)
This function allocates a workspace for a Levin u-transform of N
terms, without error estimation. The size of the workspace is
O(3n).
-- Function: void gsl_sum_levin_utrunc_free
(gsl_sum_levin_utrunc_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: int gsl_sum_levin_utrunc_accel (const double * ARRAY,
size_t ARRAY_SIZE, gsl_sum_levin_utrunc_workspace * W, double
* SUM_ACCEL, double * ABSERR_TRUNC)
This function takes the terms of a series in ARRAY of size
ARRAY_SIZE and computes the extrapolated limit of the series using
a Levin u-transform. Additional working space must be provided in
W. The extrapolated sum is stored in SUM_ACCEL. The actual
term-by-term sum is returned in 'w->sum_plain'. The algorithm
terminates when the difference between two successive
extrapolations reaches a minimum or is sufficiently small. The
difference between these two values is used as estimate of the
error and is stored in ABSERR_TRUNC. To improve the reliability of
the algorithm the extrapolated values are replaced by moving
averages when calculating the truncation error, smoothing out any
fluctuations.
31.3 Examples
=============
The following code calculates an estimate of \zeta(2) = \pi^2 / 6 using
the series,
\zeta(2) = 1 + 1/2^2 + 1/3^2 + 1/4^2 + ...
After N terms the error in the sum is O(1/N), making direct summation of
the series converge slowly.
#include
#include
#include
#define N 20
int
main (void)
{
double t[N];
double sum_accel, err;
double sum = 0;
int n;
gsl_sum_levin_u_workspace * w
= gsl_sum_levin_u_alloc (N);
const double zeta_2 = M_PI * M_PI / 6.0;
/* terms for zeta(2) = \sum_{n=1}^{\infty} 1/n^2 */
for (n = 0; n < N; n++)
{
double np1 = n + 1.0;
t[n] = 1.0 / (np1 * np1);
sum += t[n];
}
gsl_sum_levin_u_accel (t, N, w, &sum_accel, &err);
printf ("term-by-term sum = % .16f using %d terms\n",
sum, N);
printf ("term-by-term sum = % .16f using %zu terms\n",
w->sum_plain, w->terms_used);
printf ("exact value = % .16f\n", zeta_2);
printf ("accelerated sum = % .16f using %zu terms\n",
sum_accel, w->terms_used);
printf ("estimated error = % .16f\n", err);
printf ("actual error = % .16f\n",
sum_accel - zeta_2);
gsl_sum_levin_u_free (w);
return 0;
}
The output below shows that the Levin u-transform is able to obtain an
estimate of the sum to 1 part in 10^10 using the first eleven terms of
the series. The error estimate returned by the function is also
accurate, giving the correct number of significant digits.
$ ./a.out
term-by-term sum = 1.5961632439130233 using 20 terms
term-by-term sum = 1.5759958390005426 using 13 terms
exact value = 1.6449340668482264
accelerated sum = 1.6449340669228176 using 13 terms
estimated error = 0.0000000000888360
actual error = 0.0000000000745912
Note that a direct summation of this series would require 10^10 terms to
achieve the same precision as the accelerated sum does in 13 terms.
31.4 References and Further Reading
===================================
The algorithms used by these functions are described in the following
papers,
T. Fessler, W.F. Ford, D.A. Smith, HURRY: An acceleration algorithm
for scalar sequences and series 'ACM Transactions on Mathematical
Software', 9(3):346-354, 1983. and Algorithm 602 9(3):355-357,
1983.
The theory of the u-transform was presented by Levin,
D. Levin, Development of Non-Linear Transformations for Improving
Convergence of Sequences, 'Intern. J. Computer Math.' B3:371-388,
1973.
A review paper on the Levin Transform is available online,
Herbert H. H. Homeier, Scalar Levin-Type Sequence Transformations,
.
32 Wavelet Transforms
*********************
This chapter describes functions for performing Discrete Wavelet
Transforms (DWTs). The library includes wavelets for real data in both
one and two dimensions. The wavelet functions are declared in the
header files 'gsl_wavelet.h' and 'gsl_wavelet2d.h'.
32.1 Definitions
================
The continuous wavelet transform and its inverse are defined by the
relations,
w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt
and,
f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds
where the basis functions \psi_{s,\tau} are obtained by scaling and
translation from a single function, referred to as the "mother wavelet".
The discrete version of the wavelet transform acts on equally-spaced
samples, with fixed scaling and translation steps (s, \tau). The
frequency and time axes are sampled "dyadically" on scales of 2^j
through a level parameter j. The resulting family of functions
{\psi_{j,n}} constitutes an orthonormal basis for square-integrable
signals.
The discrete wavelet transform is an O(N) algorithm, and is also
referred to as the "fast wavelet transform".
32.2 Initialization
===================
The 'gsl_wavelet' structure contains the filter coefficients defining
the wavelet and any associated offset parameters.
-- Function: gsl_wavelet * gsl_wavelet_alloc (const gsl_wavelet_type *
T, size_t K)
This function allocates and initializes a wavelet object of type T.
The parameter K selects the specific member of the wavelet family.
A null pointer is returned if insufficient memory is available or
if a unsupported member is selected.
The following wavelet types are implemented:
-- Wavelet: gsl_wavelet_daubechies
-- Wavelet: gsl_wavelet_daubechies_centered
This is the Daubechies wavelet family of maximum phase with k/2
vanishing moments. The implemented wavelets are k=4, 6, ..., 20,
with K even.
-- Wavelet: gsl_wavelet_haar
-- Wavelet: gsl_wavelet_haar_centered
This is the Haar wavelet. The only valid choice of k for the Haar
wavelet is k=2.
-- Wavelet: gsl_wavelet_bspline
-- Wavelet: gsl_wavelet_bspline_centered
This is the biorthogonal B-spline wavelet family of order (i,j).
The implemented values of k = 100*i + j are 103, 105, 202, 204,
206, 208, 301, 303, 305 307, 309.
The centered forms of the wavelets align the coefficients of the various
sub-bands on edges. Thus the resulting visualization of the
coefficients of the wavelet transform in the phase plane is easier to
understand.
-- Function: const char * gsl_wavelet_name (const gsl_wavelet * W)
This function returns a pointer to the name of the wavelet family
for W.
-- Function: void gsl_wavelet_free (gsl_wavelet * W)
This function frees the wavelet object W.
The 'gsl_wavelet_workspace' structure contains scratch space of the
same size as the input data and is used to hold intermediate results
during the transform.
-- Function: gsl_wavelet_workspace * gsl_wavelet_workspace_alloc
(size_t N)
This function allocates a workspace for the discrete wavelet
transform. To perform a one-dimensional transform on N elements, a
workspace of size N must be provided. For two-dimensional
transforms of N-by-N matrices it is sufficient to allocate a
workspace of size N, since the transform operates on individual
rows and columns. A null pointer is returned if insufficient
memory is available.
-- Function: void gsl_wavelet_workspace_free (gsl_wavelet_workspace *
WORK)
This function frees the allocated workspace WORK.
32.3 Transform Functions
========================
This sections describes the actual functions performing the discrete
wavelet transform. Note that the transforms use periodic boundary
conditions. If the signal is not periodic in the sample length then
spurious coefficients will appear at the beginning and end of each level
of the transform.
32.3.1 Wavelet transforms in one dimension
------------------------------------------
-- Function: int gsl_wavelet_transform (const gsl_wavelet * W, double *
DATA, size_t STRIDE, size_t N, gsl_wavelet_direction DIR,
gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet_transform_forward (const gsl_wavelet * W,
double * DATA, size_t STRIDE, size_t N, gsl_wavelet_workspace
* WORK)
-- Function: int gsl_wavelet_transform_inverse (const gsl_wavelet * W,
double * DATA, size_t STRIDE, size_t N, gsl_wavelet_workspace
* WORK)
These functions compute in-place forward and inverse discrete
wavelet transforms of length N with stride STRIDE on the array
DATA. The length of the transform N is restricted to powers of
two. For the 'transform' version of the function the argument DIR
can be either 'forward' (+1) or 'backward' (-1). A workspace WORK
of length N must be provided.
For the forward transform, the elements of the original array are
replaced by the discrete wavelet transform f_i -> w_{j,k} in a
packed triangular storage layout, where J is the index of the level
j = 0 ... J-1 and K is the index of the coefficient within each
level, k = 0 ... (2^j)-1. The total number of levels is J =
\log_2(n). The output data has the following form,
(s_{-1,0}, d_{0,0}, d_{1,0}, d_{1,1}, d_{2,0}, ...,
d_{j,k}, ..., d_{J-1,2^{J-1}-1})
where the first element is the smoothing coefficient s_{-1,0},
followed by the detail coefficients d_{j,k} for each level j. The
backward transform inverts these coefficients to obtain the
original data.
These functions return a status of 'GSL_SUCCESS' upon successful
completion. 'GSL_EINVAL' is returned if N is not an integer power
of 2 or if insufficient workspace is provided.
32.3.2 Wavelet transforms in two dimension
------------------------------------------
The library provides functions to perform two-dimensional discrete
wavelet transforms on square matrices. The matrix dimensions must be an
integer power of two. There are two possible orderings of the rows and
columns in the two-dimensional wavelet transform, referred to as the
"standard" and "non-standard" forms.
The "standard" transform performs a complete discrete wavelet
transform on the rows of the matrix, followed by a separate complete
discrete wavelet transform on the columns of the resulting
row-transformed matrix. This procedure uses the same ordering as a
two-dimensional Fourier transform.
The "non-standard" transform is performed in interleaved passes on
the rows and columns of the matrix for each level of the transform. The
first level of the transform is applied to the matrix rows, and then to
the matrix columns. This procedure is then repeated across the rows and
columns of the data for the subsequent levels of the transform, until
the full discrete wavelet transform is complete. The non-standard form
of the discrete wavelet transform is typically used in image analysis.
The functions described in this section are declared in the header
file 'gsl_wavelet2d.h'.
-- Function: int gsl_wavelet2d_transform (const gsl_wavelet * W, double
* DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
gsl_wavelet_direction DIR, gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_transform_forward (const gsl_wavelet *
W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_transform_inverse (const gsl_wavelet *
W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
gsl_wavelet_workspace * WORK)
These functions compute two-dimensional in-place forward and
inverse discrete wavelet transforms in standard form on the array
DATA stored in row-major form with dimensions SIZE1 and SIZE2 and
physical row length TDA. The dimensions must be equal (square
matrix) and are restricted to powers of two. For the 'transform'
version of the function the argument DIR can be either 'forward'
(+1) or 'backward' (-1). A workspace WORK of the appropriate size
must be provided. On exit, the appropriate elements of the array
DATA are replaced by their two-dimensional wavelet transform.
The functions return a status of 'GSL_SUCCESS' upon successful
completion. 'GSL_EINVAL' is returned if SIZE1 and SIZE2 are not
equal and integer powers of 2, or if insufficient workspace is
provided.
-- Function: int gsl_wavelet2d_transform_matrix (const gsl_wavelet * W,
gsl_matrix * M, gsl_wavelet_direction DIR,
gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_transform_matrix_forward (const
gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_transform_matrix_inverse (const
gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
These functions compute the two-dimensional in-place wavelet
transform on a matrix A.
-- Function: int gsl_wavelet2d_nstransform (const gsl_wavelet * W,
double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
gsl_wavelet_direction DIR, gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_nstransform_forward (const gsl_wavelet *
W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_nstransform_inverse (const gsl_wavelet *
W, double * DATA, size_t TDA, size_t SIZE1, size_t SIZE2,
gsl_wavelet_workspace * WORK)
These functions compute the two-dimensional wavelet transform in
non-standard form.
-- Function: int gsl_wavelet2d_nstransform_matrix (const gsl_wavelet *
W, gsl_matrix * M, gsl_wavelet_direction DIR,
gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_nstransform_matrix_forward (const
gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
-- Function: int gsl_wavelet2d_nstransform_matrix_inverse (const
gsl_wavelet * W, gsl_matrix * M, gsl_wavelet_workspace * WORK)
These functions compute the non-standard form of the
two-dimensional in-place wavelet transform on a matrix A.
32.4 Examples
=============
The following program demonstrates the use of the one-dimensional
wavelet transform functions. It computes an approximation to an input
signal (of length 256) using the 20 largest components of the wavelet
transform, while setting the others to zero.
#include
#include
#include
#include
int
main (int argc, char **argv)
{
(void)(argc); /* avoid unused parameter warning */
int i, n = 256, nc = 20;
double *data = malloc (n * sizeof (double));
double *abscoeff = malloc (n * sizeof (double));
size_t *p = malloc (n * sizeof (size_t));
FILE * f;
gsl_wavelet *w;
gsl_wavelet_workspace *work;
w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
work = gsl_wavelet_workspace_alloc (n);
f = fopen (argv[1], "r");
for (i = 0; i < n; i++)
{
fscanf (f, "%lg", &data[i]);
}
fclose (f);
gsl_wavelet_transform_forward (w, data, 1, n, work);
for (i = 0; i < n; i++)
{
abscoeff[i] = fabs (data[i]);
}
gsl_sort_index (p, abscoeff, 1, n);
for (i = 0; (i + nc) < n; i++)
data[p[i]] = 0;
gsl_wavelet_transform_inverse (w, data, 1, n, work);
for (i = 0; i < n; i++)
{
printf ("%g\n", data[i]);
}
gsl_wavelet_free (w);
gsl_wavelet_workspace_free (work);
free (data);
free (abscoeff);
free (p);
return 0;
}
The output can be used with the GNU plotutils 'graph' program,
$ ./a.out ecg.dat > dwt.txt
$ graph -T ps -x 0 256 32 -h 0.3 -a dwt.txt > dwt.ps
32.5 References and Further Reading
===================================
The mathematical background to wavelet transforms is covered in the
original lectures by Daubechies,
Ingrid Daubechies. Ten Lectures on Wavelets. 'CBMS-NSF Regional
Conference Series in Applied Mathematics' (1992), SIAM, ISBN
0898712742.
An easy to read introduction to the subject with an emphasis on the
application of the wavelet transform in various branches of science is,
Paul S. Addison. 'The Illustrated Wavelet Transform Handbook'.
Institute of Physics Publishing (2002), ISBN 0750306920.
For extensive coverage of signal analysis by wavelets, wavelet packets
and local cosine bases see,
S. G. Mallat. 'A wavelet tour of signal processing' (Second
edition). Academic Press (1999), ISBN 012466606X.
The concept of multiresolution analysis underlying the wavelet transform
is described in,
S. G. Mallat. Multiresolution Approximations and Wavelet
Orthonormal Bases of L^2(R). 'Transactions of the American
Mathematical Society', 315(1), 1989, 69-87.
S. G. Mallat. A Theory for Multiresolution Signal
Decomposition--The Wavelet Representation. 'IEEE Transactions on
Pattern Analysis and Machine Intelligence', 11, 1989, 674-693.
The coefficients for the individual wavelet families implemented by the
library can be found in the following papers,
I. Daubechies. Orthonormal Bases of Compactly Supported Wavelets.
'Communications on Pure and Applied Mathematics', 41 (1988)
909-996.
A. Cohen, I. Daubechies, and J.-C. Feauveau. Biorthogonal Bases of
Compactly Supported Wavelets. 'Communications on Pure and Applied
Mathematics', 45 (1992) 485-560.
The PhysioNet archive of physiological datasets can be found online at
and is described in the following paper,
Goldberger et al. PhysioBank, PhysioToolkit, and PhysioNet:
Components of a New Research Resource for Complex Physiologic
Signals. 'Circulation' 101(23):e215-e220 2000.
33 Discrete Hankel Transforms
*****************************
This chapter describes functions for performing Discrete Hankel
Transforms (DHTs). The functions are declared in the header file
'gsl_dht.h'.
33.1 Definitions
================
The discrete Hankel transform acts on a vector of sampled data, where
the samples are assumed to have been taken at points related to the
zeros of a Bessel function of fixed order; compare this to the case of
the discrete Fourier transform, where samples are taken at points
related to the zeroes of the sine or cosine function.
Starting with its definition, the Hankel transform (or Bessel
transform) of order \nu of a function f with \nu > -1/2 is defined as
(see Johnson, 1987 and Lemoine, 1994)
F_\nu(u) = \int_0^\infty f(t) J_\nu(u t) t dt
If the integral exists, F_\nu is called the Hankel transformation of f.
The reverse transform is given by
f(t) = \int_0^\infty F_\nu(u) J_\nu(u t) u du ,
where \int_0^\infty f(t) t^{1/2} dt must exist and be absolutely
convergent, and where f(t) satisfies Dirichlet's conditions (of limited
total fluctuations) in the interval [0,\infty].
Now the discrete Hankel transform works on a discrete function f,
which is sampled on points n=1...M located at positions
t_n=(j_{\nu,n}/j_{\nu,M}) X in real space and at u_n=j_{\nu,n}/X in
reciprocal space. Here, j_{\nu,m} are the m-th zeros of the Bessel
function J_\nu(x) arranged in ascending order. Moreover, the discrete
functions are assumed to be band limited, so f(t_n)=0 and F(u_n)=0 for
n>M. Accordingly, the function f is defined on the interval [0,X].
Following the work of Johnson, 1987 and Lemoine, 1994, the discrete
Hankel transform is given by
F_\nu(u_m) = (2 X^2 / j_(\nu,M)^2)
\sum_{k=1}^{M-1} f(j_(\nu,k) X/j_(\nu,M))
(J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,k))^2).
It is this discrete expression which defines the discrete Hankel
transform calculated by GSL. In GSL, forward and backward transforms are
defined equally and calculate F_\nu(u_m). Following Johnson, the
backward transform reads
f(t_k) = (2 / X^2)
\sum_{m=1}^{M-1} F(j_(\nu,m)/X)
(J_\nu(j_(\nu,m) j_(\nu,k) / j_(\nu,M)) / J_(\nu+1)(j_(\nu,m))^2).
Obviously, using the forward transform instead of the backward transform
gives an additional factor X^4/j_{\nu,M}^2=t_m^2/u_m^2.
The kernel in the summation above defines the matrix of the
\nu-Hankel transform of size M-1. The coefficients of this matrix,
being dependent on \nu and M, must be precomputed and stored; the
'gsl_dht' object encapsulates this data. The allocation function
'gsl_dht_alloc' returns a 'gsl_dht' object which must be properly
initialized with 'gsl_dht_init' before it can be used to perform
transforms on data sample vectors, for fixed \nu and M, using the
'gsl_dht_apply' function. The implementation allows to define the
length X of the fundamental interval, for convenience, while discrete
Hankel transforms are often defined on the unit interval instead of
[0,X].
Notice that by assumption f(t) vanishes at the endpoints of the
interval, consistent with the inversion formula and the sampling formula
given above. Therefore, this transform corresponds to an orthogonal
expansion in eigenfunctions of the Dirichlet problem for the Bessel
differential equation.
33.2 Functions
==============
-- Function: gsl_dht * gsl_dht_alloc (size_t SIZE)
This function allocates a Discrete Hankel transform object of size
SIZE.
-- Function: int gsl_dht_init (gsl_dht * T, double NU, double XMAX)
This function initializes the transform T for the given values of
NU and XMAX.
-- Function: gsl_dht * gsl_dht_new (size_t SIZE, double NU, double
XMAX)
This function allocates a Discrete Hankel transform object of size
SIZE and initializes it for the given values of NU and XMAX.
-- Function: void gsl_dht_free (gsl_dht * T)
This function frees the transform T.
-- Function: int gsl_dht_apply (const gsl_dht * T, double * F_IN,
double * F_OUT)
This function applies the transform T to the array F_IN whose size
is equal to the size of the transform. The result is stored in the
array F_OUT which must be of the same length.
Applying this function to its output gives the original data
multiplied by (1/j_(\nu,M))^2, up to numerical errors.
-- Function: double gsl_dht_x_sample (const gsl_dht * T, int N)
This function returns the value of the N-th sample point in the
unit interval, (j_{\nu,n+1}/j_{\nu,M}) X. These are the points
where the function f(t) is assumed to be sampled.
-- Function: double gsl_dht_k_sample (const gsl_dht * T, int N)
This function returns the value of the N-th sample point in
"k-space", j_{\nu,n+1}/X.
33.3 References and Further Reading
===================================
The algorithms used by these functions are described in the following
papers,
H. Fisk Johnson, Comp. Phys. Comm. 43, 181 (1987).
D. Lemoine, J. Chem. Phys. 101, 3936 (1994).
34 One dimensional Root-Finding
*******************************
This chapter describes routines for finding roots of arbitrary
one-dimensional functions. The library provides low level components
for a variety of iterative solvers and convergence tests. These can be
combined by the user to achieve the desired solution, with full access
to the intermediate steps of the iteration. Each class of methods uses
the same framework, so that you can switch between solvers at runtime
without needing to recompile your program. Each instance of a solver
keeps track of its own state, allowing the solvers to be used in
multi-threaded programs.
The header file 'gsl_roots.h' contains prototypes for the root
finding functions and related declarations.
34.1 Overview
=============
One-dimensional root finding algorithms can be divided into two classes,
"root bracketing" and "root polishing". Algorithms which proceed by
bracketing a root are guaranteed to converge. Bracketing algorithms
begin with a bounded region known to contain a root. The size of this
bounded region is reduced, iteratively, until it encloses the root to a
desired tolerance. This provides a rigorous error estimate for the
location of the root.
The technique of "root polishing" attempts to improve an initial
guess to the root. These algorithms converge only if started "close
enough" to a root, and sacrifice a rigorous error bound for speed. By
approximating the behavior of a function in the vicinity of a root they
attempt to find a higher order improvement of an initial guess. When
the behavior of the function is compatible with the algorithm and a good
initial guess is available a polishing algorithm can provide rapid
convergence.
In GSL both types of algorithm are available in similar frameworks.
The user provides a high-level driver for the algorithms, and the
library provides the individual functions necessary for each of the
steps. There are three main phases of the iteration. The steps are,
* initialize solver state, S, for algorithm T
* update S using the iteration T
* test S for convergence, and repeat iteration if necessary
The state for bracketing solvers is held in a 'gsl_root_fsolver' struct.
The updating procedure uses only function evaluations (not derivatives).
The state for root polishing solvers is held in a 'gsl_root_fdfsolver'
struct. The updates require both the function and its derivative (hence
the name 'fdf') to be supplied by the user.
34.2 Caveats
============
Note that root finding functions can only search for one root at a time.
When there are several roots in the search area, the first root to be
found will be returned; however it is difficult to predict which of the
roots this will be. _In most cases, no error will be reported if you
try to find a root in an area where there is more than one._
Care must be taken when a function may have a multiple root (such as
f(x) = (x-x_0)^2 or f(x) = (x-x_0)^3). It is not possible to use
root-bracketing algorithms on even-multiplicity roots. For these
algorithms the initial interval must contain a zero-crossing, where the
function is negative at one end of the interval and positive at the
other end. Roots with even-multiplicity do not cross zero, but only
touch it instantaneously. Algorithms based on root bracketing will
still work for odd-multiplicity roots (e.g. cubic, quintic, ...). Root
polishing algorithms generally work with higher multiplicity roots, but
at a reduced rate of convergence. In these cases the "Steffenson
algorithm" can be used to accelerate the convergence of multiple roots.
While it is not absolutely required that f have a root within the
search region, numerical root finding functions should not be used
haphazardly to check for the _existence_ of roots. There are better
ways to do this. Because it is easy to create situations where
numerical root finders can fail, it is a bad idea to throw a root finder
at a function you do not know much about. In general it is best to
examine the function visually by plotting before searching for a root.
34.3 Initializing the Solver
============================
-- Function: gsl_root_fsolver * gsl_root_fsolver_alloc (const
gsl_root_fsolver_type * T)
This function returns a pointer to a newly allocated instance of a
solver of type T. For example, the following code creates an
instance of a bisection solver,
const gsl_root_fsolver_type * T
= gsl_root_fsolver_bisection;
gsl_root_fsolver * s
= gsl_root_fsolver_alloc (T);
If there is insufficient memory to create the solver then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
-- Function: gsl_root_fdfsolver * gsl_root_fdfsolver_alloc (const
gsl_root_fdfsolver_type * T)
This function returns a pointer to a newly allocated instance of a
derivative-based solver of type T. For example, the following code
creates an instance of a Newton-Raphson solver,
const gsl_root_fdfsolver_type * T
= gsl_root_fdfsolver_newton;
gsl_root_fdfsolver * s
= gsl_root_fdfsolver_alloc (T);
If there is insufficient memory to create the solver then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
-- Function: int gsl_root_fsolver_set (gsl_root_fsolver * S,
gsl_function * F, double X_LOWER, double X_UPPER)
This function initializes, or reinitializes, an existing solver S
to use the function F and the initial search interval [X_LOWER,
X_UPPER].
-- Function: int gsl_root_fdfsolver_set (gsl_root_fdfsolver * S,
gsl_function_fdf * FDF, double ROOT)
This function initializes, or reinitializes, an existing solver S
to use the function and derivative FDF and the initial guess ROOT.
-- Function: void gsl_root_fsolver_free (gsl_root_fsolver * S)
-- Function: void gsl_root_fdfsolver_free (gsl_root_fdfsolver * S)
These functions free all the memory associated with the solver S.
-- Function: const char * gsl_root_fsolver_name (const gsl_root_fsolver
* S)
-- Function: const char * gsl_root_fdfsolver_name (const
gsl_root_fdfsolver * S)
These functions return a pointer to the name of the solver. For
example,
printf ("s is a '%s' solver\n",
gsl_root_fsolver_name (s));
would print something like 's is a 'bisection' solver'.
34.4 Providing the function to solve
====================================
You must provide a continuous function of one variable for the root
finders to operate on, and, sometimes, its first derivative. In order
to allow for general parameters the functions are defined by the
following data types:
-- Data Type: gsl_function
This data type defines a general function with parameters.
'double (* function) (double X, void * PARAMS)'
this function should return the value f(x,params) for argument
X and parameters PARAMS
'void * params'
a pointer to the parameters of the function
Here is an example for the general quadratic function,
f(x) = a x^2 + b x + c
with a = 3, b = 2, c = 1. The following code defines a 'gsl_function'
'F' which you could pass to a root finder as a function pointer:
struct my_f_params { double a; double b; double c; };
double
my_f (double x, void * p) {
struct my_f_params * params
= (struct my_f_params *)p;
double a = (params->a);
double b = (params->b);
double c = (params->c);
return (a * x + b) * x + c;
}
gsl_function F;
struct my_f_params params = { 3.0, 2.0, 1.0 };
F.function = &my_f;
F.params = ¶ms;
The function f(x) can be evaluated using the macro 'GSL_FN_EVAL(&F,x)'
defined in 'gsl_math.h'.
-- Data Type: gsl_function_fdf
This data type defines a general function with parameters and its
first derivative.
'double (* f) (double X, void * PARAMS)'
this function should return the value of f(x,params) for
argument X and parameters PARAMS
'double (* df) (double X, void * PARAMS)'
this function should return the value of the derivative of F
with respect to X, f'(x,params), for argument X and parameters
PARAMS
'void (* fdf) (double X, void * PARAMS, double * F, double * DF)'
this function should set the values of the function F to
f(x,params) and its derivative DF to f'(x,params) for argument
X and parameters PARAMS. This function provides an
optimization of the separate functions for f(x) and f'(x)--it
is always faster to compute the function and its derivative at
the same time.
'void * params'
a pointer to the parameters of the function
Here is an example where f(x) = 2\exp(2x):
double
my_f (double x, void * params)
{
return exp (2 * x);
}
double
my_df (double x, void * params)
{
return 2 * exp (2 * x);
}
void
my_fdf (double x, void * params,
double * f, double * df)
{
double t = exp (2 * x);
*f = t;
*df = 2 * t; /* uses existing value */
}
gsl_function_fdf FDF;
FDF.f = &my_f;
FDF.df = &my_df;
FDF.fdf = &my_fdf;
FDF.params = 0;
The function f(x) can be evaluated using the macro
'GSL_FN_FDF_EVAL_F(&FDF,x)' and the derivative f'(x) can be evaluated
using the macro 'GSL_FN_FDF_EVAL_DF(&FDF,x)'. Both the function y =
f(x) and its derivative dy = f'(x) can be evaluated at the same time
using the macro 'GSL_FN_FDF_EVAL_F_DF(&FDF,x,y,dy)'. The macro stores
f(x) in its Y argument and f'(x) in its DY argument--both of these
should be pointers to 'double'.
34.5 Search Bounds and Guesses
==============================
You provide either search bounds or an initial guess; this section
explains how search bounds and guesses work and how function arguments
control them.
A guess is simply an x value which is iterated until it is within the
desired precision of a root. It takes the form of a 'double'.
Search bounds are the endpoints of an interval which is iterated
until the length of the interval is smaller than the requested
precision. The interval is defined by two values, the lower limit and
the upper limit. Whether the endpoints are intended to be included in
the interval or not depends on the context in which the interval is
used.
34.6 Iteration
==============
The following functions drive the iteration of each algorithm. Each
function performs one iteration to update the state of any solver of the
corresponding type. The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.
-- Function: int gsl_root_fsolver_iterate (gsl_root_fsolver * S)
-- Function: int gsl_root_fdfsolver_iterate (gsl_root_fdfsolver * S)
These functions perform a single iteration of the solver S. If the
iteration encounters an unexpected problem then an error code will
be returned,
'GSL_EBADFUNC'
the iteration encountered a singular point where the function
or its derivative evaluated to 'Inf' or 'NaN'.
'GSL_EZERODIV'
the derivative of the function vanished at the iteration
point, preventing the algorithm from continuing without a
division by zero.
The solver maintains a current best estimate of the root at all
times. The bracketing solvers also keep track of the current best
interval bounding the root. This information can be accessed with the
following auxiliary functions,
-- Function: double gsl_root_fsolver_root (const gsl_root_fsolver * S)
-- Function: double gsl_root_fdfsolver_root (const gsl_root_fdfsolver *
S)
These functions return the current estimate of the root for the
solver S.
-- Function: double gsl_root_fsolver_x_lower (const gsl_root_fsolver *
S)
-- Function: double gsl_root_fsolver_x_upper (const gsl_root_fsolver *
S)
These functions return the current bracketing interval for the
solver S.
34.7 Search Stopping Parameters
===============================
A root finding procedure should stop when one of the following
conditions is true:
* A root has been found to within the user-specified precision.
* A user-specified maximum number of iterations has been reached.
* An error has occurred.
The handling of these conditions is under user control. The functions
below allow the user to test the precision of the current result in
several standard ways.
-- Function: int gsl_root_test_interval (double X_LOWER, double
X_UPPER, double EPSABS, double EPSREL)
This function tests for the convergence of the interval [X_LOWER,
X_UPPER] with absolute error EPSABS and relative error EPSREL. The
test returns 'GSL_SUCCESS' if the following condition is achieved,
|a - b| < epsabs + epsrel min(|a|,|b|)
when the interval x = [a,b] does not include the origin. If the
interval includes the origin then \min(|a|,|b|) is replaced by zero
(which is the minimum value of |x| over the interval). This
ensures that the relative error is accurately estimated for roots
close to the origin.
This condition on the interval also implies that any estimate of
the root r in the interval satisfies the same condition with
respect to the true root r^*,
|r - r^*| < epsabs + epsrel r^*
assuming that the true root r^* is contained within the interval.
-- Function: int gsl_root_test_delta (double X1, double X0, double
EPSABS, double EPSREL)
This function tests for the convergence of the sequence ..., X0, X1
with absolute error EPSABS and relative error EPSREL. The test
returns 'GSL_SUCCESS' if the following condition is achieved,
|x_1 - x_0| < epsabs + epsrel |x_1|
and returns 'GSL_CONTINUE' otherwise.
-- Function: int gsl_root_test_residual (double F, double EPSABS)
This function tests the residual value F against the absolute error
bound EPSABS. The test returns 'GSL_SUCCESS' if the following
condition is achieved,
|f| < epsabs
and returns 'GSL_CONTINUE' otherwise. This criterion is suitable
for situations where the precise location of the root, x, is
unimportant provided a value can be found where the residual,
|f(x)|, is small enough.
34.8 Root Bracketing Algorithms
===============================
The root bracketing algorithms described in this section require an
initial interval which is guaranteed to contain a root--if a and b are
the endpoints of the interval then f(a) must differ in sign from f(b).
This ensures that the function crosses zero at least once in the
interval. If a valid initial interval is used then these algorithm
cannot fail, provided the function is well-behaved.
Note that a bracketing algorithm cannot find roots of even degree,
since these do not cross the x-axis.
-- Solver: gsl_root_fsolver_bisection
The "bisection algorithm" is the simplest method of bracketing the
roots of a function. It is the slowest algorithm provided by the
library, with linear convergence.
On each iteration, the interval is bisected and the value of the
function at the midpoint is calculated. The sign of this value is
used to determine which half of the interval does not contain a
root. That half is discarded to give a new, smaller interval
containing the root. This procedure can be continued indefinitely
until the interval is sufficiently small.
At any time the current estimate of the root is taken as the
midpoint of the interval.
-- Solver: gsl_root_fsolver_falsepos
The "false position algorithm" is a method of finding roots based
on linear interpolation. Its convergence is linear, but it is
usually faster than bisection.
On each iteration a line is drawn between the endpoints (a,f(a))
and (b,f(b)) and the point where this line crosses the x-axis taken
as a "midpoint". The value of the function at this point is
calculated and its sign is used to determine which side of the
interval does not contain a root. That side is discarded to give a
new, smaller interval containing the root. This procedure can be
continued indefinitely until the interval is sufficiently small.
The best estimate of the root is taken from the linear
interpolation of the interval on the current iteration.
-- Solver: gsl_root_fsolver_brent
The "Brent-Dekker method" (referred to here as "Brent's method")
combines an interpolation strategy with the bisection algorithm.
This produces a fast algorithm which is still robust.
On each iteration Brent's method approximates the function using an
interpolating curve. On the first iteration this is a linear
interpolation of the two endpoints. For subsequent iterations the
algorithm uses an inverse quadratic fit to the last three points,
for higher accuracy. The intercept of the interpolating curve with
the x-axis is taken as a guess for the root. If it lies within the
bounds of the current interval then the interpolating point is
accepted, and used to generate a smaller interval. If the
interpolating point is not accepted then the algorithm falls back
to an ordinary bisection step.
The best estimate of the root is taken from the most recent
interpolation or bisection.
34.9 Root Finding Algorithms using Derivatives
==============================================
The root polishing algorithms described in this section require an
initial guess for the location of the root. There is no absolute
guarantee of convergence--the function must be suitable for this
technique and the initial guess must be sufficiently close to the root
for it to work. When these conditions are satisfied then convergence is
quadratic.
These algorithms make use of both the function and its derivative.
-- Derivative Solver: gsl_root_fdfsolver_newton
Newton's Method is the standard root-polishing algorithm. The
algorithm begins with an initial guess for the location of the
root. On each iteration, a line tangent to the function f is drawn
at that position. The point where this line crosses the x-axis
becomes the new guess. The iteration is defined by the following
sequence,
x_{i+1} = x_i - f(x_i)/f'(x_i)
Newton's method converges quadratically for single roots, and
linearly for multiple roots.
-- Derivative Solver: gsl_root_fdfsolver_secant
The "secant method" is a simplified version of Newton's method
which does not require the computation of the derivative on every
step.
On its first iteration the algorithm begins with Newton's method,
using the derivative to compute a first step,
x_1 = x_0 - f(x_0)/f'(x_0)
Subsequent iterations avoid the evaluation of the derivative by
replacing it with a numerical estimate, the slope of the line
through the previous two points,
x_{i+1} = x_i f(x_i) / f'_{est} where
f'_{est} = (f(x_i) - f(x_{i-1})/(x_i - x_{i-1})
When the derivative does not change significantly in the vicinity
of the root the secant method gives a useful saving.
Asymptotically the secant method is faster than Newton's method
whenever the cost of evaluating the derivative is more than 0.44
times the cost of evaluating the function itself. As with all
methods of computing a numerical derivative the estimate can suffer
from cancellation errors if the separation of the points becomes
too small.
On single roots, the method has a convergence of order (1 + \sqrt
5)/2 (approximately 1.62). It converges linearly for multiple
roots.
-- Derivative Solver: gsl_root_fdfsolver_steffenson
The "Steffenson Method"(1) provides the fastest convergence of all
the routines. It combines the basic Newton algorithm with an
Aitken "delta-squared" acceleration. If the Newton iterates are
x_i then the acceleration procedure generates a new sequence R_i,
R_i = x_i - (x_{i+1} - x_i)^2 / (x_{i+2} - 2 x_{i+1} + x_{i})
which converges faster than the original sequence under reasonable
conditions. The new sequence requires three terms before it can
produce its first value so the method returns accelerated values on
the second and subsequent iterations. On the first iteration it
returns the ordinary Newton estimate. The Newton iterate is also
returned if the denominator of the acceleration term ever becomes
zero.
As with all acceleration procedures this method can become unstable
if the function is not well-behaved.
---------- Footnotes ----------
(1) J.F. Steffensen (1873-1961). The spelling used in the name of
the function is slightly incorrect, but has been preserved to avoid
incompatibility.
34.10 Examples
==============
For any root finding algorithm we need to prepare the function to be
solved. For this example we will use the general quadratic equation
described earlier. We first need a header file ('demo_fn.h') to define
the function parameters,
struct quadratic_params
{
double a, b, c;
};
double quadratic (double x, void *params);
double quadratic_deriv (double x, void *params);
void quadratic_fdf (double x, void *params,
double *y, double *dy);
We place the function definitions in a separate file ('demo_fn.c'),
double
quadratic (double x, void *params)
{
struct quadratic_params *p
= (struct quadratic_params *) params;
double a = p->a;
double b = p->b;
double c = p->c;
return (a * x + b) * x + c;
}
double
quadratic_deriv (double x, void *params)
{
struct quadratic_params *p
= (struct quadratic_params *) params;
double a = p->a;
double b = p->b;
return 2.0 * a * x + b;
}
void
quadratic_fdf (double x, void *params,
double *y, double *dy)
{
struct quadratic_params *p
= (struct quadratic_params *) params;
double a = p->a;
double b = p->b;
double c = p->c;
*y = (a * x + b) * x + c;
*dy = 2.0 * a * x + b;
}
The first program uses the function solver 'gsl_root_fsolver_brent' for
Brent's method and the general quadratic defined above to solve the
following equation,
x^2 - 5 = 0
with solution x = \sqrt 5 = 2.236068...
#include
#include
#include
#include
#include "demo_fn.h"
#include "demo_fn.c"
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_root_fsolver_type *T;
gsl_root_fsolver *s;
double r = 0, r_expected = sqrt (5.0);
double x_lo = 0.0, x_hi = 5.0;
gsl_function F;
struct quadratic_params params = {1.0, 0.0, -5.0};
F.function = &quadratic;
F.params = ¶ms;
T = gsl_root_fsolver_brent;
s = gsl_root_fsolver_alloc (T);
gsl_root_fsolver_set (s, &F, x_lo, x_hi);
printf ("using %s method\n",
gsl_root_fsolver_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n",
"iter", "lower", "upper", "root",
"err", "err(est)");
do
{
iter++;
status = gsl_root_fsolver_iterate (s);
r = gsl_root_fsolver_root (s);
x_lo = gsl_root_fsolver_x_lower (s);
x_hi = gsl_root_fsolver_x_upper (s);
status = gsl_root_test_interval (x_lo, x_hi,
0, 0.001);
if (status == GSL_SUCCESS)
printf ("Converged:\n");
printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
iter, x_lo, x_hi,
r, r - r_expected,
x_hi - x_lo);
}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_root_fsolver_free (s);
return status;
}
Here are the results of the iterations,
$ ./a.out
using brent method
iter [ lower, upper] root err err(est)
1 [1.0000000, 5.0000000] 1.0000000 -1.2360680 4.0000000
2 [1.0000000, 3.0000000] 3.0000000 +0.7639320 2.0000000
3 [2.0000000, 3.0000000] 2.0000000 -0.2360680 1.0000000
4 [2.2000000, 3.0000000] 2.2000000 -0.0360680 0.8000000
5 [2.2000000, 2.2366300] 2.2366300 +0.0005621 0.0366300
Converged:
6 [2.2360634, 2.2366300] 2.2360634 -0.0000046 0.0005666
If the program is modified to use the bisection solver instead of
Brent's method, by changing 'gsl_root_fsolver_brent' to
'gsl_root_fsolver_bisection' the slower convergence of the Bisection
method can be observed,
$ ./a.out
using bisection method
iter [ lower, upper] root err err(est)
1 [0.0000000, 2.5000000] 1.2500000 -0.9860680 2.5000000
2 [1.2500000, 2.5000000] 1.8750000 -0.3610680 1.2500000
3 [1.8750000, 2.5000000] 2.1875000 -0.0485680 0.6250000
4 [2.1875000, 2.5000000] 2.3437500 +0.1076820 0.3125000
5 [2.1875000, 2.3437500] 2.2656250 +0.0295570 0.1562500
6 [2.1875000, 2.2656250] 2.2265625 -0.0095055 0.0781250
7 [2.2265625, 2.2656250] 2.2460938 +0.0100258 0.0390625
8 [2.2265625, 2.2460938] 2.2363281 +0.0002601 0.0195312
9 [2.2265625, 2.2363281] 2.2314453 -0.0046227 0.0097656
10 [2.2314453, 2.2363281] 2.2338867 -0.0021813 0.0048828
11 [2.2338867, 2.2363281] 2.2351074 -0.0009606 0.0024414
Converged:
12 [2.2351074, 2.2363281] 2.2357178 -0.0003502 0.0012207
The next program solves the same function using a derivative solver
instead.
#include
#include
#include
#include
#include "demo_fn.h"
#include "demo_fn.c"
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_root_fdfsolver_type *T;
gsl_root_fdfsolver *s;
double x0, x = 5.0, r_expected = sqrt (5.0);
gsl_function_fdf FDF;
struct quadratic_params params = {1.0, 0.0, -5.0};
FDF.f = &quadratic;
FDF.df = &quadratic_deriv;
FDF.fdf = &quadratic_fdf;
FDF.params = ¶ms;
T = gsl_root_fdfsolver_newton;
s = gsl_root_fdfsolver_alloc (T);
gsl_root_fdfsolver_set (s, &FDF, x);
printf ("using %s method\n",
gsl_root_fdfsolver_name (s));
printf ("%-5s %10s %10s %10s\n",
"iter", "root", "err", "err(est)");
do
{
iter++;
status = gsl_root_fdfsolver_iterate (s);
x0 = x;
x = gsl_root_fdfsolver_root (s);
status = gsl_root_test_delta (x, x0, 0, 1e-3);
if (status == GSL_SUCCESS)
printf ("Converged:\n");
printf ("%5d %10.7f %+10.7f %10.7f\n",
iter, x, x - r_expected, x - x0);
}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_root_fdfsolver_free (s);
return status;
}
Here are the results for Newton's method,
$ ./a.out
using newton method
iter root err err(est)
1 3.0000000 +0.7639320 -2.0000000
2 2.3333333 +0.0972654 -0.6666667
3 2.2380952 +0.0020273 -0.0952381
Converged:
4 2.2360689 +0.0000009 -0.0020263
Note that the error can be estimated more accurately by taking the
difference between the current iterate and next iterate rather than the
previous iterate. The other derivative solvers can be investigated by
changing 'gsl_root_fdfsolver_newton' to 'gsl_root_fdfsolver_secant' or
'gsl_root_fdfsolver_steffenson'.
34.11 References and Further Reading
====================================
For information on the Brent-Dekker algorithm see the following two
papers,
R. P. Brent, "An algorithm with guaranteed convergence for finding
a zero of a function", 'Computer Journal', 14 (1971) 422-425
J. C. P. Bus and T. J. Dekker, "Two Efficient Algorithms with
Guaranteed Convergence for Finding a Zero of a Function", 'ACM
Transactions of Mathematical Software', Vol. 1 No. 4 (1975) 330-345
35 One dimensional Minimization
*******************************
This chapter describes routines for finding minima of arbitrary
one-dimensional functions. The library provides low level components
for a variety of iterative minimizers and convergence tests. These can
be combined by the user to achieve the desired solution, with full
access to the intermediate steps of the algorithms. Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program. Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs.
The header file 'gsl_min.h' contains prototypes for the minimization
functions and related declarations. To use the minimization algorithms
to find the maximum of a function simply invert its sign.
35.1 Overview
=============
The minimization algorithms begin with a bounded region known to contain
a minimum. The region is described by a lower bound a and an upper
bound b, with an estimate of the location of the minimum x.
The value of the function at x must be less than the value of the
function at the ends of the interval,
f(a) > f(x) < f(b)
This condition guarantees that a minimum is contained somewhere within
the interval. On each iteration a new point x' is selected using one of
the available algorithms. If the new point is a better estimate of the
minimum, i.e. where f(x') < f(x), then the current estimate of the
minimum x is updated. The new point also allows the size of the bounded
interval to be reduced, by choosing the most compact set of points which
satisfies the constraint f(a) > f(x) < f(b). The interval is reduced
until it encloses the true minimum to a desired tolerance. This
provides a best estimate of the location of the minimum and a rigorous
error estimate.
Several bracketing algorithms are available within a single
framework. The user provides a high-level driver for the algorithm, and
the library provides the individual functions necessary for each of the
steps. There are three main phases of the iteration. The steps are,
* initialize minimizer state, S, for algorithm T
* update S using the iteration T
* test S for convergence, and repeat iteration if necessary
The state for the minimizers is held in a 'gsl_min_fminimizer' struct.
The updating procedure uses only function evaluations (not derivatives).
35.2 Caveats
============
Note that minimization functions can only search for one minimum at a
time. When there are several minima in the search area, the first
minimum to be found will be returned; however it is difficult to predict
which of the minima this will be. _In most cases, no error will be
reported if you try to find a minimum in an area where there is more
than one._
With all minimization algorithms it can be difficult to determine the
location of the minimum to full numerical precision. The behavior of
the function in the region of the minimum x^* can be approximated by a
Taylor expansion,
y = f(x^*) + (1/2) f''(x^*) (x - x^*)^2
and the second term of this expansion can be lost when added to the
first term at finite precision. This magnifies the error in locating
x^*, making it proportional to \sqrt \epsilon (where \epsilon is the
relative accuracy of the floating point numbers). For functions with
higher order minima, such as x^4, the magnification of the error is
correspondingly worse. The best that can be achieved is to converge to
the limit of numerical accuracy in the function values, rather than the
location of the minimum itself.
35.3 Initializing the Minimizer
===============================
-- Function: gsl_min_fminimizer * gsl_min_fminimizer_alloc (const
gsl_min_fminimizer_type * T)
This function returns a pointer to a newly allocated instance of a
minimizer of type T. For example, the following code creates an
instance of a golden section minimizer,
const gsl_min_fminimizer_type * T
= gsl_min_fminimizer_goldensection;
gsl_min_fminimizer * s
= gsl_min_fminimizer_alloc (T);
If there is insufficient memory to create the minimizer then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
-- Function: int gsl_min_fminimizer_set (gsl_min_fminimizer * S,
gsl_function * F, double X_MINIMUM, double X_LOWER, double
X_UPPER)
This function sets, or resets, an existing minimizer S to use the
function F and the initial search interval [X_LOWER, X_UPPER], with
a guess for the location of the minimum X_MINIMUM.
If the interval given does not contain a minimum, then the function
returns an error code of 'GSL_EINVAL'.
-- Function: int gsl_min_fminimizer_set_with_values (gsl_min_fminimizer
* S, gsl_function * F, double X_MINIMUM, double F_MINIMUM,
double X_LOWER, double F_LOWER, double X_UPPER, double
F_UPPER)
This function is equivalent to 'gsl_min_fminimizer_set' but uses
the values F_MINIMUM, F_LOWER and F_UPPER instead of computing
'f(x_minimum)', 'f(x_lower)' and 'f(x_upper)'.
-- Function: void gsl_min_fminimizer_free (gsl_min_fminimizer * S)
This function frees all the memory associated with the minimizer S.
-- Function: const char * gsl_min_fminimizer_name (const
gsl_min_fminimizer * S)
This function returns a pointer to the name of the minimizer. For
example,
printf ("s is a '%s' minimizer\n",
gsl_min_fminimizer_name (s));
would print something like 's is a 'brent' minimizer'.
35.4 Providing the function to minimize
=======================================
You must provide a continuous function of one variable for the
minimizers to operate on. In order to allow for general parameters the
functions are defined by a 'gsl_function' data type (*note Providing the
function to solve::).
35.5 Iteration
==============
The following functions drive the iteration of each algorithm. Each
function performs one iteration to update the state of any minimizer of
the corresponding type. The same functions work for all minimizers so
that different methods can be substituted at runtime without
modifications to the code.
-- Function: int gsl_min_fminimizer_iterate (gsl_min_fminimizer * S)
This function performs a single iteration of the minimizer S. If
the iteration encounters an unexpected problem then an error code
will be returned,
'GSL_EBADFUNC'
the iteration encountered a singular point where the function
evaluated to 'Inf' or 'NaN'.
'GSL_FAILURE'
the algorithm could not improve the current best approximation
or bounding interval.
The minimizer maintains a current best estimate of the position of
the minimum at all times, and the current interval bounding the minimum.
This information can be accessed with the following auxiliary functions,
-- Function: double gsl_min_fminimizer_x_minimum (const
gsl_min_fminimizer * S)
This function returns the current estimate of the position of the
minimum for the minimizer S.
-- Function: double gsl_min_fminimizer_x_upper (const
gsl_min_fminimizer * S)
-- Function: double gsl_min_fminimizer_x_lower (const
gsl_min_fminimizer * S)
These functions return the current upper and lower bound of the
interval for the minimizer S.
-- Function: double gsl_min_fminimizer_f_minimum (const
gsl_min_fminimizer * S)
-- Function: double gsl_min_fminimizer_f_upper (const
gsl_min_fminimizer * S)
-- Function: double gsl_min_fminimizer_f_lower (const
gsl_min_fminimizer * S)
These functions return the value of the function at the current
estimate of the minimum and at the upper and lower bounds of the
interval for the minimizer S.
35.6 Stopping Parameters
========================
A minimization procedure should stop when one of the following
conditions is true:
* A minimum has been found to within the user-specified precision.
* A user-specified maximum number of iterations has been reached.
* An error has occurred.
The handling of these conditions is under user control. The function
below allows the user to test the precision of the current result.
-- Function: int gsl_min_test_interval (double X_LOWER, double X_UPPER,
double EPSABS, double EPSREL)
This function tests for the convergence of the interval [X_LOWER,
X_UPPER] with absolute error EPSABS and relative error EPSREL. The
test returns 'GSL_SUCCESS' if the following condition is achieved,
|a - b| < epsabs + epsrel min(|a|,|b|)
when the interval x = [a,b] does not include the origin. If the
interval includes the origin then \min(|a|,|b|) is replaced by zero
(which is the minimum value of |x| over the interval). This
ensures that the relative error is accurately estimated for minima
close to the origin.
This condition on the interval also implies that any estimate of
the minimum x_m in the interval satisfies the same condition with
respect to the true minimum x_m^*,
|x_m - x_m^*| < epsabs + epsrel x_m^*
assuming that the true minimum x_m^* is contained within the
interval.
35.7 Minimization Algorithms
============================
The minimization algorithms described in this section require an initial
interval which is guaranteed to contain a minimum--if a and b are the
endpoints of the interval and x is an estimate of the minimum then f(a)
> f(x) < f(b). This ensures that the function has at least one minimum
somewhere in the interval. If a valid initial interval is used then
these algorithm cannot fail, provided the function is well-behaved.
-- Minimizer: gsl_min_fminimizer_goldensection
The "golden section algorithm" is the simplest method of bracketing
the minimum of a function. It is the slowest algorithm provided by
the library, with linear convergence.
On each iteration, the algorithm first compares the subintervals
from the endpoints to the current minimum. The larger subinterval
is divided in a golden section (using the famous ratio (3-\sqrt
5)/2 = 0.3819660...) and the value of the function at this new
point is calculated. The new value is used with the constraint
f(a') > f(x') < f(b') to a select new interval containing the
minimum, by discarding the least useful point. This procedure can
be continued indefinitely until the interval is sufficiently small.
Choosing the golden section as the bisection ratio can be shown to
provide the fastest convergence for this type of algorithm.
-- Minimizer: gsl_min_fminimizer_brent
The "Brent minimization algorithm" combines a parabolic
interpolation with the golden section algorithm. This produces a
fast algorithm which is still robust.
The outline of the algorithm can be summarized as follows: on each
iteration Brent's method approximates the function using an
interpolating parabola through three existing points. The minimum
of the parabola is taken as a guess for the minimum. If it lies
within the bounds of the current interval then the interpolating
point is accepted, and used to generate a smaller interval. If the
interpolating point is not accepted then the algorithm falls back
to an ordinary golden section step. The full details of Brent's
method include some additional checks to improve convergence.
-- Minimizer: gsl_min_fminimizer_quad_golden
This is a variant of Brent's algorithm which uses the safeguarded
step-length algorithm of Gill and Murray.
35.8 Examples
=============
The following program uses the Brent algorithm to find the minimum of
the function f(x) = \cos(x) + 1, which occurs at x = \pi. The starting
interval is (0,6), with an initial guess for the minimum of 2.
#include
#include
#include
#include
double fn1 (double x, void * params)
{
(void)(params); /* avoid unused parameter warning */
return cos(x) + 1.0;
}
int
main (void)
{
int status;
int iter = 0, max_iter = 100;
const gsl_min_fminimizer_type *T;
gsl_min_fminimizer *s;
double m = 2.0, m_expected = M_PI;
double a = 0.0, b = 6.0;
gsl_function F;
F.function = &fn1;
F.params = 0;
T = gsl_min_fminimizer_brent;
s = gsl_min_fminimizer_alloc (T);
gsl_min_fminimizer_set (s, &F, m, a, b);
printf ("using %s method\n",
gsl_min_fminimizer_name (s));
printf ("%5s [%9s, %9s] %9s %10s %9s\n",
"iter", "lower", "upper", "min",
"err", "err(est)");
printf ("%5d [%.7f, %.7f] %.7f %+.7f %.7f\n",
iter, a, b,
m, m - m_expected, b - a);
do
{
iter++;
status = gsl_min_fminimizer_iterate (s);
m = gsl_min_fminimizer_x_minimum (s);
a = gsl_min_fminimizer_x_lower (s);
b = gsl_min_fminimizer_x_upper (s);
status
= gsl_min_test_interval (a, b, 0.001, 0.0);
if (status == GSL_SUCCESS)
printf ("Converged:\n");
printf ("%5d [%.7f, %.7f] "
"%.7f %+.7f %.7f\n",
iter, a, b,
m, m - m_expected, b - a);
}
while (status == GSL_CONTINUE && iter < max_iter);
gsl_min_fminimizer_free (s);
return status;
}
Here are the results of the minimization procedure.
$ ./a.out
using brent method
iter [ lower, upper] min err err(est)
0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
1 [2.0000000, 6.0000000] 3.5278640 +0.3862713 4.0000000
2 [2.0000000, 3.5278640] 3.1748217 +0.0332290 1.5278640
3 [2.0000000, 3.1748217] 3.1264576 -0.0151351 1.1748217
4 [3.1264576, 3.1748217] 3.1414743 -0.0001183 0.0483641
5 [3.1414743, 3.1748217] 3.1415930 +0.0000004 0.0333474
Converged:
6 [3.1414743, 3.1415930] 3.1415927 +0.0000000 0.0001187
35.9 References and Further Reading
===================================
Further information on Brent's algorithm is available in the following
book,
Richard Brent, 'Algorithms for minimization without derivatives',
Prentice-Hall (1973), republished by Dover in paperback (2002),
ISBN 0-486-41998-3.
36 Multidimensional Root-Finding
********************************
This chapter describes functions for multidimensional root-finding
(solving nonlinear systems with n equations in n unknowns). The library
provides low level components for a variety of iterative solvers and
convergence tests. These can be combined by the user to achieve the
desired solution, with full access to the intermediate steps of the
iteration. Each class of methods uses the same framework, so that you
can switch between solvers at runtime without needing to recompile your
program. Each instance of a solver keeps track of its own state,
allowing the solvers to be used in multi-threaded programs. The solvers
are based on the original Fortran library MINPACK.
The header file 'gsl_multiroots.h' contains prototypes for the
multidimensional root finding functions and related declarations.
36.1 Overview
=============
The problem of multidimensional root finding requires the simultaneous
solution of n equations, f_i, in n variables, x_i,
f_i (x_1, ..., x_n) = 0 for i = 1 ... n.
In general there are no bracketing methods available for n dimensional
systems, and no way of knowing whether any solutions exist. All
algorithms proceed from an initial guess using a variant of the Newton
iteration,
x -> x' = x - J^{-1} f(x)
where x, f are vector quantities and J is the Jacobian matrix J_{ij} = d
f_i / d x_j. Additional strategies can be used to enlarge the region of
convergence. These include requiring a decrease in the norm |f| on each
step proposed by Newton's method, or taking steepest-descent steps in
the direction of the negative gradient of |f|.
Several root-finding algorithms are available within a single
framework. The user provides a high-level driver for the algorithms,
and the library provides the individual functions necessary for each of
the steps. There are three main phases of the iteration. The steps
are,
* initialize solver state, S, for algorithm T
* update S using the iteration T
* test S for convergence, and repeat iteration if necessary
The evaluation of the Jacobian matrix can be problematic, either because
programming the derivatives is intractable or because computation of the
n^2 terms of the matrix becomes too expensive. For these reasons the
algorithms provided by the library are divided into two classes
according to whether the derivatives are available or not.
The state for solvers with an analytic Jacobian matrix is held in a
'gsl_multiroot_fdfsolver' struct. The updating procedure requires both
the function and its derivatives to be supplied by the user.
The state for solvers which do not use an analytic Jacobian matrix is
held in a 'gsl_multiroot_fsolver' struct. The updating procedure uses
only function evaluations (not derivatives). The algorithms estimate
the matrix J or J^{-1} by approximate methods.
36.2 Initializing the Solver
============================
The following functions initialize a multidimensional solver, either
with or without derivatives. The solver itself depends only on the
dimension of the problem and the algorithm and can be reused for
different problems.
-- Function: gsl_multiroot_fsolver * gsl_multiroot_fsolver_alloc (const
gsl_multiroot_fsolver_type * T, size_t N)
This function returns a pointer to a newly allocated instance of a
solver of type T for a system of N dimensions. For example, the
following code creates an instance of a hybrid solver, to solve a
3-dimensional system of equations.
const gsl_multiroot_fsolver_type * T
= gsl_multiroot_fsolver_hybrid;
gsl_multiroot_fsolver * s
= gsl_multiroot_fsolver_alloc (T, 3);
If there is insufficient memory to create the solver then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
-- Function: gsl_multiroot_fdfsolver * gsl_multiroot_fdfsolver_alloc
(const gsl_multiroot_fdfsolver_type * T, size_t N)
This function returns a pointer to a newly allocated instance of a
derivative solver of type T for a system of N dimensions. For
example, the following code creates an instance of a Newton-Raphson
solver, for a 2-dimensional system of equations.
const gsl_multiroot_fdfsolver_type * T
= gsl_multiroot_fdfsolver_newton;
gsl_multiroot_fdfsolver * s =
gsl_multiroot_fdfsolver_alloc (T, 2);
If there is insufficient memory to create the solver then the
function returns a null pointer and the error handler is invoked
with an error code of 'GSL_ENOMEM'.
-- Function: int gsl_multiroot_fsolver_set (gsl_multiroot_fsolver * S,
gsl_multiroot_function * F, const gsl_vector * X)
-- Function: int gsl_multiroot_fdfsolver_set (gsl_multiroot_fdfsolver *
S, gsl_multiroot_function_fdf * FDF, const gsl_vector * X)
These functions set, or reset, an existing solver S to use the
function F or function and derivative FDF, and the initial guess X.
Note that the initial position is copied from X, this argument is
not modified by subsequent iterations.
-- Function: void gsl_multiroot_fsolver_free (gsl_multiroot_fsolver *
S)
-- Function: void gsl_multiroot_fdfsolver_free (gsl_multiroot_fdfsolver
* S)
These functions free all the memory associated with the solver S.
-- Function: const char * gsl_multiroot_fsolver_name (const
gsl_multiroot_fsolver * S)
-- Function: const char * gsl_multiroot_fdfsolver_name (const
gsl_multiroot_fdfsolver * S)
These functions return a pointer to the name of the solver. For
example,
printf ("s is a '%s' solver\n",
gsl_multiroot_fdfsolver_name (s));
would print something like 's is a 'newton' solver'.
36.3 Providing the function to solve
====================================
You must provide n functions of n variables for the root finders to
operate on. In order to allow for general parameters the functions are
defined by the following data types:
-- Data Type: gsl_multiroot_function
This data type defines a general system of functions with
parameters.
'int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
this function should store the vector result f(x,params) in F
for argument X and parameters PARAMS, returning an appropriate
error code if the function cannot be computed.
'size_t n'
the dimension of the system, i.e. the number of components of
the vectors X and F.
'void * params'
a pointer to the parameters of the function.
Here is an example using Powell's test function,
f_1(x) = A x_0 x_1 - 1,
f_2(x) = exp(-x_0) + exp(-x_1) - (1 + 1/A)
with A = 10^4. The following code defines a 'gsl_multiroot_function'
system 'F' which you could pass to a solver:
struct powell_params { double A; };
int
powell (gsl_vector * x, void * p, gsl_vector * f) {
struct powell_params * params
= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, (exp(-x0) + exp(-x1)
- (1.0 + 1.0/A)));
return GSL_SUCCESS
}
gsl_multiroot_function F;
struct powell_params params = { 10000.0 };
F.f = &powell;
F.n = 2;
F.params = ¶ms;
-- Data Type: gsl_multiroot_function_fdf
This data type defines a general system of functions with
parameters and the corresponding Jacobian matrix of derivatives,
'int (* f) (const gsl_vector * X, void * PARAMS, gsl_vector * F)'
this function should store the vector result f(x,params) in F
for argument X and parameters PARAMS, returning an appropriate
error code if the function cannot be computed.
'int (* df) (const gsl_vector * X, void * PARAMS, gsl_matrix * J)'
this function should store the N-by-N matrix result J_ij = d
f_i(x,params) / d x_j in J for argument X and parameters
PARAMS, returning an appropriate error code if the function
cannot be computed.
'int (* fdf) (const gsl_vector * X, void * PARAMS, gsl_vector * F, gsl_matrix * J)'
This function should set the values of the F and J as above,
for arguments X and parameters PARAMS. This function provides
an optimization of the separate functions for f(x) and
J(x)--it is always faster to compute the function and its
derivative at the same time.
'size_t n'
the dimension of the system, i.e. the number of components of
the vectors X and F.
'void * params'
a pointer to the parameters of the function.
The example of Powell's test function defined above can be extended to
include analytic derivatives using the following code,
int
powell_df (gsl_vector * x, void * p, gsl_matrix * J)
{
struct powell_params * params
= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
gsl_matrix_set (J, 1, 0, -exp(-x0));
gsl_matrix_set (J, 1, 1, -exp(-x1));
return GSL_SUCCESS
}
int
powell_fdf (gsl_vector * x, void * p,
gsl_matrix * f, gsl_matrix * J) {
struct powell_params * params
= (struct powell_params *)p;
const double A = (params->A);
const double x0 = gsl_vector_get(x,0);
const double x1 = gsl_vector_get(x,1);
const double u0 = exp(-x0);
const double u1 = exp(-x1);
gsl_vector_set (f, 0, A * x0 * x1 - 1);
gsl_vector_set (f, 1, u0 + u1 - (1 + 1/A));
gsl_matrix_set (J, 0, 0, A * x1);
gsl_matrix_set (J, 0, 1, A * x0);
gsl_matrix_set (J, 1, 0, -u0);
gsl_matrix_set (J, 1, 1, -u1);
return GSL_SUCCESS
}
gsl_multiroot_function_fdf FDF;
FDF.f = &powell_f;
FDF.df = &powell_df;
FDF.fdf = &powell_fdf;
FDF.n = 2;
FDF.params = 0;
Note that the function 'powell_fdf' is able to reuse existing terms from
the function when calculating the Jacobian, thus saving time.
36.4 Iteration
==============
The following functions drive the iteration of each algorithm. Each
function performs one iteration to update the state of any solver of the
corresponding type. The same functions work for all solvers so that
different methods can be substituted at runtime without modifications to
the code.
-- Function: int gsl_multiroot_fsolver_iterate (gsl_multiroot_fsolver *
S)
-- Function: int gsl_multiroot_fdfsolver_iterate
(gsl_multiroot_fdfsolver * S)
These functions perform a single iteration of the solver S. If the
iteration encounters an unexpected problem then an error code will
be returned,
'GSL_EBADFUNC'
the iteration encountered a singular point where the function
or its derivative evaluated to 'Inf' or 'NaN'.
'GSL_ENOPROG'
the iteration is not making any progress, preventing the
algorithm from continuing.
The solver maintains a current best estimate of the root 's->x' and
its function value 's->f' at all times. This information can be
accessed with the following auxiliary functions,
-- Function: gsl_vector * gsl_multiroot_fsolver_root (const
gsl_multiroot_fsolver * S)
-- Function: gsl_vector * gsl_multiroot_fdfsolver_root (const
gsl_multiroot_fdfsolver * S)
These functions return the current estimate of the root for the
solver S, given by 's->x'.
-- Function: gsl_vector * gsl_multiroot_fsolver_f (const
gsl_multiroot_fsolver * S)
-- Function: gsl_vector * gsl_multiroot_fdfsolver_f (const
gsl_multiroot_fdfsolver * S)
These functions return the function value f(x) at the current
estimate of the root for the solver S, given by 's->f'.
-- Function: gsl_vector * gsl_multiroot_fsolver_dx (const
gsl_multiroot_fsolver * S)
-- Function: gsl_vector * gsl_multiroot_fdfsolver_dx (const
gsl_multiroot_fdfsolver * S)
These functions return the last step dx taken by the solver S,
given by 's->dx'.
36.5 Search Stopping Parameters
===============================
A root finding procedure should stop when one of the following
conditions is true:
* A multidimensional root has been found to within the user-specified
precision.
* A user-specified maximum number of iterations has been reached.
* An error has occurred.
The handling of these conditions is under user control. The functions
below allow the user to test the precision of the current result in
several standard ways.
-- Function: int gsl_multiroot_test_delta (const gsl_vector * DX, const
gsl_vector * X, double EPSABS, double EPSREL)
This function tests for the convergence of the sequence by
comparing the last step DX with the absolute error EPSABS and
relative error EPSREL to the current position X. The test returns
'GSL_SUCCESS' if the following condition is achieved,
|dx_i| < epsabs + epsrel |x_i|
for each component of X and returns 'GSL_CONTINUE' otherwise.
-- Function: int gsl_multiroot_test_residual (const gsl_vector * F,
double EPSABS)
This function tests the residual value F against the absolute error
bound EPSABS. The test returns 'GSL_SUCCESS' if the following
condition is achieved,
\sum_i |f_i| < epsabs
and returns 'GSL_CONTINUE' otherwise. This criterion is suitable
for situations where the precise location of the root, x, is
unimportant provided a value can be found where the residual is
small enough.
36.6 Algorithms using Derivatives
=================================
The root finding algorithms described in this section make use of both
the function and its derivative. They require an initial guess for the
location of the root, but there is no absolute guarantee of
convergence--the function must be suitable for this technique and the
initial guess must be sufficiently close to the root for it to work.
When the conditions are satisfied then convergence is quadratic.
-- Derivative Solver: gsl_multiroot_fdfsolver_hybridsj
This is a modified version of Powell's Hybrid method as implemented
in the HYBRJ algorithm in MINPACK. Minpack was written by Jorge J.
More', Burton S. Garbow and Kenneth E. Hillstrom. The Hybrid
algorithm retains the fast convergence of Newton's method but will
also reduce the residual when Newton's method is unreliable.
The algorithm uses a generalized trust region to keep each step
under control. In order to be accepted a proposed new position x'
must satisfy the condition |D (x' - x)| < \delta, where D is a
diagonal scaling matrix and \delta is the size of the trust region.
The components of D are computed internally, using the column norms
of the Jacobian to estimate the sensitivity of the residual to each
component of x. This improves the behavior of the algorithm for
badly scaled functions.
On each iteration the algorithm first determines the standard
Newton step by solving the system J dx = - f. If this step falls
inside the trust region it is used as a trial step in the next
stage. If not, the algorithm uses the linear combination of the
Newton and gradient directions which is predicted to minimize the
norm of the function while staying inside the trust region,
dx = - \alpha J^{-1} f(x) - \beta \nabla |f(x)|^2.
This combination of Newton and gradient directions is referred to
as a "dogleg step".
The proposed step is now tested by evaluating the function at the
resulting point, x'. If the step reduces the norm of the function
sufficiently then it is accepted and size of the trust region is
increased. If the proposed step fails to improve the solution then
the size of the trust region is decreased and another trial step is
computed.
The speed of the algorithm is increased by computing the changes to
the Jacobian approximately, using a rank-1 update. If two
successive attempts fail to reduce the residual then the full
Jacobian is recomputed. The algorithm also monitors the progress
of the solution and returns an error if several steps fail to make
any improvement,
'GSL_ENOPROG'
the iteration is not making any progress, preventing the
algorithm from continuing.
'GSL_ENOPROGJ'
re-evaluations of the Jacobian indicate that the iteration is
not making any progress, preventing the algorithm from
continuing.
-- Derivative Solver: gsl_multiroot_fdfsolver_hybridj
This algorithm is an unscaled version of 'hybridsj'. The steps are
controlled by a spherical trust region |x' - x| < \delta, instead
of a generalized region. This can be useful if the generalized
region estimated by 'hybridsj' is inappropriate.
-- Derivative Solver: gsl_multiroot_fdfsolver_newton
Newton's Method is the standard root-polishing algorithm. The
algorithm begins with an initial guess for the location of the
solution. On each iteration a linear approximation to the function
F is used to estimate the step which will zero all the components
of the residual. The iteration is defined by the following
sequence,
x -> x' = x - J^{-1} f(x)
where the Jacobian matrix J is computed from the derivative
functions provided by F. The step dx is obtained by solving the
linear system,
J dx = - f(x)
using LU decomposition. If the Jacobian matrix is singular, an
error code of 'GSL_EDOM' is returned.
-- Derivative Solver: gsl_multiroot_fdfsolver_gnewton
This is a modified version of Newton's method which attempts to
improve global convergence by requiring every step to reduce the
Euclidean norm of the residual, |f(x)|. If the Newton step leads
to an increase in the norm then a reduced step of relative size,
t = (\sqrt(1 + 6 r) - 1) / (3 r)
is proposed, with r being the ratio of norms |f(x')|^2/|f(x)|^2.
This procedure is repeated until a suitable step size is found.
36.7 Algorithms without Derivatives
===================================
The algorithms described in this section do not require any derivative
information to be supplied by the user. Any derivatives needed are
approximated by finite differences. Note that if the
finite-differencing step size chosen by these routines is inappropriate,
an explicit user-supplied numerical derivative can always be used with
the algorithms described in the previous section.
-- Solver: gsl_multiroot_fsolver_hybrids
This is a version of the Hybrid algorithm which replaces calls to
the Jacobian function by its finite difference approximation. The
finite difference approximation is computed using
'gsl_multiroots_fdjac' with a relative step size of
'GSL_SQRT_DBL_EPSILON'. Note that this step size will not be
suitable for all problems.
-- Solver: gsl_multiroot_fsolver_hybrid
This is a finite difference version of the Hybrid algorithm without
internal scaling.
-- Solver: gsl_multiroot_fsolver_dnewton
The "discrete Newton algorithm" is the simplest method of solving a
multidimensional system. It uses the Newton iteration
x -> x - J^{-1} f(x)
where the Jacobian matrix J is approximated by taking finite
differences of the function F. The approximation scheme used by
this implementation is,
J_{ij} = (f_i(x + \delta_j) - f_i(x)) / \delta_j
where \delta_j is a step of size \sqrt\epsilon |x_j| with \epsilon
being the machine precision (\epsilon \approx 2.22 \times 10^-16).
The order of convergence of Newton's algorithm is quadratic, but
the finite differences require n^2 function evaluations on each
iteration. The algorithm may become unstable if the finite
differences are not a good approximation to the true derivatives.
-- Solver: gsl_multiroot_fsolver_broyden
The "Broyden algorithm" is a version of the discrete Newton
algorithm which attempts to avoids the expensive update of the
Jacobian matrix on each iteration. The changes to the Jacobian are
also approximated, using a rank-1 update,
J^{-1} \to J^{-1} - (J^{-1} df - dx) dx^T J^{-1} / dx^T J^{-1} df
where the vectors dx and df are the changes in x and f. On the
first iteration the inverse Jacobian is estimated using finite
differences, as in the discrete Newton algorithm.
This approximation gives a fast update but is unreliable if the
changes are not small, and the estimate of the inverse Jacobian
becomes worse as time passes. The algorithm has a tendency to
become unstable unless it starts close to the root. The Jacobian
is refreshed if this instability is detected (consult the source
for details).
This algorithm is included only for demonstration purposes, and is
not recommended for serious use.
36.8 Examples
=============
The multidimensional solvers are used in a similar way to the
one-dimensional root finding algorithms. This first example
demonstrates the 'hybrids' scaled-hybrid algorithm, which does not
require derivatives. The program solves the Rosenbrock system of
equations,
f_1 (x, y) = a (1 - x)
f_2 (x, y) = b (y - x^2)
with a = 1, b = 10. The solution of this system lies at (x,y) = (1,1)
in a narrow valley.
The first stage of the program is to define the system of equations,
#include
#include
#include
#include
struct rparams
{
double a;
double b;
};
int
rosenbrock_f (const gsl_vector * x, void *params,
gsl_vector * f)
{
double a = ((struct rparams *) params)->a;
double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);
const double x1 = gsl_vector_get (x, 1);
const double y0 = a * (1 - x0);
const double y1 = b * (x1 - x0 * x0);
gsl_vector_set (f, 0, y0);
gsl_vector_set (f, 1, y1);
return GSL_SUCCESS;
}
The main program begins by creating the function object 'f', with the
arguments '(x,y)' and parameters '(a,b)'. The solver 's' is initialized
to use this function, with the 'hybrids' method.
int
main (void)
{
const gsl_multiroot_fsolver_type *T;
gsl_multiroot_fsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function f = {&rosenbrock_f, n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fsolver_hybrids;
s = gsl_multiroot_fsolver_alloc (T, 2);
gsl_multiroot_fsolver_set (s, &f, x);
print_state (iter, s);
do
{
iter++;
status = gsl_multiroot_fsolver_iterate (s);
print_state (iter, s);
if (status) /* check if solver is stuck */
break;
status =
gsl_multiroot_test_residual (s->f, 1e-7);
}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fsolver_free (s);
gsl_vector_free (x);
return 0;
}
Note that it is important to check the return status of each solver
step, in case the algorithm becomes stuck. If an error condition is
detected, indicating that the algorithm cannot proceed, then the error
can be reported to the user, a new starting point chosen or a different
algorithm used.
The intermediate state of the solution is displayed by the following
function. The solver state contains the vector 's->x' which is the
current position, and the vector 's->f' with corresponding function
values.
int
print_state (size_t iter, gsl_multiroot_fsolver * s)
{
printf ("iter = %3u x = % .3f % .3f "
"f(x) = % .3e % .3e\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
gsl_vector_get (s->f, 0),
gsl_vector_get (s->f, 1));
}
Here are the results of running the program. The algorithm is started
at (-10,-5) far from the solution. Since the solution is hidden in a
narrow valley the earliest steps follow the gradient of the function
downhill, in an attempt to reduce the large value of the residual. Once
the root has been approximately located, on iteration 8, the Newton
behavior takes over and convergence is very rapid.
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 1 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 2 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01
iter = 3 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01
iter = 4 x = -3.976 24.827 f(x) = 4.976e+00 9.020e+01
iter = 5 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01
iter = 6 x = -1.274 -5.680 f(x) = 2.274e+00 -7.302e+01
iter = 7 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00
iter = 8 x = 0.249 0.298 f(x) = 7.511e-01 2.359e+00
iter = 9 x = 1.000 0.878 f(x) = 1.268e-10 -1.218e+00
iter = 10 x = 1.000 0.989 f(x) = 1.124e-11 -1.080e-01
iter = 11 x = 1.000 1.000 f(x) = 0.000e+00 0.000e+00
status = success
Note that the algorithm does not update the location on every iteration.
Some iterations are used to adjust the trust-region parameter, after
trying a step which was found to be divergent, or to recompute the
Jacobian, when poor convergence behavior is detected.
The next example program adds derivative information, in order to
accelerate the solution. There are two derivative functions
'rosenbrock_df' and 'rosenbrock_fdf'. The latter computes both the
function and its derivative simultaneously. This allows the
optimization of any common terms. For simplicity we substitute calls to
the separate 'f' and 'df' functions at this point in the code below.
int
rosenbrock_df (const gsl_vector * x, void *params,
gsl_matrix * J)
{
const double a = ((struct rparams *) params)->a;
const double b = ((struct rparams *) params)->b;
const double x0 = gsl_vector_get (x, 0);
const double df00 = -a;
const double df01 = 0;
const double df10 = -2 * b * x0;
const double df11 = b;
gsl_matrix_set (J, 0, 0, df00);
gsl_matrix_set (J, 0, 1, df01);
gsl_matrix_set (J, 1, 0, df10);
gsl_matrix_set (J, 1, 1, df11);
return GSL_SUCCESS;
}
int
rosenbrock_fdf (const gsl_vector * x, void *params,
gsl_vector * f, gsl_matrix * J)
{
rosenbrock_f (x, params, f);
rosenbrock_df (x, params, J);
return GSL_SUCCESS;
}
The main program now makes calls to the corresponding 'fdfsolver'
versions of the functions,
int
main (void)
{
const gsl_multiroot_fdfsolver_type *T;
gsl_multiroot_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = 2;
struct rparams p = {1.0, 10.0};
gsl_multiroot_function_fdf f = {&rosenbrock_f,
&rosenbrock_df,
&rosenbrock_fdf,
n, &p};
double x_init[2] = {-10.0, -5.0};
gsl_vector *x = gsl_vector_alloc (n);
gsl_vector_set (x, 0, x_init[0]);
gsl_vector_set (x, 1, x_init[1]);
T = gsl_multiroot_fdfsolver_gnewton;
s = gsl_multiroot_fdfsolver_alloc (T, n);
gsl_multiroot_fdfsolver_set (s, &f, x);
print_state (iter, s);
do
{
iter++;
status = gsl_multiroot_fdfsolver_iterate (s);
print_state (iter, s);
if (status)
break;
status = gsl_multiroot_test_residual (s->f, 1e-7);
}
while (status == GSL_CONTINUE && iter < 1000);
printf ("status = %s\n", gsl_strerror (status));
gsl_multiroot_fdfsolver_free (s);
gsl_vector_free (x);
return 0;
}
The addition of derivative information to the 'hybrids' solver does not
make any significant difference to its behavior, since it able to
approximate the Jacobian numerically with sufficient accuracy. To
illustrate the behavior of a different derivative solver we switch to
'gnewton'. This is a traditional Newton solver with the constraint that
it scales back its step if the full step would lead "uphill". Here is
the output for the 'gnewton' algorithm,
iter = 0 x = -10.000 -5.000 f(x) = 1.100e+01 -1.050e+03
iter = 1 x = -4.231 -65.317 f(x) = 5.231e+00 -8.321e+02
iter = 2 x = 1.000 -26.358 f(x) = -8.882e-16 -2.736e+02
iter = 3 x = 1.000 1.000 f(x) = -2.220e-16 -4.441e-15
status = success
The convergence is much more rapid, but takes a wide excursion out to
the point (-4.23,-65.3). This could cause the algorithm to go astray in
a realistic application. The hybrid algorithm follows the downhill path
to the solution more reliably.
36.9 References and Further Reading
===================================
The original version of the Hybrid method is described in the following
articles by Powell,
M.J.D. Powell, "A Hybrid Method for Nonlinear Equations" (Chap 6, p
87-114) and "A Fortran Subroutine for Solving systems of Nonlinear
Algebraic Equations" (Chap 7, p 115-161), in 'Numerical Methods for
Nonlinear Algebraic Equations', P. Rabinowitz, editor. Gordon and
Breach, 1970.
The following papers are also relevant to the algorithms described in
this section,
J.J. More', M.Y. Cosnard, "Numerical Solution of Nonlinear
Equations", 'ACM Transactions on Mathematical Software', Vol 5, No
1, (1979), p 64-85
C.G. Broyden, "A Class of Methods for Solving Nonlinear
Simultaneous Equations", 'Mathematics of Computation', Vol 19
(1965), p 577-593
J.J. More', B.S. Garbow, K.E. Hillstrom, "Testing Unconstrained
Optimization Software", ACM Transactions on Mathematical Software,
Vol 7, No 1 (1981), p 17-41
37 Multidimensional Minimization
********************************
This chapter describes routines for finding minima of arbitrary
multidimensional functions. The library provides low level components
for a variety of iterative minimizers and convergence tests. These can
be combined by the user to achieve the desired solution, while providing
full access to the intermediate steps of the algorithms. Each class of
methods uses the same framework, so that you can switch between
minimizers at runtime without needing to recompile your program. Each
instance of a minimizer keeps track of its own state, allowing the
minimizers to be used in multi-threaded programs. The minimization
algorithms can be used to maximize a function by inverting its sign.
The header file 'gsl_multimin.h' contains prototypes for the
minimization functions and related declarations.
37.1 Overview
=============
The problem of multidimensional minimization requires finding a point x
such that the scalar function,
f(x_1, ..., x_n)
takes a value which is lower than at any neighboring point. For smooth
functions the gradient g = \nabla f vanishes at the minimum. In general
there are no bracketing methods available for the minimization of
n-dimensional functions. The algorithms proceed from an initial guess
using a search algorithm which attempts to move in a downhill direction.
Algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance. The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true n-dimensional minimum is
found.
Algorithms which do not require the gradient of the function use
different strategies. For example, the Nelder-Mead Simplex algorithm
maintains n+1 trial parameter vectors as the vertices of a n-dimensional
simplex. On each iteration it tries to improve the worst vertex of the
simplex by geometrical transformations. The iterations are continued
until the overall size of the simplex has decreased sufficiently.
Both types of algorithms use a standard framework. The user provides
a high-level driver for the algorithms, and the library provides the
individual functions necessary for each of the steps. There are three
main phases of the iteration. The steps are,
* initialize minimizer state, S, for algorithm T
* update S using the iteration T
* test S for convergence, and repeat iteration if necessary
Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself. The state for the minimizers is held in a
'gsl_multimin_fdfminimizer' struct or a 'gsl_multimin_fminimizer'
struct.
37.2 Caveats
============
Note that the minimization algorithms can only search for one local
minimum at a time. When there are several local minima in the search
area, the first minimum to be found will be returned; however it is
difficult to predict which of the minima this will be. In most cases,
no error will be reported if you try to find a local minimum in an area
where there is more than one.
It is also important to note that the minimization algorithms find
local minima; there is no way to determine whether a minimum is a global
minimum of the function in question.
37.3 Initializing the Multidimensional Minimizer
================================================
The following function initializes a multidimensional minimizer. The
minimizer itself depends only on the dimension of the problem and the
algorithm and can be reused for different problems.
-- Function: gsl_multimin_fdfminimizer *
gsl_multimin_fdfminimizer_alloc (const
gsl_multimin_fdfminimizer_type * T, size_t N)
-- Function: gsl_multimin_fminimizer * gsl_multimin_fminimizer_alloc
(const gsl_multimin_fminimizer_type * T, size_t N)
This function returns a pointer to a newly allocated instance of a
minimizer of type T for an N-dimension function. If there is
insufficient memory to create the minimizer then the function
returns a null pointer and the error handler is invoked with an
error code of 'GSL_ENOMEM'.
-- Function: int gsl_multimin_fdfminimizer_set
(gsl_multimin_fdfminimizer * S, gsl_multimin_function_fdf *
FDF, const gsl_vector * X, double STEP_SIZE, double TOL)
-- Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer *
S, gsl_multimin_function * F, const gsl_vector * X, const
gsl_vector * STEP_SIZE)
The function 'gsl_multimin_fdfminimizer_set' initializes the
minimizer S to minimize the function FDF starting from the initial
point X. The size of the first trial step is given by STEP_SIZE.
The accuracy of the line minimization is specified by TOL. The
precise meaning of this parameter depends on the method used.
Typically the line minimization is considered successful if the
gradient of the function g is orthogonal to the current search
direction p to a relative accuracy of TOL, where dot(p,g) < tol |p|
|g|. A TOL value of 0.1 is suitable for most purposes, since line
minimization only needs to be carried out approximately. Note that
setting TOL to zero will force the use of "exact" line-searches,
which are extremely expensive.
The function 'gsl_multimin_fminimizer_set' initializes the
minimizer S to minimize the function F, starting from the initial
point X. The size of the initial trial steps is given in vector
STEP_SIZE. The precise meaning of this parameter depends on the
method used.
-- Function: void gsl_multimin_fdfminimizer_free
(gsl_multimin_fdfminimizer * S)
-- Function: void gsl_multimin_fminimizer_free (gsl_multimin_fminimizer
* S)
This function frees all the memory associated with the minimizer S.
-- Function: const char * gsl_multimin_fdfminimizer_name (const
gsl_multimin_fdfminimizer * S)
-- Function: const char * gsl_multimin_fminimizer_name (const
gsl_multimin_fminimizer * S)
This function returns a pointer to the name of the minimizer. For
example,
printf ("s is a '%s' minimizer\n",
gsl_multimin_fdfminimizer_name (s));
would print something like 's is a 'conjugate_pr' minimizer'.
37.4 Providing a function to minimize
=====================================
You must provide a parametric function of n variables for the minimizers
to operate on. You may also need to provide a routine which calculates
the gradient of the function and a third routine which calculates both
the function value and the gradient together. In order to allow for
general parameters the functions are defined by the following data
types:
-- Data Type: gsl_multimin_function_fdf
This data type defines a general function of n variables with
parameters and the corresponding gradient vector of derivatives,
'double (* f) (const gsl_vector * X, void * PARAMS)'
this function should return the result f(x,params) for
argument X and parameters PARAMS. If the function cannot be
computed, an error value of 'GSL_NAN' should be returned.
'void (* df) (const gsl_vector * X, void * PARAMS, gsl_vector * G)'
this function should store the N-dimensional gradient g_i = d
f(x,params) / d x_i in the vector G for argument X and
parameters PARAMS, returning an appropriate error code if the
function cannot be computed.
'void (* fdf) (const gsl_vector * X, void * PARAMS, double * f, gsl_vector * G)'
This function should set the values of the F and G as above,
for arguments X and parameters PARAMS. This function provides
an optimization of the separate functions for f(x) and
g(x)--it is always faster to compute the function and its
derivative at the same time.
'size_t n'
the dimension of the system, i.e. the number of components of
the vectors X.
'void * params'
a pointer to the parameters of the function.
-- Data Type: gsl_multimin_function
This data type defines a general function of n variables with
parameters,
'double (* f) (const gsl_vector * X, void * PARAMS)'
this function should return the result f(x,params) for
argument X and parameters PARAMS. If the function cannot be
computed, an error value of 'GSL_NAN' should be returned.
'size_t n'
the dimension of the system, i.e. the number of components of
the vectors X.
'void * params'
a pointer to the parameters of the function.
The following example function defines a simple two-dimensional
paraboloid with five parameters,
/* Paraboloid centered on (p[0],p[1]), with
scale factors (p[2],p[3]) and minimum p[4] */
double
my_f (const gsl_vector *v, void *params)
{
double x, y;
double *p = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
return p[2] * (x - p[0]) * (x - p[0]) +
p[3] * (y - p[1]) * (y - p[1]) + p[4];
}
/* The gradient of f, df = (df/dx, df/dy). */
void
my_df (const gsl_vector *v, void *params,
gsl_vector *df)
{
double x, y;
double *p = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
gsl_vector_set(df, 0, 2.0 * p[2] * (x - p[0]));
gsl_vector_set(df, 1, 2.0 * p[3] * (y - p[1]));
}
/* Compute both f and df together. */
void
my_fdf (const gsl_vector *x, void *params,
double *f, gsl_vector *df)
{
*f = my_f(x, params);
my_df(x, params, df);
}
The function can be initialized using the following code,
gsl_multimin_function_fdf my_func;
/* Paraboloid center at (1,2), scale factors (10, 20),
minimum value 30 */
double p[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };
my_func.n = 2; /* number of function components */
my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.params = (void *)p;
37.5 Iteration
==============
The following function drives the iteration of each algorithm. The
function performs one iteration to update the state of the minimizer.
The same function works for all minimizers so that different methods can
be substituted at runtime without modifications to the code.
-- Function: int gsl_multimin_fdfminimizer_iterate
(gsl_multimin_fdfminimizer * S)
-- Function: int gsl_multimin_fminimizer_iterate
(gsl_multimin_fminimizer * S)
These functions perform a single iteration of the minimizer S. If
the iteration encounters an unexpected problem then an error code
will be returned. The error code 'GSL_ENOPROG' signifies that the
minimizer is unable to improve on its current estimate, either due
to numerical difficulty or because a genuine local minimum has been
reached.
The minimizer maintains a current best estimate of the minimum at all
times. This information can be accessed with the following auxiliary
functions,
-- Function: gsl_vector * gsl_multimin_fdfminimizer_x (const
gsl_multimin_fdfminimizer * S)
-- Function: gsl_vector * gsl_multimin_fminimizer_x (const
gsl_multimin_fminimizer * S)
-- Function: double gsl_multimin_fdfminimizer_minimum (const
gsl_multimin_fdfminimizer * S)
-- Function: double gsl_multimin_fminimizer_minimum (const
gsl_multimin_fminimizer * S)
-- Function: gsl_vector * gsl_multimin_fdfminimizer_gradient (const
gsl_multimin_fdfminimizer * S)
-- Function: gsl_vector * gsl_multimin_fdfminimizer_dx (const
gsl_multimin_fdfminimizer * S)
-- Function: double gsl_multimin_fminimizer_size (const
gsl_multimin_fminimizer * S)
These functions return the current best estimate of the location of
the minimum, the value of the function at that point, its gradient,
the last step increment of the estimate, and minimizer specific
characteristic size for the minimizer S.
-- Function: int gsl_multimin_fdfminimizer_restart
(gsl_multimin_fdfminimizer * S)
This function resets the minimizer S to use the current point as a
new starting point.
37.6 Stopping Criteria
======================
A minimization procedure should stop when one of the following
conditions is true:
* A minimum has been found to within the user-specified precision.
* A user-specified maximum number of iterations has been reached.
* An error has occurred.
The handling of these conditions is under user control. The functions
below allow the user to test the precision of the current result.
-- Function: int gsl_multimin_test_gradient (const gsl_vector * G,
double EPSABS)
This function tests the norm of the gradient G against the absolute
tolerance EPSABS. The gradient of a multidimensional function goes
to zero at a minimum. The test returns 'GSL_SUCCESS' if the
following condition is achieved,
|g| < epsabs
and returns 'GSL_CONTINUE' otherwise. A suitable choice of EPSABS
can be made from the desired accuracy in the function for small
variations in x. The relationship between these quantities is
given by \delta f = g \delta x.
-- Function: int gsl_multimin_test_size (const double SIZE, double
EPSABS)
This function tests the minimizer specific characteristic size (if
applicable to the used minimizer) against absolute tolerance
EPSABS. The test returns 'GSL_SUCCESS' if the size is smaller than
tolerance, otherwise 'GSL_CONTINUE' is returned.
37.7 Algorithms with Derivatives
================================
There are several minimization methods available. The best choice of
algorithm depends on the problem. The algorithms described in this
section use the value of the function and its gradient at each
evaluation point.
-- Minimizer: gsl_multimin_fdfminimizer_conjugate_fr
This is the Fletcher-Reeves conjugate gradient algorithm. The
conjugate gradient algorithm proceeds as a succession of line
minimizations. The sequence of search directions is used to build
up an approximation to the curvature of the function in the
neighborhood of the minimum.
An initial search direction P is chosen using the gradient, and
line minimization is carried out in that direction. The accuracy
of the line minimization is specified by the parameter TOL. The
minimum along this line occurs when the function gradient G and the
search direction P are orthogonal. The line minimization
terminates when dot(p,g) < tol |p| |g|. The search direction is
updated using the Fletcher-Reeves formula p' = g' - \beta g where
\beta=-|g'|^2/|g|^2, and the line minimization is then repeated for
the new search direction.
-- Minimizer: gsl_multimin_fdfminimizer_conjugate_pr
This is the Polak-Ribiere conjugate gradient algorithm. It is
similar to the Fletcher-Reeves method, differing only in the choice
of the coefficient \beta. Both methods work well when the
evaluation point is close enough to the minimum of the objective
function that it is well approximated by a quadratic hypersurface.
-- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs2
-- Minimizer: gsl_multimin_fdfminimizer_vector_bfgs
These methods use the vector Broyden-Fletcher-Goldfarb-Shanno
(BFGS) algorithm. This is a quasi-Newton method which builds up an
approximation to the second derivatives of the function f using the
difference between successive gradient vectors. By combining the
first and second derivatives the algorithm is able to take
Newton-type steps towards the function minimum, assuming quadratic
behavior in that region.
The 'bfgs2' version of this minimizer is the most efficient version
available, and is a faithful implementation of the line
minimization scheme described in Fletcher's 'Practical Methods of
Optimization', Algorithms 2.6.2 and 2.6.4. It supersedes the
original 'bfgs' routine and requires substantially fewer function
and gradient evaluations. The user-supplied tolerance TOL
corresponds to the parameter \sigma used by Fletcher. A value of
0.1 is recommended for typical use (larger values correspond to
less accurate line searches).
-- Minimizer: gsl_multimin_fdfminimizer_steepest_descent
The steepest descent algorithm follows the downhill gradient of the
function at each step. When a downhill step is successful the
step-size is increased by a factor of two. If the downhill step
leads to a higher function value then the algorithm backtracks and
the step size is decreased using the parameter TOL. A suitable
value of TOL for most applications is 0.1. The steepest descent
method is inefficient and is included only for demonstration
purposes.
37.8 Algorithms without Derivatives
===================================
The algorithms described in this section use only the value of the
function at each evaluation point.
-- Minimizer: gsl_multimin_fminimizer_nmsimplex2
-- Minimizer: gsl_multimin_fminimizer_nmsimplex
These methods use the Simplex algorithm of Nelder and Mead.
Starting from the initial vector X = p_0, the algorithm constructs
an additional n vectors p_i using the step size vector s =
STEP_SIZE as follows:
p_0 = (x_0, x_1, ... , x_n)
p_1 = (x_0 + s_0, x_1, ... , x_n)
p_2 = (x_0, x_1 + s_1, ... , x_n)
... = ...
p_n = (x_0, x_1, ... , x_n + s_n)
These vectors form the n+1 vertices of a simplex in n dimensions.
On each iteration the algorithm uses simple geometrical
transformations to update the vector corresponding to the highest
function value. The geometric transformations are reflection,
reflection followed by expansion, contraction and multiple
contraction. Using these transformations the simplex moves through
the space towards the minimum, where it contracts itself.
After each iteration, the best vertex is returned. Note, that due
to the nature of the algorithm not every step improves the current
best parameter vector. Usually several iterations are required.
The minimizer-specific characteristic size is calculated as the
average distance from the geometrical center of the simplex to all
its vertices. This size can be used as a stopping criteria, as the
simplex contracts itself near the minimum. The size is returned by
the function 'gsl_multimin_fminimizer_size'.
The 'nmsimplex2' version of this minimiser is a new O(N) operations
implementation of the earlier O(N^2) operations 'nmsimplex'
minimiser. It uses the same underlying algorithm, but the simplex
updates are computed more efficiently for high-dimensional
problems. In addition, the size of simplex is calculated as the
RMS distance of each vertex from the center rather than the mean
distance, allowing a linear update of this quantity on each step.
The memory usage is O(N^2) for both algorithms.
-- Minimizer: gsl_multimin_fminimizer_nmsimplex2rand
This method is a variant of 'nmsimplex2' which initialises the
simplex around the starting point X using a randomly-oriented set
of basis vectors instead of the fixed coordinate axes. The final
dimensions of the simplex are scaled along the coordinate axes by
the vector STEP_SIZE. The randomisation uses a simple
deterministic generator so that repeated calls to
'gsl_multimin_fminimizer_set' for a given solver object will vary
the orientation in a well-defined way.
37.9 Examples
=============
This example program finds the minimum of the paraboloid function
defined earlier. The location of the minimum is offset from the origin
in x and y, and the function value at the minimum is non-zero. The main
program is given below, it requires the example function given earlier
in this chapter.
int
main (void)
{
size_t iter = 0;
int status;
const gsl_multimin_fdfminimizer_type *T;
gsl_multimin_fdfminimizer *s;
/* Position of the minimum (1,2), scale factors
10,20, height 30. */
double par[5] = { 1.0, 2.0, 10.0, 20.0, 30.0 };
gsl_vector *x;
gsl_multimin_function_fdf my_func;
my_func.n = 2;
my_func.f = my_f;
my_func.df = my_df;
my_func.fdf = my_fdf;
my_func.params = par;
/* Starting point, x = (5,7) */
x = gsl_vector_alloc (2);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
T = gsl_multimin_fdfminimizer_conjugate_fr;
s = gsl_multimin_fdfminimizer_alloc (T, 2);
gsl_multimin_fdfminimizer_set (s, &my_func, x, 0.01, 1e-4);
do
{
iter++;
status = gsl_multimin_fdfminimizer_iterate (s);
if (status)
break;
status = gsl_multimin_test_gradient (s->gradient, 1e-3);
if (status == GSL_SUCCESS)
printf ("Minimum found at:\n");
printf ("%5d %.5f %.5f %10.5f\n", iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
s->f);
}
while (status == GSL_CONTINUE && iter < 100);
gsl_multimin_fdfminimizer_free (s);
gsl_vector_free (x);
return 0;
}
The initial step-size is chosen as 0.01, a conservative estimate in this
case, and the line minimization parameter is set at 0.0001. The program
terminates when the norm of the gradient has been reduced below 0.001.
The output of the program is shown below,
x y f
1 4.99629 6.99072 687.84780
2 4.98886 6.97215 683.55456
3 4.97400 6.93501 675.01278
4 4.94429 6.86073 658.10798
5 4.88487 6.71217 625.01340
6 4.76602 6.41506 561.68440
7 4.52833 5.82083 446.46694
8 4.05295 4.63238 261.79422
9 3.10219 2.25548 75.49762
10 2.85185 1.62963 67.03704
11 2.19088 1.76182 45.31640
12 0.86892 2.02622 30.18555
Minimum found at:
13 1.00000 2.00000 30.00000
Note that the algorithm gradually increases the step size as it
successfully moves downhill, as can be seen by plotting the successive
points.
The conjugate gradient algorithm finds the minimum on its second
direction because the function is purely quadratic. Additional
iterations would be needed for a more complicated function.
Here is another example using the Nelder-Mead Simplex algorithm to
minimize the same example object function, as above.
int
main(void)
{
double par[5] = {1.0, 2.0, 10.0, 20.0, 30.0};
const gsl_multimin_fminimizer_type *T =
gsl_multimin_fminimizer_nmsimplex2;
gsl_multimin_fminimizer *s = NULL;
gsl_vector *ss, *x;
gsl_multimin_function minex_func;
size_t iter = 0;
int status;
double size;
/* Starting point */
x = gsl_vector_alloc (2);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
/* Set initial step sizes to 1 */
ss = gsl_vector_alloc (2);
gsl_vector_set_all (ss, 1.0);
/* Initialize method and iterate */
minex_func.n = 2;
minex_func.f = my_f;
minex_func.params = par;
s = gsl_multimin_fminimizer_alloc (T, 2);
gsl_multimin_fminimizer_set (s, &minex_func, x, ss);
do
{
iter++;
status = gsl_multimin_fminimizer_iterate(s);
if (status)
break;
size = gsl_multimin_fminimizer_size (s);
status = gsl_multimin_test_size (size, 1e-2);
if (status == GSL_SUCCESS)
{
printf ("converged to minimum at\n");
}
printf ("%5d %10.3e %10.3e f() = %7.3f size = %.3f\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
s->fval, size);
}
while (status == GSL_CONTINUE && iter < 100);
gsl_vector_free(x);
gsl_vector_free(ss);
gsl_multimin_fminimizer_free (s);
return status;
}
The minimum search stops when the Simplex size drops to 0.01. The
output is shown below.
1 6.500e+00 5.000e+00 f() = 512.500 size = 1.130
2 5.250e+00 4.000e+00 f() = 290.625 size = 1.409
3 5.250e+00 4.000e+00 f() = 290.625 size = 1.409
4 5.500e+00 1.000e+00 f() = 252.500 size = 1.409
5 2.625e+00 3.500e+00 f() = 101.406 size = 1.847
6 2.625e+00 3.500e+00 f() = 101.406 size = 1.847
7 0.000e+00 3.000e+00 f() = 60.000 size = 1.847
8 2.094e+00 1.875e+00 f() = 42.275 size = 1.321
9 2.578e-01 1.906e+00 f() = 35.684 size = 1.069
10 5.879e-01 2.445e+00 f() = 35.664 size = 0.841
11 1.258e+00 2.025e+00 f() = 30.680 size = 0.476
12 1.258e+00 2.025e+00 f() = 30.680 size = 0.367
13 1.093e+00 1.849e+00 f() = 30.539 size = 0.300
14 8.830e-01 2.004e+00 f() = 30.137 size = 0.172
15 8.830e-01 2.004e+00 f() = 30.137 size = 0.126
16 9.582e-01 2.060e+00 f() = 30.090 size = 0.106
17 1.022e+00 2.004e+00 f() = 30.005 size = 0.063
18 1.022e+00 2.004e+00 f() = 30.005 size = 0.043
19 1.022e+00 2.004e+00 f() = 30.005 size = 0.043
20 1.022e+00 2.004e+00 f() = 30.005 size = 0.027
21 1.022e+00 2.004e+00 f() = 30.005 size = 0.022
22 9.920e-01 1.997e+00 f() = 30.001 size = 0.016
23 9.920e-01 1.997e+00 f() = 30.001 size = 0.013
converged to minimum at
24 9.920e-01 1.997e+00 f() = 30.001 size = 0.008
The simplex size first increases, while the simplex moves towards the
minimum. After a while the size begins to decrease as the simplex
contracts around the minimum.
37.10 References and Further Reading
====================================
The conjugate gradient and BFGS methods are described in detail in the
following book,
R. Fletcher, 'Practical Methods of Optimization (Second Edition)'
Wiley (1987), ISBN 0471915475.
A brief description of multidimensional minimization algorithms and
more recent references can be found in,
C.W. Ueberhuber, 'Numerical Computation (Volume 2)', Chapter 14,
Section 4.4 "Minimization Methods", p. 325-335, Springer (1997),
ISBN 3-540-62057-5.
The simplex algorithm is described in the following paper,
J.A. Nelder and R. Mead, 'A simplex method for function
minimization', Computer Journal vol. 7 (1965), 308-313.
38 Least-Squares Fitting
************************
This chapter describes routines for performing least squares fits to
experimental data using linear combinations of functions. The data may
be weighted or unweighted, i.e. with known or unknown errors. For
weighted data the functions compute the best fit parameters and their
associated covariance matrix. For unweighted data the covariance matrix
is estimated from the scatter of the points, giving a
variance-covariance matrix.
The functions are divided into separate versions for simple one- or
two-parameter regression and multiple-parameter fits.
38.1 Overview
=============
Least-squares fits are found by minimizing \chi^2 (chi-squared), the
weighted sum of squared residuals over n experimental datapoints (x_i,
y_i) for the model Y(c,x),
\chi^2 = \sum_i w_i (y_i - Y(c, x_i))^2
The p parameters of the model are c = {c_0, c_1, ...}. The weight
factors w_i are given by w_i = 1/\sigma_i^2, where \sigma_i is the
experimental error on the data-point y_i. The errors are assumed to be
Gaussian and uncorrelated. For unweighted data the chi-squared sum is
computed without any weight factors.
The fitting routines return the best-fit parameters c and their p
\times p covariance matrix. The covariance matrix measures the
statistical errors on the best-fit parameters resulting from the errors
on the data, \sigma_i, and is defined as C_{ab} = <\delta c_a \delta
c_b> where < > denotes an average over the Gaussian error distributions
of the underlying datapoints.
The covariance matrix is calculated by error propagation from the
data errors \sigma_i. The change in a fitted parameter \delta c_a
caused by a small change in the data \delta y_i is given by
\delta c_a = \sum_i (dc_a/dy_i) \delta y_i
allowing the covariance matrix to be written in terms of the errors on
the data,
C_{ab} = \sum_{i,j} (dc_a/dy_i) (dc_b/dy_j) <\delta y_i \delta y_j>
For uncorrelated data the fluctuations of the underlying datapoints
satisfy <\delta y_i \delta y_j> = \sigma_i^2 \delta_{ij}, giving a
corresponding parameter covariance matrix of
C_{ab} = \sum_i (1/w_i) (dc_a/dy_i) (dc_b/dy_i)
When computing the covariance matrix for unweighted data, i.e. data
with unknown errors, the weight factors w_i in this sum are replaced by
the single estimate w = 1/\sigma^2, where \sigma^2 is the computed
variance of the residuals about the best-fit model, \sigma^2 = \sum (y_i
- Y(c,x_i))^2 / (n-p). This is referred to as the "variance-covariance
matrix".
The standard deviations of the best-fit parameters are given by the
square root of the corresponding diagonal elements of the covariance
matrix, \sigma_{c_a} = \sqrt{C_{aa}}. The correlation coefficient of
the fit parameters c_a and c_b is given by \rho_{ab} = C_{ab} /
\sqrt{C_{aa} C_{bb}}.
38.2 Linear regression
======================
The functions in this section are used to fit simple one or two
parameter linear regression models. The functions are declared in the
header file 'gsl_fit.h'.
38.2.1 Linear regression with a constant term
---------------------------------------------
The functions described in this section can be used to perform
least-squares fits to a straight line model, Y(c,x) = c_0 + c_1 x.
-- Function: int gsl_fit_linear (const double * X, const size_t
XSTRIDE, const double * Y, const size_t YSTRIDE, size_t N,
double * C0, double * C1, double * COV00, double * COV01,
double * COV11, double * SUMSQ)
This function computes the best-fit linear regression coefficients
(C0,C1) of the model Y = c_0 + c_1 X for the dataset (X, Y), two
vectors of length N with strides XSTRIDE and YSTRIDE. The errors
on Y are assumed unknown so the variance-covariance matrix for the
parameters (C0, C1) is estimated from the scatter of the points
around the best-fit line and returned via the parameters (COV00,
COV01, COV11). The sum of squares of the residuals from the
best-fit line is returned in SUMSQ. Note: the correlation
coefficient of the data can be computed using
'gsl_stats_correlation' (*note Correlation::), it does not depend
on the fit.
-- Function: int gsl_fit_wlinear (const double * X, const size_t
XSTRIDE, const double * W, const size_t WSTRIDE, const double
* Y, const size_t YSTRIDE, size_t N, double * C0, double * C1,
double * COV00, double * COV01, double * COV11, double *
CHISQ)
This function computes the best-fit linear regression coefficients
(C0,C1) of the model Y = c_0 + c_1 X for the weighted dataset (X,
Y), two vectors of length N with strides XSTRIDE and YSTRIDE. The
vector W, of length N and stride WSTRIDE, specifies the weight of
each datapoint. The weight is the reciprocal of the variance for
each datapoint in Y.
The covariance matrix for the parameters (C0, C1) is computed using
the weights and returned via the parameters (COV00, COV01, COV11).
The weighted sum of squares of the residuals from the best-fit
line, \chi^2, is returned in CHISQ.
-- Function: int gsl_fit_linear_est (double X, double C0, double C1,
double COV00, double COV01, double COV11, double * Y, double *
Y_ERR)
This function uses the best-fit linear regression coefficients C0,
C1 and their covariance COV00, COV01, COV11 to compute the fitted
function Y and its standard deviation Y_ERR for the model Y = c_0 +
c_1 X at the point X.
38.2.2 Linear regression without a constant term
------------------------------------------------
The functions described in this section can be used to perform
least-squares fits to a straight line model without a constant term, Y =
c_1 X.
-- Function: int gsl_fit_mul (const double * X, const size_t XSTRIDE,
const double * Y, const size_t YSTRIDE, size_t N, double * C1,
double * COV11, double * SUMSQ)
This function computes the best-fit linear regression coefficient
C1 of the model Y = c_1 X for the datasets (X, Y), two vectors of
length N with strides XSTRIDE and YSTRIDE. The errors on Y are
assumed unknown so the variance of the parameter C1 is estimated
from the scatter of the points around the best-fit line and
returned via the parameter COV11. The sum of squares of the
residuals from the best-fit line is returned in SUMSQ.
-- Function: int gsl_fit_wmul (const double * X, const size_t XSTRIDE,
const double * W, const size_t WSTRIDE, const double * Y,
const size_t YSTRIDE, size_t N, double * C1, double * COV11,
double * SUMSQ)
This function computes the best-fit linear regression coefficient
C1 of the model Y = c_1 X for the weighted datasets (X, Y), two
vectors of length N with strides XSTRIDE and YSTRIDE. The vector
W, of length N and stride WSTRIDE, specifies the weight of each
datapoint. The weight is the reciprocal of the variance for each
datapoint in Y.
The variance of the parameter C1 is computed using the weights and
returned via the parameter COV11. The weighted sum of squares of
the residuals from the best-fit line, \chi^2, is returned in CHISQ.
-- Function: int gsl_fit_mul_est (double X, double C1, double COV11,
double * Y, double * Y_ERR)
This function uses the best-fit linear regression coefficient C1
and its covariance COV11 to compute the fitted function Y and its
standard deviation Y_ERR for the model Y = c_1 X at the point X.
38.3 Multi-parameter regression
===============================
This section describes routines which perform least squares fits to a
linear model by minimizing the cost function
\chi^2 = \sum_i w_i (y_i - \sum_j X_ij c_j)^2 = || y - Xc ||_W^2
where y is a vector of n observations, X is an n-by-p matrix of
predictor variables, c is a vector of the p unknown best-fit parameters
to be estimated, and ||r||_W^2 = r^T W r. The matrix W =
diag(w_1,w_2,...,w_n) defines the weights or uncertainties of the
observation vector.
This formulation can be used for fits to any number of functions
and/or variables by preparing the n-by-p matrix X appropriately. For
example, to fit to a p-th order polynomial in X, use the following
matrix,
X_{ij} = x_i^j
where the index i runs over the observations and the index j runs from 0
to p-1.
To fit to a set of p sinusoidal functions with fixed frequencies
\omega_1, \omega_2, ..., \omega_p, use,
X_{ij} = sin(\omega_j x_i)
To fit to p independent variables x_1, x_2, ..., x_p, use,
X_{ij} = x_j(i)
where x_j(i) is the i-th value of the predictor variable x_j.
The solution of the general linear least-squares system requires an
additional working space for intermediate results, such as the singular
value decomposition of the matrix X.
These functions are declared in the header file 'gsl_multifit.h'.
-- Function: gsl_multifit_linear_workspace * gsl_multifit_linear_alloc
(const size_t N, const size_t P)
This function allocates a workspace for fitting a model to a
maximum of N observations using a maximum of P parameters. The
user may later supply a smaller least squares system if desired.
The size of the workspace is O(np + p^2).
-- Function: void gsl_multifit_linear_free
(gsl_multifit_linear_workspace * WORK)
This function frees the memory associated with the workspace W.
-- Function: int gsl_multifit_linear_svd (const gsl_matrix * X,
gsl_multifit_linear_workspace * WORK)
This function performs a singular value decomposition of the matrix
X and stores the SVD factors internally in WORK.
-- Function: int gsl_multifit_linear_bsvd (const gsl_matrix * X,
gsl_multifit_linear_workspace * WORK)
This function performs a singular value decomposition of the matrix
X and stores the SVD factors internally in WORK. The matrix X is
first balanced by applying column scaling factors to improve the
accuracy of the singular values.
-- Function: int gsl_multifit_linear (const gsl_matrix * X, const
gsl_vector * Y, gsl_vector * C, gsl_matrix * COV, double *
CHISQ, gsl_multifit_linear_workspace * WORK)
This function computes the best-fit parameters C of the model y = X
c for the observations Y and the matrix of predictor variables X,
using the preallocated workspace provided in WORK. The p-by-p
variance-covariance matrix of the model parameters COV is set to
\sigma^2 (X^T X)^{-1}, where \sigma is the standard deviation of
the fit residuals. The sum of squares of the residuals from the
best-fit, \chi^2, is returned in CHISQ. If the coefficient of
determination is desired, it can be computed from the expression
R^2 = 1 - \chi^2 / TSS, where the total sum of squares (TSS) of the
observations Y may be computed from 'gsl_stats_tss'.
The best-fit is found by singular value decomposition of the matrix
X using the modified Golub-Reinsch SVD algorithm, with column
scaling to improve the accuracy of the singular values. Any
components which have zero singular value (to machine precision)
are discarded from the fit.
-- Function: int gsl_multifit_linear_tsvd (const gsl_matrix * X, const
gsl_vector * Y, const double TOL, gsl_vector * C, gsl_matrix *
COV, double * CHISQ, size_t * RANK,
gsl_multifit_linear_workspace * WORK)
This function computes the best-fit parameters C of the model y = X
c for the observations Y and the matrix of predictor variables X,
using a truncated SVD expansion. Singular values which satisfy s_i
\le tol \times s_0 are discarded from the fit, where s_0 is the
largest singular value. The p-by-p variance-covariance matrix of
the model parameters COV is set to \sigma^2 (X^T X)^{-1}, where
\sigma is the standard deviation of the fit residuals. The sum of
squares of the residuals from the best-fit, \chi^2, is returned in
CHISQ. The effective rank (number of singular values used in
solution) is returned in RANK. If the coefficient of determination
is desired, it can be computed from the expression R^2 = 1 - \chi^2
/ TSS, where the total sum of squares (TSS) of the observations Y
may be computed from 'gsl_stats_tss'.
-- Function: int gsl_multifit_wlinear (const gsl_matrix * X, const
gsl_vector * W, const gsl_vector * Y, gsl_vector * C,
gsl_matrix * COV, double * CHISQ,
gsl_multifit_linear_workspace * WORK)
This function computes the best-fit parameters C of the weighted
model y = X c for the observations Y with weights W and the matrix
of predictor variables X, using the preallocated workspace provided
in WORK. The p-by-p covariance matrix of the model parameters COV
is computed as (X^T W X)^{-1}. The weighted sum of squares of the
residuals from the best-fit, \chi^2, is returned in CHISQ. If the
coefficient of determination is desired, it can be computed from
the expression R^2 = 1 - \chi^2 / WTSS, where the weighted total
sum of squares (WTSS) of the observations Y may be computed from
'gsl_stats_wtss'.
-- Function: int gsl_multifit_wlinear_tsvd (const gsl_matrix * X, const
gsl_vector * W, const gsl_vector * Y, const double TOL,
gsl_vector * C, gsl_matrix * COV, double * CHISQ, size_t *
RANK, gsl_multifit_linear_workspace * WORK)
This function computes the best-fit parameters C of the weighted
model y = X c for the observations Y with weights W and the matrix
of predictor variables X, using a truncated SVD expansion.
Singular values which satisfy s_i \le tol \times s_0 are discarded
from the fit, where s_0 is the largest singular value. The p-by-p
covariance matrix of the model parameters COV is computed as (X^T W
X)^{-1}. The weighted sum of squares of the residuals from the
best-fit, \chi^2, is returned in CHISQ. The effective rank of the
system (number of singular values used in the solution) is returned
in RANK. If the coefficient of determination is desired, it can be
computed from the expression R^2 = 1 - \chi^2 / WTSS, where the
weighted total sum of squares (WTSS) of the observations Y may be
computed from 'gsl_stats_wtss'.
-- Function: int gsl_multifit_linear_est (const gsl_vector * X, const
gsl_vector * C, const gsl_matrix * COV, double * Y, double *
Y_ERR)
This function uses the best-fit multilinear regression coefficients
C and their covariance matrix COV to compute the fitted function
value Y and its standard deviation Y_ERR for the model y = x.c at
the point X.
-- Function: int gsl_multifit_linear_residuals (const gsl_matrix * X,
const gsl_vector * Y, const gsl_vector * C, gsl_vector * R)
This function computes the vector of residuals r = y - X c for the
observations Y, coefficients C and matrix of predictor variables X.
-- Function: size_t gsl_multifit_linear_rank (const double TOL, const
gsl_multifit_linear_workspace * WORK)
This function returns the rank of the matrix X which must first
have its singular value decomposition computed. The rank is
computed by counting the number of singular values \sigma_j which
satisfy \sigma_j > tol \times \sigma_0, where \sigma_0 is the
largest singular value.
38.4 Regularized regression
===========================
Ordinary weighted least squares models seek a solution vector c which
minimizes the residual
\chi^2 = || y - Xc ||_W^2
where y is the n-by-1 observation vector, X is the n-by-p design
matrix, c is the p-by-1 solution vector, W = diag(w_1,...,w_n) is the
data weighting matrix, and ||r||_W^2 = r^T W r. In cases where the
least squares matrix X is ill-conditioned, small perturbations (ie:
noise) in the observation vector could lead to widely different solution
vectors c. One way of dealing with ill-conditioned matrices is to use a
"truncated SVD" in which small singular values, below some given
tolerance, are discarded from the solution. The truncated SVD method is
available using the functions 'gsl_multifit_linear_tsvd' and
'gsl_multifit_wlinear_tsvd'. Another way to help solve ill-posed
problems is to include a regularization term in the least squares
minimization
\chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2
for a suitably chosen regularization parameter \lambda and matrix L.
This type of regularization is known as Tikhonov, or ridge, regression.
In some applications, L is chosen as the identity matrix, giving
preference to solution vectors c with smaller norms. Including this
regularization term leads to the explicit "normal equations" solution
c = ( X^T W X + \lambda^2 L^T L )^-1 X^T W y
which reduces to the ordinary least squares solution when L = 0. In
practice, it is often advantageous to transform a regularized least
squares system into the form
\chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2
This is known as the Tikhonov "standard form" and has the normal
equations solution \tilde{c} = \left( \tilde{X}^T \tilde{X} + \lambda^2
I \right)^{-1} \tilde{X}^T \tilde{y}. For an m-by-p matrix L which is
full rank and has m >= p (ie: L is square or has more rows than
columns), we can calculate the "thin" QR decomposition of L, and note
that ||L c|| = ||R c|| since the Q factor will not change the norm.
Since R is p-by-p, we can then use the transformation
X~ = sqrt(W) X R^-1
y~ = sqrt(W) y
c~ = R c
to achieve the standard form. For a rectangular matrix L with m < p,
a more sophisticated approach is needed (see Hansen 1998, chapter 2.3).
In practice, the normal equations solution above is not desirable due to
numerical instabilities, and so the system is solved using the singular
value decomposition of the matrix \tilde{X}. The matrix L is often
chosen as the identity matrix, or as a first or second finite difference
operator, to ensure a smoothly varying coefficient vector c, or as a
diagonal matrix to selectively damp each model parameter differently.
If L \ne I, the user must first convert the least squares problem to
standard form using 'gsl_multifit_linear_stdform1' or
'gsl_multifit_linear_stdform2', solve the system, and then backtransform
the solution vector to recover the solution of the original problem (see
'gsl_multifit_linear_genform1' and 'gsl_multifit_linear_genform2').
In many regularization problems, care must be taken when choosing the
regularization parameter \lambda. Since both the residual norm ||y - X
c|| and solution norm ||L c|| are being minimized, the parameter \lambda
represents a tradeoff between minimizing either the residuals or the
solution vector. A common tool for visualizing the comprimise between
the minimization of these two quantities is known as the L-curve. The
L-curve is a log-log plot of the residual norm ||y - X c|| on the
horizontal axis and the solution norm ||L c|| on the vertical axis.
This curve nearly always as an L shaped appearance, with a distinct
corner separating the horizontal and vertical sections of the curve.
The regularization parameter corresponding to this corner is often
chosen as the optimal value. GSL provides routines to calculate the
L-curve for all relevant regularization parameters as well as locating
the corner.
Another method of choosing the regularization parameter is known as
Generalized Cross Validation (GCV). This method is based on the idea
that if an arbitrary element y_i is left out of the right hand side, the
resulting regularized solution should predict this element accurately.
This leads to choosing the parameter \lambda which minimizes the GCV
function
G(\lambda) = (||y - X c_{\lambda}||^2) / Tr(I_n - X X^I)^2
where X_{\lambda}^I is the matrix which relates the solution
c_{\lambda} to the right hand side y, ie: c_{\lambda} = X_{\lambda}^I y.
GSL provides routines to compute the GCV curve and its minimum.
For most applications, the steps required to solve a regularized least
squares problem are as follows:
1. Construct the least squares system (X, y, W, L)
2. Transform the system to standard form (\tilde{X},\tilde{y}). This
step can be skipped if L = I_p and W = I_n.
3. Calculate the SVD of \tilde{X}.
4. Determine an appropriate regularization parameter \lambda (using
for example L-curve or GCV analysis).
5. Solve the standard form system using the chosen \lambda and the SVD
of \tilde{X}.
6. Backtransform the standard form solution \tilde{c} to recover the
original solution vector c.
-- Function: int gsl_multifit_linear_stdform1 (const gsl_vector * L,
const gsl_matrix * X, const gsl_vector * Y, gsl_matrix * XS,
gsl_vector * YS, gsl_multifit_linear_workspace * WORK)
-- Function: int gsl_multifit_linear_wstdform1 (const gsl_vector * L,
const gsl_matrix * X, const gsl_vector * W, const gsl_vector *
Y, gsl_matrix * XS, gsl_vector * YS,
gsl_multifit_linear_workspace * WORK)
These functions define a regularization matrix L =
diag(l_0,l_1,...,l_{p-1}). The diagonal matrix element l_i is
provided by the ith element of the input vector L. The n-by-p
least squares matrix X and vector Y of length n are then converted
to standard form as described above and the parameters
(\tilde{X},\tilde{y}) are stored in XS and YS on output. XS and YS
have the same dimensions as X and Y. Optional data weights may be
supplied in the vector W of length n. In order to apply this
transformation, L^{-1} must exist and so none of the l_i may be
zero. After the standard form system has been solved, use
'gsl_multifit_linear_genform1' to recover the original solution
vector. It is allowed to have X = XS and Y = YS for an in-place
transform. In order to perform a weighted regularized fit with L =
I, the user may call 'gsl_multifit_linear_applyW' to convert to
standard form.
-- Function: int gsl_multifit_linear_L_decomp (gsl_matrix * L,
gsl_vector * TAU)
This function factors the m-by-p regularization matrix L into a
form needed for the later transformation to standard form. L may
have any number of rows m. If m \ge p the QR decomposition of L is
computed and stored in L on output. If m < p, the QR decomposition
of L^T is computed and stored in L on output. On output, the
Householder scalars are stored in the vector TAU of size MIN(m,p).
These outputs will be used by 'gsl_multifit_linear_wstdform2' to
complete the transformation to standard form.
-- Function: int gsl_multifit_linear_stdform2 (const gsl_matrix * LQR,
const gsl_vector * LTAU, const gsl_matrix * X, const
gsl_vector * Y, gsl_matrix * XS, gsl_vector * YS, gsl_matrix *
M, gsl_multifit_linear_workspace * WORK)
-- Function: int gsl_multifit_linear_wstdform2 (const gsl_matrix * LQR,
const gsl_vector * LTAU, const gsl_matrix * X, const
gsl_vector * W, const gsl_vector * Y, gsl_matrix * XS,
gsl_vector * YS, gsl_matrix * M, gsl_multifit_linear_workspace
* WORK)
These functions convert the least squares system (X,Y,W,L) to
standard form (\tilde{X},\tilde{y}) which are stored in XS and YS
respectively. The m-by-p regularization matrix L is specified by
the inputs LQR and LTAU, which are outputs from
'gsl_multifit_linear_L_decomp'. The dimensions of the standard
form parameters (\tilde{X},\tilde{y}) depend on whether m is larger
or less than p. For m \ge p, XS is n-by-p, YS is n-by-1, and M is
not used. For m < p, XS is (n - p + m)-by-m, YS is (n - p +
m)-by-1, and M is additional n-by-p workspace, which is required to
recover the original solution vector after the system has been
solved (see 'gsl_multifit_linear_genform2'). Optional data weights
may be supplied in the vector W of length n, where W = diag(w).
-- Function: int gsl_multifit_linear_solve (const double LAMBDA, const
gsl_matrix * XS, const gsl_vector * YS, gsl_vector * CS,
double * RNORM, double * SNORM, gsl_multifit_linear_workspace
* WORK)
This function computes the regularized best-fit parameters
\tilde{c} which minimize the cost function \chi^2 = || \tilde{y} -
\tilde{X} \tilde{c} ||^2 + \lambda^2 || \tilde{c} ||^2 which is in
standard form. The least squares system must therefore be
converted to standard form prior to calling this function. The
observation vector \tilde{y} is provided in YS and the matrix of
predictor variables \tilde{X} in XS. The solution vector \tilde{c}
is returned in CS, which has length min(m,p). The SVD of XS must
be computed prior to calling this function, using
'gsl_multifit_linear_svd'. The regularization parameter \lambda is
provided in LAMBDA. The residual norm || \tilde{y} - \tilde{X}
\tilde{c} || = ||y - X c||_W is returned in RNORM. The solution
norm || \tilde{c} || = ||L c|| is returned in SNORM.
-- Function: int gsl_multifit_linear_genform1 (const gsl_vector * L,
const gsl_vector * CS, gsl_vector * C,
gsl_multifit_linear_workspace * WORK)
After a regularized system has been solved with L =
diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
standard form solution vector CS to recover the solution vector of
the original problem C. The diagonal matrix elements l_i are
provided in the vector L. It is allowed to have C = CS for an
in-place transform.
-- Function: int gsl_multifit_linear_genform2 (const gsl_matrix * LQR,
const gsl_vector * LTAU, const gsl_matrix * X, const
gsl_vector * Y, const gsl_vector * CS, const gsl_matrix * M,
gsl_vector * C, gsl_multifit_linear_workspace * WORK)
-- Function: int gsl_multifit_linear_wgenform2 (const gsl_matrix * LQR,
const gsl_vector * LTAU, const gsl_matrix * X, const
gsl_vector * W, const gsl_vector * Y, const gsl_vector * CS,
const gsl_matrix * M, gsl_vector * C,
gsl_multifit_linear_workspace * WORK)
After a regularized system has been solved with a general
rectangular matrix L, specified by (LQR,LTAU), this function
backtransforms the standard form solution CS to recover the
solution vector of the original problem, which is stored in C, of
length p. The original least squares matrix and observation vector
are provided in X and Y respectively. M is the matrix computed by
'gsl_multifit_linear_stdform2'. For weighted fits, the weight
vector W must also be supplied.
-- Function: int gsl_multifit_linear_applyW (const gsl_matrix * X,
const gsl_vector * W, const gsl_vector * Y, gsl_matrix * WX,
gsl_vector * WY)
For weighted least squares systems with L = I, this function may be
used to convert the system to standard form by applying the weight
matrix W = diag(W) to the least squares matrix X and observation
vector Y. On output, WX is equal to W^{1/2} X and WY is equal to
W^{1/2} y. It is allowed for WX = X and WY = Y for an in-place
transform.
-- Function: int gsl_multifit_linear_lcurve (const gsl_vector * Y,
gsl_vector * REG_PARAM, gsl_vector * RHO, gsl_vector * ETA,
gsl_multifit_linear_workspace * WORK)
This function computes the L-curve for a least squares system using
the right hand side vector Y and the SVD decomposition of the least
squares matrix X, which must be provided to
'gsl_multifit_linear_svd' prior to calling this function. The
output vectors REG_PARAM, RHO, and ETA must all be the same size,
and will contain the regularization parameters \lambda_i, residual
norms ||y - X c_i||, and solution norms || L c_i || which compose
the L-curve, where c_i is the regularized solution vector
corresponding to \lambda_i. The user may determine the number of
points on the L-curve by adjusting the size of these input arrays.
The regularization parameters \lambda_i are estimated from the
singular values of X, and chosen to represent the most relevant
portion of the L-curve.
-- Function: int gsl_multifit_linear_lcorner (const gsl_vector * RHO,
const gsl_vector * ETA, size_t * IDX)
This function attempts to locate the corner of the L-curve (||y - X
c||, ||L c||) defined by the RHO and ETA input arrays respectively.
The corner is defined as the point of maximum curvature of the
L-curve in log-log scale. The RHO and ETA arrays can be outputs of
'gsl_multifit_linear_lcurve'. The algorithm used simply fits a
circle to 3 consecutive points on the L-curve and uses the circle's
radius to determine the curvature at the middle point. Therefore,
the input array sizes must be \ge 3. With more points provided for
the L-curve, a better estimate of the curvature can be obtained.
The array index corresponding to maximum curvature (ie: the corner)
is returned in IDX. If the input arrays contain colinear points,
this function could fail and return 'GSL_EINVAL'.
-- Function: int gsl_multifit_linear_lcorner2 (const gsl_vector *
REG_PARAM, const gsl_vector * ETA, size_t * IDX)
This function attempts to locate the corner of an alternate L-curve
(\lambda^2, ||L c||^2) studied by Rezghi and Hosseini, 2009. This
alternate L-curve can provide better estimates of the
regularization parameter for smooth solution vectors. The
regularization parameters \lambda and solution norms ||L c|| are
provided in the REG_PARAM and ETA input arrays respectively. The
corner is defined as the point of maximum curvature of this
alternate L-curve in linear scale. The REG_PARAM and ETA arrays
can be outputs of 'gsl_multifit_linear_lcurve'. The algorithm used
simply fits a circle to 3 consecutive points on the L-curve and
uses the circle's radius to determine the curvature at the middle
point. Therefore, the input array sizes must be \ge 3. With more
points provided for the L-curve, a better estimate of the curvature
can be obtained. The array index corresponding to maximum
curvature (ie: the corner) is returned in IDX. If the input arrays
contain colinear points, this function could fail and return
'GSL_EINVAL'.
-- Function: int gsl_multifit_linear_gcv_init(const gsl_vector * Y,
gsl_vector * REG_PARAM, gsl_vector * UTY, double * DELTA0,
gsl_multifit_linear_workspace * WORK)
This function performs some initialization in preparation for
computing the GCV curve and its minimum. The right hand side
vector is provided in Y. On output, REG_PARAM is set to a vector
of regularization parameters in decreasing order and may be of any
size. The vector UTY of size p is set to U^T y. The parameter
DELTA0 is needed for subsequent steps of the GCV calculation.
-- Function: int gsl_multifit_linear_gcv_curve(const gsl_vector *
REG_PARAM, const gsl_vector * UTY, const double DELTA0,
gsl_vector * G, gsl_multifit_linear_workspace * WORK)
This funtion calculates the GCV curve G(\lambda) and stores it in G
on output, which must be the same size as REG_PARAM. The inputs
REG_PARAM, UTY and DELTA0 are computed in
'gsl_multifit_linear_gcv_init'.
-- Function: int gsl_multifit_linear_gcv_min(const gsl_vector *
REG_PARAM, const gsl_vector * UTY, const gsl_vector * G, const
double DELTA0, double * LAMBDA, gsl_multifit_linear_workspace
* WORK)
This function computes the value of the regularization parameter
which minimizes the GCV curve G(\lambda) and stores it in LAMBDA.
The input G is calculated by 'gsl_multifit_linear_gcv_curve' and
the inputs REG_PARAM, UTY and DELTA0 are computed by
'gsl_multifit_linear_gcv_init'.
-- Function: double gsl_multifit_linear_gcv_calc(const double LAMBDA,
const gsl_vector * UTY, const double DELTA0,
gsl_multifit_linear_workspace * WORK)
This function returns the value of the GCV curve G(\lambda)
corresponding to the input LAMBDA.
-- Function: int gsl_multifit_linear_gcv(const gsl_vector * Y,
gsl_vector * REG_PARAM, gsl_vector * G, double * LAMBDA,
double * G_LAMBDA, gsl_multifit_linear_workspace * WORK)
This function combines the steps 'gcv_init', 'gcv_curve', and
'gcv_min' defined above into a single function. The input Y is the
right hand side vector. On output, REG_PARAM and G, which must be
the same size, are set to vectors of \lambda and G(\lambda) values
respectively. The output LAMBDA is set to the optimal value of
\lambda which minimizes the GCV curve. The minimum value of the
GCV curve is returned in G_LAMBDA.
-- Function: int gsl_multifit_linear_Lk (const size_t P, const size_t
K, gsl_matrix * L)
This function computes the discrete approximation to the derivative
operator L_k of order K on a regular grid of P points and stores it
in L. The dimensions of L are (p-k)-by-p.
-- Function: int gsl_multifit_linear_Lsobolev (const size_t P, const
size_t KMAX, const gsl_vector * ALPHA, gsl_matrix * L,
gsl_multifit_linear_workspace * WORK)
This function computes the regularization matrix L corresponding to
the weighted Sobolov norm ||L c||^2 = \sum_k \alpha_k^2 ||L_k c||^2
where L_k approximates the derivative operator of order k. This
regularization norm can be useful in applications where it is
necessary to smooth several derivatives of the solution. P is the
number of model parameters, KMAX is the highest derivative to
include in the summation above, and ALPHA is the vector of weights
of size KMAX + 1, where ALPHA[k] = \alpha_k is the weight assigned
to the derivative of order k. The output matrix L is size P-by-P
and upper triangular.
-- Function: double gsl_multifit_linear_rcond (const
gsl_multifit_linear_workspace * WORK)
This function returns the reciprocal condition number of the least
squares matrix X, defined as the ratio of the smallest and largest
singular values, rcond = \sigma_{min}/\sigma_{max}. The routine
'gsl_multifit_linear_svd' must first be called to compute the SVD
of X.
38.5 Robust linear regression
=============================
Ordinary least squares (OLS) models are often heavily influenced by the
presence of outliers. Outliers are data points which do not follow the
general trend of the other observations, although there is strictly no
precise definition of an outlier. Robust linear regression refers to
regression algorithms which are robust to outliers. The most common
type of robust regression is M-estimation. The general M-estimator
minimizes the objective function
\sum_i \rho(e_i) = \sum_i \rho (y_i - Y(c, x_i))
where e_i = y_i - Y(c, x_i) is the residual of the ith data point,
and \rho(e_i) is a function which should have the following properties:
\rho(e) \ge 0
\rho(0) = 0
\rho(-e) = \rho(e)
\rho(e_1) > \rho(e_2) for |e_1| > |e_2|
The special case of ordinary least squares is given by \rho(e_i) =
e_i^2. Letting \psi = \rho' be the derivative of \rho, differentiating
the objective function with respect to the coefficients c and setting
the partial derivatives to zero produces the system of equations
\sum_i \psi(e_i) X_i = 0
where X_i is a vector containing row i of the design matrix X. Next,
we define a weight function w(e) = \psi(e)/e, and let w_i = w(e_i):
\sum_i w_i e_i X_i = 0
This system of equations is equivalent to solving a weighted ordinary
least squares problem, minimizing \chi^2 = \sum_i w_i e_i^2. The
weights however, depend on the residuals e_i, which depend on the
coefficients c, which depend on the weights. Therefore, an iterative
solution is used, called Iteratively Reweighted Least Squares (IRLS).
1. Compute initial estimates of the coefficients c^{(0)} using
ordinary least squares
2. For iteration k, form the residuals e_i^{(k)} = (y_i - X_i
c^{(k-1)})/(t \sigma^{(k)} \sqrt{1 - h_i}), where t is a tuning
constant depending on the choice of \psi, and h_i are the
statistical leverages (diagonal elements of the matrix X (X^T
X)^{-1} X^T). Including t and h_i in the residual calculation has
been shown to improve the convergence of the method. The residual
standard deviation is approximated as \sigma^{(k)} = MAD / 0.6745,
where MAD is the Median-Absolute-Deviation of the n-p largest
residuals from the previous iteration.
3. Compute new weights w_i^{(k)} = \psi(e_i^{(k)})/e_i^{(k)}.
4. Compute new coefficients c^{(k)} by solving the weighted least
squares problem with weights w_i^{(k)}.
5. Steps 2 through 4 are iterated until the coefficients converge or
until some maximum iteration limit is reached. Coefficients are
tested for convergence using the critera:
|c_i^(k) - c_i^(k-1)| \le \epsilon \times max(|c_i^(k)|, |c_i^(k-1)|)
for all 0 \le i < p where \epsilon is a small tolerance factor.
The key to this method lies in selecting the function \psi(e_i) to
assign smaller weights to large residuals, and larger weights to smaller
residuals. As the iteration proceeds, outliers are assigned smaller and
smaller weights, eventually having very little or no effect on the
fitted model.
-- Function: gsl_multifit_robust_workspace * gsl_multifit_robust_alloc
(const gsl_multifit_robust_type * T, const size_t N, const
size_t P)
This function allocates a workspace for fitting a model to N
observations using P parameters. The size of the workspace is O(np
+ p^2). The type T specifies the function \psi and can be selected
from the following choices.
-- Robust type: gsl_multifit_robust_default
This specifies the 'gsl_multifit_robust_bisquare' type (see
below) and is a good general purpose choice for robust
regression.
-- Robust type: gsl_multifit_robust_bisquare
This is Tukey's biweight (bisquare) function and is a good
general purpose choice for robust regression. The weight
function is given by
w(e) = (1 - e^2)^2
and the default tuning constant is t = 4.685.
-- Robust type: gsl_multifit_robust_cauchy
This is Cauchy's function, also known as the Lorentzian
function. This function does not guarantee a unique solution,
meaning different choices of the coefficient vector C could
minimize the objective function. Therefore this option should
be used with care. The weight function is given by
w(e) = 1 / (1 + e^2)
and the default tuning constant is t = 2.385.
-- Robust type: gsl_multifit_robust_fair
This is the fair \rho function, which guarantees a unique
solution and has continuous derivatives to three orders. The
weight function is given by
w(e) = 1 / (1 + |e|)
and the default tuning constant is t = 1.400.
-- Robust type: gsl_multifit_robust_huber
This specifies Huber's \rho function, which is a parabola in
the vicinity of zero and increases linearly for a given
threshold |e| > t. This function is also considered an
excellent general purpose robust estimator, however,
occasional difficulties can be encountered due to the
discontinuous first derivative of the \psi function. The
weight function is given by
w(e) = 1/max(1,|e|)
and the default tuning constant is t = 1.345.
-- Robust type: gsl_multifit_robust_ols
This specifies the ordinary least squares solution, which can
be useful for quickly checking the difference between the
various robust and OLS solutions. The weight function is
given by
w(e) = 1
and the default tuning constant is t = 1.
-- Robust type: gsl_multifit_robust_welsch
This specifies the Welsch function which can perform well in
cases where the residuals have an exponential distribution.
The weight function is given by
w(e) = \exp(-e^2)
and the default tuning constant is t = 2.985.
-- Function: void gsl_multifit_robust_free
(gsl_multifit_robust_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: const char * gsl_multifit_robust_name (const
gsl_multifit_robust_workspace * W)
This function returns the name of the robust type T specified to
'gsl_multifit_robust_alloc'.
-- Function: int gsl_multifit_robust_tune (const double TUNE,
gsl_multifit_robust_workspace * W)
This function sets the tuning constant t used to adjust the
residuals at each iteration to TUNE. Decreasing the tuning
constant increases the downweight assigned to large residuals,
while increasing the tuning constant decreases the downweight
assigned to large residuals.
-- Function: int gsl_multifit_robust_maxiter (const size_t MAXITER,
gsl_multifit_robust_workspace * W)
This function sets the maximum number of iterations in the
iteratively reweighted least squares algorithm to MAXITER. By
default, this value is set to 100 by 'gsl_multifit_robust_alloc'.
-- Function: int gsl_multifit_robust_weights (const gsl_vector * R,
gsl_vector * WTS, gsl_multifit_robust_workspace * W)
This function assigns weights to the vector WTS using the residual
vector R and previously specified weighting function. The output
weights are given by wts_i = w(r_i / (t \sigma)), where the
weighting functions w are detailed in 'gsl_multifit_robust_alloc'.
\sigma is an estimate of the residual standard deviation based on
the Median-Absolute-Deviation and t is the tuning constant. This
function is useful if the user wishes to implement their own robust
regression rather than using the supplied 'gsl_multifit_robust'
routine below.
-- Function: int gsl_multifit_robust (const gsl_matrix * X, const
gsl_vector * Y, gsl_vector * C, gsl_matrix * COV,
gsl_multifit_robust_workspace * W)
This function computes the best-fit parameters C of the model y = X
c for the observations Y and the matrix of predictor variables X,
attemping to reduce the influence of outliers using the algorithm
outlined above. The p-by-p variance-covariance matrix of the model
parameters COV is estimated as \sigma^2 (X^T X)^{-1}, where \sigma
is an approximation of the residual standard deviation using the
theory of robust regression. Special care must be taken when
estimating \sigma and other statistics such as R^2, and so these
are computed internally and are available by calling the function
'gsl_multifit_robust_statistics'.
If the coefficients do not converge within the maximum iteration
limit, the function returns 'GSL_EMAXITER'. In this case, the
current estimates of the coefficients and covariance matrix are
returned in C and COV and the internal fit statistics are computed
with these estimates.
-- Function: int gsl_multifit_robust_est (const gsl_vector * X, const
gsl_vector * C, const gsl_matrix * COV, double * Y, double *
Y_ERR)
This function uses the best-fit robust regression coefficients C
and their covariance matrix COV to compute the fitted function
value Y and its standard deviation Y_ERR for the model y = x.c at
the point X.
-- Function: int gsl_multifit_robust_residuals (const gsl_matrix * X,
const gsl_vector * Y, const gsl_vector * C, gsl_vector * R,
gsl_multifit_robust_workspace * W)
This function computes the vector of studentized residuals r_i =
{y_i - (X c)_i \over \sigma \sqrt{1 - h_i}} for the observations Y,
coefficients C and matrix of predictor variables X. The routine
'gsl_multifit_robust' must first be called to compute the
statisical leverages h_i of the matrix X and residual standard
deviation estimate \sigma.
-- Function: gsl_multifit_robust_stats gsl_multifit_robust_statistics
(const gsl_multifit_robust_workspace * W)
This function returns a structure containing relevant statistics
from a robust regression. The function 'gsl_multifit_robust' must
be called first to perform the regression and calculate these
statistics. The returned 'gsl_multifit_robust_stats' structure
contains the following fields.
double 'sigma_ols' This contains the standard deviation of the
residuals as computed from ordinary least squares (OLS).
double 'sigma_mad' This contains an estimate of the standard
deviation of the final residuals using the
Median-Absolute-Deviation statistic
double 'sigma_rob' This contains an estimate of the standard
deviation of the final residuals from the theory of robust
regression (see Street et al, 1988).
double 'sigma' This contains an estimate of the standard
deviation of the final residuals by attemping to reconcile
'sigma_rob' and 'sigma_ols' in a reasonable way.
double 'Rsq' This contains the R^2 coefficient of
determination statistic using the estimate 'sigma'.
double 'adj_Rsq' This contains the adjusted R^2 coefficient of
determination statistic using the estimate 'sigma'.
double 'rmse' This contains the root mean squared error of the
final residuals
double 'sse' This contains the residual sum of squares taking
into account the robust covariance matrix.
size_t 'dof' This contains the number of degrees of freedom n
- p
size_t 'numit' Upon successful convergence, this contains the
number of iterations performed
gsl_vector * 'weights' This contains the final weight vector
of length N
gsl_vector * 'r' This contains the final residual vector of
length N, r = y - X c
38.6 Large dense linear systems
===============================
This module is concerned with solving large dense least squares systems
X c = y where the n-by-p matrix X has n >> p (ie: many more rows than
columns). This type of matrix is called a "tall skinny" matrix, and for
some applications, it may not be possible to fit the entire matrix in
memory at once to use the standard SVD approach. Therefore, the
algorithms in this module are designed to allow the user to construct
smaller blocks of the matrix X and accumulate those blocks into the
larger system one at a time. The algorithms in this module never need
to store the entire matrix X in memory. The large linear least squares
routines support data weights and Tikhonov regularization, and are
designed to minimize the residual
\chi^2 = || y - Xc ||_W^2 + \lambda^2 || L c ||^2
where y is the n-by-1 observation vector, X is the n-by-p design
matrix, c is the p-by-1 solution vector, W = diag(w_1,...,w_n) is the
data weighting matrix, L is an m-by-p regularization matrix, \lambda is
a regularization parameter, and ||r||_W^2 = r^T W r. In the discussion
which follows, we will assume that the system has been converted into
Tikhonov standard form,
\chi^2 = || y~ - X~ c~ ||^2 + \lambda^2 || c~ ||^2
and we will drop the tilde characters from the various parameters.
For a discussion of the transformation to standard form *note
Regularized regression::.
The basic idea is to partition the matrix X and observation vector y
as
[ X_1 ] c = [ y_1 ]
[ X_2 ] [ y_2 ]
[ X_3 ] [ y_3 ]
[ ... ] [ ... ]
[ X_k ] [ y_k ]
into k blocks, where each block (X_i,y_i) may have any number of
rows, but each X_i has p columns. The sections below describe the
methods available for solving this partitioned system. The functions
are declared in the header file 'gsl_multilarge.h'.
38.6.1 Normal Equations Approach
--------------------------------
The normal equations approach to the large linear least squares problem
described above is popular due to its speed and simplicity. Since the
normal equations solution to the problem is given by
c = ( X^T X + \lambda^2 I )^-1 X^T y
only the p-by-p matrix X^T X and p-by-1 vector X^T y need to be
stored. Using the partition scheme described above, these are given by
X^T X = \sum_i X_i^T X_i
X^T y = \sum_i X_i^T y_i
Since the matrix X^T X is symmetric, only half of it needs to be
calculated. Once all of the blocks (X_i,y_i) have been accumulated into
the final X^T X and X^T y, the system can be solved with a Cholesky
factorization of the X^T X matrix. If the Cholesky factorization fails
(occasionally due to numerical rounding errors), a QR decomposition is
then used. In both cases, the X^T X matrix is first transformed via a
diagonal scaling transformation to attempt to reduce its condition
number as much as possible to recover a more accurate solution vector.
The normal equations approach is the fastest method for solving the
large least squares problem, and is accurate for well-conditioned
matrices X. However, for ill-conditioned matrices, as is often the case
for large systems, this method can suffer from numerical instabilities
(see Trefethen and Bau, 1997). The number of operations for this method
is O(np^2 + {1 \over 3}p^3).
38.6.2 Tall Skinny QR (TSQR) Approach
-------------------------------------
An algorithm which has better numerical stability for ill-conditioned
problems is known as the Tall Skinny QR (TSQR) method. This method is
based on computing the thin QR decomposition of the least squares matrix
X = Q R, where Q is an n-by-p matrix with orthogonal columns, and R is a
p-by-p upper triangular matrix. Once these factors are calculated, the
residual becomes
\chi^2 = || Q^T y - R c ||^2 + \lambda^2 || c ||^2
which can be written as the matrix equation
[ R ; \lambda I ] c = [ Q^T b ; 0 ]
The matrix on the left hand side is now a much smaller 2p-by-p matrix
which can be solved with a standard SVD approach. The Q matrix is just
as large as the original matrix X, however it does not need to be
explicitly constructed. The TSQR algorithm computes only the p-by-p
matrix R and the p-by-1 vector Q^T y, and updates these quantities as
new blocks are added to the system. Each time a new block of rows
(X_i,y_i) is added, the algorithm performs a QR decomposition of the
matrix
[ R_(i-1) ; X_i ]
where R_{i-1} is the upper triangular R factor for the matrix
[ X_1 ; ... ; X_(i-1) ]
This QR decomposition is done efficiently taking into account the
sparse structure of R_{i-1}. See Demmel et al, 2008 for more details on
how this is accomplished. The number of operations for this method is
O(2np^2 - {2 \over 3}p^3).
38.6.3 Large Dense Linear Systems Solution Steps
------------------------------------------------
The typical steps required to solve large regularized linear least
squares problems are as follows:
1. Choose the regularization matrix L.
2. Construct a block of rows of the least squares matrix, right hand
side vector, and weight vector (X_i, y_i, w_i).
3. Transform the block to standard form (\tilde{X_i},\tilde{y_i}).
This step can be skipped if L = I and W = I.
4. Accumulate the standard form block (\tilde{X_i},\tilde{y_i}) into
the system.
5. Repeat steps 2-4 until the entire matrix and right hand side vector
have been accumulated.
6. Determine an appropriate regularization parameter \lambda (using
for example L-curve analysis).
7. Solve the standard form system using the chosen \lambda.
8. Backtransform the standard form solution \tilde{c} to recover the
original solution vector c.
38.6.4 Large Dense Linear Least Squares Routines
------------------------------------------------
-- Function: gsl_multilarge_linear_workspace *
gsl_multilarge_linear_alloc (const gsl_multilarge_linear_type
* T, const size_t P)
This function allocates a workspace for solving large linear least
squares systems. The least squares matrix X has P columns, but may
have any number of rows. The parameter T specifies the method to
be used for solving the large least squares system and may be
selected from the following choices
-- Multilarge type: gsl_multilarge_linear_normal
This specifies the normal equations approach for solving the
least squares system. This method is suitable in cases where
performance is critical and it is known that the least squares
matrix X is well conditioned. The size of this workspace is
O(p^2).
-- Multilarge type: gsl_multilarge_linear_tsqr
This specifies the sequential Tall Skinny QR (TSQR) approach
for solving the least squares system. This method is a good
general purpose choice for large systems, but requires about
twice as many operations as the normal equations method for n
>> p. The size of this workspace is O(p^2).
-- Function: void gsl_multilarge_linear_free
(gsl_multilarge_linear_workspace * W)
This function frees the memory associated with the workspace W.
-- Function: const char * gsl_multilarge_linear_name
(gsl_multilarge_linear_workspace * W)
This function returns a string pointer to the name of the
multilarge solver.
-- Function: int gsl_multilarge_linear_reset
(gsl_multilarge_linear_workspace * W)
This function resets the workspace W so it can begin to accumulate
a new least squares system.
-- Function: int gsl_multilarge_linear_stdform1 (const gsl_vector * L,
const gsl_matrix * X, const gsl_vector * Y, gsl_matrix * XS,
gsl_vector * YS, gsl_multilarge_linear_workspace * WORK)
-- Function: int gsl_multilarge_linear_wstdform1 (const gsl_vector * L,
const gsl_matrix * X, const gsl_vector * W, const gsl_vector *
Y, gsl_matrix * XS, gsl_vector * YS,
gsl_multilarge_linear_workspace * WORK)
These functions define a regularization matrix L =
diag(l_0,l_1,...,l_{p-1}). The diagonal matrix element l_i is
provided by the ith element of the input vector L. The block (X,Y)
is converted to standard form and the parameters
(\tilde{X},\tilde{y}) are stored in XS and YS on output. XS and YS
have the same dimensions as X and Y. Optional data weights may be
supplied in the vector W. In order to apply this transformation,
L^{-1} must exist and so none of the l_i may be zero. After the
standard form system has been solved, use
'gsl_multilarge_linear_genform1' to recover the original solution
vector. It is allowed to have X = XS and Y = YS for an in-place
transform.
-- Function: int gsl_multilarge_linear_L_decomp (gsl_matrix * L,
gsl_vector * TAU)
This function calculates the QR decomposition of the m-by-p
regularization matrix L. L must have m \ge p. On output, the
Householder scalars are stored in the vector TAU of size p. These
outputs will be used by 'gsl_multilarge_linear_wstdform2' to
complete the transformation to standard form.
-- Function: int gsl_multilarge_linear_stdform2 (const gsl_matrix *
LQR, const gsl_vector * LTAU, const gsl_matrix * X, const
gsl_vector * Y, gsl_matrix * XS, gsl_vector * YS,
gsl_multilarge_linear_workspace * WORK)
-- Function: int gsl_multilarge_linear_wstdform2 (const gsl_matrix *
LQR, const gsl_vector * LTAU, const gsl_matrix * X, const
gsl_vector * W, const gsl_vector * Y, gsl_matrix * XS,
gsl_vector * YS, gsl_multilarge_linear_workspace * WORK)
These functions convert a block of rows (X,Y,W) to standard form
(\tilde{X},\tilde{y}) which are stored in XS and YS respectively.
X, Y, and W must all have the same number of rows. The m-by-p
regularization matrix L is specified by the inputs LQR and LTAU,
which are outputs from 'gsl_multilarge_linear_L_decomp'. XS and YS
have the same dimensions as X and Y. After the standard form
system has been solved, use 'gsl_multilarge_linear_genform2' to
recover the original solution vector. Optional data weights may be
supplied in the vector W, where W = diag(w).
-- Function: int gsl_multilarge_linear_accumulate (gsl_matrix * X,
gsl_vector * Y, gsl_multilarge_linear_workspace * W)
This function accumulates the standard form block (X,y) into the
current least squares system. X and Y have the same number of
rows, which can be arbitrary. X must have p columns. For the TSQR
method, X and Y are destroyed on output. For the normal equations
method, they are both unchanged.
-- Function: int gsl_multilarge_linear_solve (const double LAMBDA,
gsl_vector * C, double * RNORM, double * SNORM,
gsl_multilarge_linear_workspace * W)
After all blocks (X_i,y_i) have been accumulated into the large
least squares system, this function will compute the solution
vector which is stored in C on output. The regularization
parameter \lambda is provided in LAMBDA. On output, RNORM contains
the residual norm ||y - X c||_W and SNORM contains the solution
norm ||L c||.
-- Function: int gsl_multilarge_linear_genform1 (const gsl_vector * L,
const gsl_vector * CS, gsl_vector * C,
gsl_multilarge_linear_workspace * WORK)
After a regularized system has been solved with L =
diag(\l_0,\l_1,...,\l_{p-1}), this function backtransforms the
standard form solution vector CS to recover the solution vector of
the original problem C. The diagonal matrix elements l_i are
provided in the vector L. It is allowed to have C = CS for an
in-place transform.
-- Function: int gsl_multilarge_linear_genform2 (const gsl_matrix *
LQR, const gsl_vector * LTAU, const gsl_vector * CS,
gsl_vector * C, gsl_multilarge_linear_workspace * WORK)
After a regularized system has been solved with a regularization
matrix L, specified by (LQR,LTAU), this function backtransforms the
standard form solution CS to recover the solution vector of the
original problem, which is stored in C, of length p.
-- Function: int gsl_multilarge_linear_lcurve (gsl_vector * REG_PARAM,
gsl_vector * RHO, gsl_vector * ETA,
gsl_multilarge_linear_workspace * WORK)
This function computes the L-curve for a large least squares system
after it has been fully accumulated into the workspace WORK. The
output vectors REG_PARAM, RHO, and ETA must all be the same size,
and will contain the regularization parameters \lambda_i, residual
norms ||y - X c_i||, and solution norms || L c_i || which compose
the L-curve, where c_i is the regularized solution vector
corresponding to \lambda_i. The user may determine the number of
points on the L-curve by adjusting the size of these input arrays.
For the TSQR method, the regularization parameters \lambda_i are
estimated from the singular values of the triangular R factor. For
the normal equations method, they are estimated from the
eigenvalues of the X^T X matrix.
-- Function: int gsl_multilarge_linear_rcond (double * RCOND,
gsl_multilarge_linear_workspace * WORK)
This function computes the reciprocal condition number, stored in
RCOND, of the least squares matrix after it has been accumulated
into the workspace WORK. For the TSQR algorithm, this is
accomplished by calculating the SVD of the R factor, which has the
same singular values as the matrix X. For the normal equations
method, this is done by computing the eigenvalues of X^T X, which
could be inaccurate for ill-conditioned matrices X.
38.7 Troubleshooting
====================
When using models based on polynomials, care should be taken when
constructing the design matrix X. If the x values are large, then the
matrix X could be ill-conditioned since its columns are powers of x,
leading to unstable least-squares solutions. In this case it can often
help to center and scale the x values using the mean and standard
deviation:
x' = (x - mu)/sigma
and then construct the X matrix using the transformed values x'.
38.8 Examples
=============
The example programs in this section demonstrate the various linear
regression methods.
38.8.1 Simple Linear Regression Example
---------------------------------------
The following program computes a least squares straight-line fit to a
simple dataset, and outputs the best-fit line and its associated one
standard-deviation error bars.
#include
#include
int
main (void)
{
int i, n = 4;
double x[4] = { 1970, 1980, 1990, 2000 };
double y[4] = { 12, 11, 14, 13 };
double w[4] = { 0.1, 0.2, 0.3, 0.4 };
double c0, c1, cov00, cov01, cov11, chisq;
gsl_fit_wlinear (x, 1, w, 1, y, 1, n,
&c0, &c1, &cov00, &cov01, &cov11,
&chisq);
printf ("# best fit: Y = %g + %g X\n", c0, c1);
printf ("# covariance matrix:\n");
printf ("# [ %g, %g\n# %g, %g]\n",
cov00, cov01, cov01, cov11);
printf ("# chisq = %g\n", chisq);
for (i = 0; i < n; i++)
printf ("data: %g %g %g\n",
x[i], y[i], 1/sqrt(w[i]));
printf ("\n");
for (i = -30; i < 130; i++)
{
double xf = x[0] + (i/100.0) * (x[n-1] - x[0]);
double yf, yf_err;
gsl_fit_linear_est (xf,
c0, c1,
cov00, cov01, cov11,
&yf, &yf_err);
printf ("fit: %g %g\n", xf, yf);
printf ("hi : %g %g\n", xf, yf + yf_err);
printf ("lo : %g %g\n", xf, yf - yf_err);
}
return 0;
}
The following commands extract the data from the output of the program
and display it using the GNU plotutils 'graph' utility,
$ ./demo > tmp
$ more tmp
# best fit: Y = -106.6 + 0.06 X
# covariance matrix:
# [ 39602, -19.9
# -19.9, 0.01]
# chisq = 0.8
$ for n in data fit hi lo ;
do
grep "^$n" tmp | cut -d: -f2 > $n ;
done
$ graph -T X -X x -Y y -y 0 20 -m 0 -S 2 -Ie data
-S 0 -I a -m 1 fit -m 2 hi -m 2 lo
38.8.2 Multi-parameter Linear Regression Example
------------------------------------------------
The following program performs a quadratic fit y = c_0 + c_1 x + c_2 x^2
to a weighted dataset using the generalised linear fitting function
'gsl_multifit_wlinear'. The model matrix X for a quadratic fit is given
by,
X = [ 1 , x_0 , x_0^2 ;
1 , x_1 , x_1^2 ;
1 , x_2 , x_2^2 ;
... , ... , ... ]
where the column of ones corresponds to the constant term c_0. The two
remaining columns corresponds to the terms c_1 x and c_2 x^2.
The program reads N lines of data in the format (X, Y, ERR) where ERR
is the error (standard deviation) in the value Y.
#include
#include
int
main (int argc, char **argv)
{
int i, n;
double xi, yi, ei, chisq;
gsl_matrix *X, *cov;
gsl_vector *y, *w, *c;
if (argc != 2)
{
fprintf (stderr,"usage: fit n < data\n");
exit (-1);
}
n = atoi (argv[1]);
X = gsl_matrix_alloc (n, 3);
y = gsl_vector_alloc (n);
w = gsl_vector_alloc (n);
c = gsl_vector_alloc (3);
cov = gsl_matrix_alloc (3, 3);
for (i = 0; i < n; i++)
{
int count = fscanf (stdin, "%lg %lg %lg",
&xi, &yi, &ei);
if (count != 3)
{
fprintf (stderr, "error reading file\n");
exit (-1);
}
printf ("%g %g +/- %g\n", xi, yi, ei);
gsl_matrix_set (X, i, 0, 1.0);
gsl_matrix_set (X, i, 1, xi);
gsl_matrix_set (X, i, 2, xi*xi);
gsl_vector_set (y, i, yi);
gsl_vector_set (w, i, 1.0/(ei*ei));
}
{
gsl_multifit_linear_workspace * work
= gsl_multifit_linear_alloc (n, 3);
gsl_multifit_wlinear (X, w, y, c, cov,
&chisq, work);
gsl_multifit_linear_free (work);
}
#define C(i) (gsl_vector_get(c,(i)))
#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
{
printf ("# best fit: Y = %g + %g X + %g X^2\n",
C(0), C(1), C(2));
printf ("# covariance matrix:\n");
printf ("[ %+.5e, %+.5e, %+.5e \n",
COV(0,0), COV(0,1), COV(0,2));
printf (" %+.5e, %+.5e, %+.5e \n",
COV(1,0), COV(1,1), COV(1,2));
printf (" %+.5e, %+.5e, %+.5e ]\n",
COV(2,0), COV(2,1), COV(2,2));
printf ("# chisq = %g\n", chisq);
}
gsl_matrix_free (X);
gsl_vector_free (y);
gsl_vector_free (w);
gsl_vector_free (c);
gsl_matrix_free (cov);
return 0;
}
A suitable set of data for fitting can be generated using the following
program. It outputs a set of points with gaussian errors from the curve
y = e^x in the region 0 < x < 2.
#include
#include
#include
int
main (void)
{
double x;
const gsl_rng_type * T;
gsl_rng * r;
gsl_rng_env_setup ();
T = gsl_rng_default;
r = gsl_rng_alloc (T);
for (x = 0.1; x < 2; x+= 0.1)
{
double y0 = exp (x);
double sigma = 0.1 * y0;
double dy = gsl_ran_gaussian (r, sigma);
printf ("%g %g %g\n", x, y0 + dy, sigma);
}
gsl_rng_free(r);
return 0;
}
The data can be prepared by running the resulting executable program,
$ GSL_RNG_TYPE=mt19937_1999 ./generate > exp.dat
$ more exp.dat
0.1 0.97935 0.110517
0.2 1.3359 0.12214
0.3 1.52573 0.134986
0.4 1.60318 0.149182
0.5 1.81731 0.164872
0.6 1.92475 0.182212
....
To fit the data use the previous program, with the number of data points
given as the first argument. In this case there are 19 data points.
$ ./fit 19 < exp.dat
0.1 0.97935 +/- 0.110517
0.2 1.3359 +/- 0.12214
...
# best fit: Y = 1.02318 + 0.956201 X + 0.876796 X^2
# covariance matrix:
[ +1.25612e-02, -3.64387e-02, +1.94389e-02
-3.64387e-02, +1.42339e-01, -8.48761e-02
+1.94389e-02, -8.48761e-02, +5.60243e-02 ]
# chisq = 23.0987
The parameters of the quadratic fit match the coefficients of the
expansion of e^x, taking into account the errors on the parameters and
the O(x^3) difference between the exponential and quadratic functions
for the larger values of x. The errors on the parameters are given by
the square-root of the corresponding diagonal elements of the covariance
matrix. The chi-squared per degree of freedom is 1.4, indicating a
reasonable fit to the data.
38.8.3 Regularized Linear Regression Example 1
----------------------------------------------
The next program demonstrates the difference between ordinary and
regularized least squares when the design matrix is near-singular. In
this program, we generate two random normally distributed variables u
and v, with v = u + noise so that u and v are nearly colinear. We then
set a third dependent variable y = u + v + noise and solve for the
coefficients c_1,c_2 of the model Y(c_1,c_2) = c_1 u + c_2 v. Since u
\approx v, the design matrix X is nearly singular, leading to unstable
ordinary least squares solutions.
Here is the program output:
matrix condition number = 1.025113e+04
=== Unregularized fit ===
best fit: y = -43.6588 u + 45.6636 v
residual norm = 31.6248
solution norm = 63.1764
chisq/dof = 1.00213
=== Regularized fit (L-curve) ===
optimal lambda: 4.51103
best fit: y = 1.00113 u + 1.0032 v
residual norm = 31.6547
solution norm = 1.41728
chisq/dof = 1.04499
=== Regularized fit (GCV) ===
optimal lambda: 0.0232029
best fit: y = -19.8367 u + 21.8417 v
residual norm = 31.6332
solution norm = 29.5051
chisq/dof = 1.00314
We see that the ordinary least squares solution is completely wrong,
while the L-curve regularized method with the optimal \lambda = 4.51103
finds the correct solution c_1 \approx c_2 \approx 1. The GCV
regularized method finds a regularization parameter \lambda = 0.0232029
which is too small to give an accurate solution, although it performs
better than OLS. The L-curve and its computed corner, as well as the GCV
curve and its minimum are plotted below.
The program is given below.
#include
#include
#include
#include
#include
#include
int
main()
{
const size_t n = 1000; /* number of observations */
const size_t p = 2; /* number of model parameters */
size_t i;
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);
for (i = 0; i < n; ++i)
{
/* generate first random variable u */
double ui = 5.0 * gsl_ran_gaussian(r, 1.0);
/* set v = u + noise */
double vi = ui + gsl_ran_gaussian(r, 0.001);
/* set y = u + v + noise */
double yi = ui + vi + gsl_ran_gaussian(r, 1.0);
/* since u =~ v, the matrix X is ill-conditioned */
gsl_matrix_set(X, i, 0, ui);
gsl_matrix_set(X, i, 1, vi);
/* rhs vector */
gsl_vector_set(y, i, yi);
}
{
const size_t npoints = 200; /* number of points on L-curve and GCV curve */
gsl_multifit_linear_workspace *w =
gsl_multifit_linear_alloc(n, p);
gsl_vector *c = gsl_vector_alloc(p); /* OLS solution */
gsl_vector *c_lcurve = gsl_vector_alloc(p); /* regularized solution (L-curve) */
gsl_vector *c_gcv = gsl_vector_alloc(p); /* regularized solution (GCV) */
gsl_vector *reg_param = gsl_vector_alloc(npoints);
gsl_vector *rho = gsl_vector_alloc(npoints); /* residual norms */
gsl_vector *eta = gsl_vector_alloc(npoints); /* solution norms */
gsl_vector *G = gsl_vector_alloc(npoints); /* GCV function values */
double lambda_l; /* optimal regularization parameter (L-curve) */
double lambda_gcv; /* optimal regularization parameter (GCV) */
double G_gcv; /* G(lambda_gcv) */
size_t reg_idx; /* index of optimal lambda */
double rcond; /* reciprocal condition number of X */
double chisq, rnorm, snorm;
/* compute SVD of X */
gsl_multifit_linear_svd(X, w);
rcond = gsl_multifit_linear_rcond(w);
fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);
/* unregularized (standard) least squares fit, lambda = 0 */
gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0);
fprintf(stderr, "=== Unregularized fit ===\n");
fprintf(stderr, "best fit: y = %g u + %g v\n",
gsl_vector_get(c, 0), gsl_vector_get(c, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate L-curve and find its corner */
gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
gsl_multifit_linear_lcorner(rho, eta, ®_idx);
/* store optimal regularization parameter */
lambda_l = gsl_vector_get(reg_param, reg_idx);
/* regularize with lambda_l */
gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (L-curve) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_l);
fprintf(stderr, "best fit: y = %g u + %g v\n",
gsl_vector_get(c_lcurve, 0), gsl_vector_get(c_lcurve, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate GCV curve and find its minimum */
gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);
/* regularize with lambda_gcv */
gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (GCV) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
fprintf(stderr, "best fit: y = %g u + %g v\n",
gsl_vector_get(c_gcv, 0), gsl_vector_get(c_gcv, 1));
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* output L-curve and GCV curve */
for (i = 0; i < npoints; ++i)
{
printf("%e %e %e %e\n",
gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i),
gsl_vector_get(G, i));
}
/* output L-curve corner point */
printf("\n\n%f %f\n",
gsl_vector_get(rho, reg_idx),
gsl_vector_get(eta, reg_idx));
/* output GCV curve corner minimum */
printf("\n\n%e %e\n",
lambda_gcv,
G_gcv);
gsl_multifit_linear_free(w);
gsl_vector_free(c);
gsl_vector_free(c_lcurve);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
gsl_vector_free(G);
}
gsl_rng_free(r);
gsl_matrix_free(X);
gsl_vector_free(y);
return 0;
}
38.8.4 Regularized Linear Regression Example 2
----------------------------------------------
The following example program minimizes the cost function
||y - X c||^2 + \lambda^2 ||x||^2
where X is the 10-by-8 Hilbert matrix whose entries are given by
X_{ij} = 1 / (i + j - 1)
and the right hand side vector is given by y =
[1,-1,1,-1,1,-1,1,-1,1,-1]^T. Solutions are computed for \lambda = 0
(unregularized) as well as for optimal parameters \lambda chosen by
analyzing the L-curve and GCV curve.
Here is the program output:
matrix condition number = 3.565872e+09
=== Unregularized fit ===
residual norm = 2.15376
solution norm = 2.92217e+09
chisq/dof = 2.31934
=== Regularized fit (L-curve) ===
optimal lambda: 7.11407e-07
residual norm = 2.60386
solution norm = 424507
chisq/dof = 3.43565
=== Regularized fit (GCV) ===
optimal lambda: 1.72278
residual norm = 3.1375
solution norm = 0.139357
chisq/dof = 4.95076
Here we see the unregularized solution results in a large solution norm
due to the ill-conditioned matrix. The L-curve solution finds a small
value of \lambda = 7.11e-7 which still results in a badly conditioned
system and a large solution norm. The GCV method finds a parameter
\lambda = 1.72 which results in a well-conditioned system and small
solution norm.
The L-curve and its computed corner, as well as the GCV curve and its
minimum are plotted below.
The program is given below.
#include
#include
#include
#include
#include
static int
hilbert_matrix(gsl_matrix * m)
{
const size_t N = m->size1;
const size_t M = m->size2;
size_t i, j;
for (i = 0; i < N; i++)
{
for (j = 0; j < M; j++)
{
gsl_matrix_set(m, i, j, 1.0/(i+j+1.0));
}
}
return GSL_SUCCESS;
}
int
main()
{
const size_t n = 10; /* number of observations */
const size_t p = 8; /* number of model parameters */
size_t i;
gsl_matrix *X = gsl_matrix_alloc(n, p);
gsl_vector *y = gsl_vector_alloc(n);
/* construct Hilbert matrix and rhs vector */
hilbert_matrix(X);
{
double val = 1.0;
for (i = 0; i < n; ++i)
{
gsl_vector_set(y, i, val);
val *= -1.0;
}
}
{
const size_t npoints = 200; /* number of points on L-curve and GCV curve */
gsl_multifit_linear_workspace *w =
gsl_multifit_linear_alloc(n, p);
gsl_vector *c = gsl_vector_alloc(p); /* OLS solution */
gsl_vector *c_lcurve = gsl_vector_alloc(p); /* regularized solution (L-curve) */
gsl_vector *c_gcv = gsl_vector_alloc(p); /* regularized solution (GCV) */
gsl_vector *reg_param = gsl_vector_alloc(npoints);
gsl_vector *rho = gsl_vector_alloc(npoints); /* residual norms */
gsl_vector *eta = gsl_vector_alloc(npoints); /* solution norms */
gsl_vector *G = gsl_vector_alloc(npoints); /* GCV function values */
double lambda_l; /* optimal regularization parameter (L-curve) */
double lambda_gcv; /* optimal regularization parameter (GCV) */
double G_gcv; /* G(lambda_gcv) */
size_t reg_idx; /* index of optimal lambda */
double rcond; /* reciprocal condition number of X */
double chisq, rnorm, snorm;
/* compute SVD of X */
gsl_multifit_linear_svd(X, w);
rcond = gsl_multifit_linear_rcond(w);
fprintf(stderr, "matrix condition number = %e\n", 1.0 / rcond);
/* unregularized (standard) least squares fit, lambda = 0 */
gsl_multifit_linear_solve(0.0, X, y, c, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0);
fprintf(stderr, "=== Unregularized fit ===\n");
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate L-curve and find its corner */
gsl_multifit_linear_lcurve(y, reg_param, rho, eta, w);
gsl_multifit_linear_lcorner(rho, eta, ®_idx);
/* store optimal regularization parameter */
lambda_l = gsl_vector_get(reg_param, reg_idx);
/* regularize with lambda_l */
gsl_multifit_linear_solve(lambda_l, X, y, c_lcurve, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_l * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (L-curve) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_l);
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* calculate GCV curve and find its minimum */
gsl_multifit_linear_gcv(y, reg_param, G, &lambda_gcv, &G_gcv, w);
/* regularize with lambda_gcv */
gsl_multifit_linear_solve(lambda_gcv, X, y, c_gcv, &rnorm, &snorm, w);
chisq = pow(rnorm, 2.0) + pow(lambda_gcv * snorm, 2.0);
fprintf(stderr, "=== Regularized fit (GCV) ===\n");
fprintf(stderr, "optimal lambda: %g\n", lambda_gcv);
fprintf(stderr, "residual norm = %g\n", rnorm);
fprintf(stderr, "solution norm = %g\n", snorm);
fprintf(stderr, "chisq/dof = %g\n", chisq / (n - p));
/* output L-curve and GCV curve */
for (i = 0; i < npoints; ++i)
{
printf("%e %e %e %e\n",
gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i),
gsl_vector_get(G, i));
}
/* output L-curve corner point */
printf("\n\n%f %f\n",
gsl_vector_get(rho, reg_idx),
gsl_vector_get(eta, reg_idx));
/* output GCV curve corner minimum */
printf("\n\n%e %e\n",
lambda_gcv,
G_gcv);
gsl_multifit_linear_free(w);
gsl_vector_free(c);
gsl_vector_free(c_lcurve);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
gsl_vector_free(G);
}
gsl_matrix_free(X);
gsl_vector_free(y);
return 0;
}
38.8.5 Robust Linear Regression Example
---------------------------------------
The next program demonstrates the advantage of robust least squares on a
dataset with outliers. The program generates linear (x,y) data pairs on
the line y = 1.45 x + 3.88, adds some random noise, and inserts 3
outliers into the dataset. Both the robust and ordinary least squares
(OLS) coefficients are computed for comparison.
#include
#include
#include
int
dofit(const gsl_multifit_robust_type *T,
const gsl_matrix *X, const gsl_vector *y,
gsl_vector *c, gsl_matrix *cov)
{
int s;
gsl_multifit_robust_workspace * work
= gsl_multifit_robust_alloc (T, X->size1, X->size2);
s = gsl_multifit_robust (X, y, c, cov, work);
gsl_multifit_robust_free (work);
return s;
}
int
main (int argc, char **argv)
{
size_t i;
size_t n;
const size_t p = 2; /* linear fit */
gsl_matrix *X, *cov;
gsl_vector *x, *y, *c, *c_ols;
const double a = 1.45; /* slope */
const double b = 3.88; /* intercept */
gsl_rng *r;
if (argc != 2)
{
fprintf (stderr,"usage: robfit n\n");
exit (-1);
}
n = atoi (argv[1]);
X = gsl_matrix_alloc (n, p);
x = gsl_vector_alloc (n);
y = gsl_vector_alloc (n);
c = gsl_vector_alloc (p);
c_ols = gsl_vector_alloc (p);
cov = gsl_matrix_alloc (p, p);
r = gsl_rng_alloc(gsl_rng_default);
/* generate linear dataset */
for (i = 0; i < n - 3; i++)
{
double dx = 10.0 / (n - 1.0);
double ei = gsl_rng_uniform(r);
double xi = -5.0 + i * dx;
double yi = a * xi + b;
gsl_vector_set (x, i, xi);
gsl_vector_set (y, i, yi + ei);
}
/* add a few outliers */
gsl_vector_set(x, n - 3, 4.7);
gsl_vector_set(y, n - 3, -8.3);
gsl_vector_set(x, n - 2, 3.5);
gsl_vector_set(y, n - 2, -6.7);
gsl_vector_set(x, n - 1, 4.1);
gsl_vector_set(y, n - 1, -6.0);
/* construct design matrix X for linear fit */
for (i = 0; i < n; ++i)
{
double xi = gsl_vector_get(x, i);
gsl_matrix_set (X, i, 0, 1.0);
gsl_matrix_set (X, i, 1, xi);
}
/* perform robust and OLS fit */
dofit(gsl_multifit_robust_ols, X, y, c_ols, cov);
dofit(gsl_multifit_robust_bisquare, X, y, c, cov);
/* output data and model */
for (i = 0; i < n; ++i)
{
double xi = gsl_vector_get(x, i);
double yi = gsl_vector_get(y, i);
gsl_vector_view v = gsl_matrix_row(X, i);
double y_ols, y_rob, y_err;
gsl_multifit_robust_est(&v.vector, c, cov, &y_rob, &y_err);
gsl_multifit_robust_est(&v.vector, c_ols, cov, &y_ols, &y_err);
printf("%g %g %g %g\n", xi, yi, y_rob, y_ols);
}
#define C(i) (gsl_vector_get(c,(i)))
#define COV(i,j) (gsl_matrix_get(cov,(i),(j)))
{
printf ("# best fit: Y = %g + %g X\n",
C(0), C(1));
printf ("# covariance matrix:\n");
printf ("# [ %+.5e, %+.5e\n",
COV(0,0), COV(0,1));
printf ("# %+.5e, %+.5e\n",
COV(1,0), COV(1,1));
}
gsl_matrix_free (X);
gsl_vector_free (x);
gsl_vector_free (y);
gsl_vector_free (c);
gsl_vector_free (c_ols);
gsl_matrix_free (cov);
gsl_rng_free(r);
return 0;
}
The output from the program is shown in the following plot.
38.8.6 Large Dense Linear Regression Example
--------------------------------------------
The following program demostrates the large dense linear least squares
solvers. This example is adapted from Trefethen and Bau, and fits the
function f(t) = \exp{(\sin^3{(10t)}}) on the interval [0,1] with a
degree 15 polynomial. The program generates n = 50000 equally spaced
points t_i on this interval, calculates the function value and adds
random noise to determine the observation value y_i. The entries of the
least squares matrix are X_{ij} = t_i^j, representing a polynomial fit.
The matrix is highly ill-conditioned, with a condition number of about
1.4 \cdot 10^{11}. The program accumulates the matrix into the least
squares system in 5 blocks, each with 10000 rows. This way the full
matrix X is never stored in memory. We solve the system with both the
normal equations and TSQR methods. The results are shown in the plot
below. In the top left plot, we see the unregularized normal equations
solution has larger error than TSQR due to the ill-conditioning of the
matrix. In the bottom left plot, we show the L-curve, which exhibits
multiple corners. In the top right panel, we plot a regularized
solution using \lambda = 10^{-6}. The TSQR and normal solutions now
agree, however they are unable to provide a good fit due to the damping.
This indicates that for some ill-conditioned problems, regularizing the
normal equations does not improve the solution. This is further
illustrated in the bottom right panel, where we plot the L-curve
calculated from the normal equations. The curve agrees with the TSQR
curve for larger damping parameters, but for small \lambda, the normal
equations approach cannot provide accurate solution vectors leading to
numerical inaccuracies in the left portion of the curve.
#include
#include
#include
#include
#include
#include
#include
#include
/* function to be fitted */
double
func(const double t)
{
double x = sin(10.0 * t);
return exp(x*x*x);
}
/* construct a row of the least squares matrix */
int
build_row(const double t, gsl_vector *row)
{
const size_t p = row->size;
double Xj = 1.0;
size_t j;
for (j = 0; j < p; ++j)
{
gsl_vector_set(row, j, Xj);
Xj *= t;
}
return 0;
}
int
solve_system(const int print_data, const gsl_multilarge_linear_type * T,
const double lambda, const size_t n, const size_t p,
gsl_vector * c)
{
const size_t nblock = 5; /* number of blocks to accumulate */
const size_t nrows = n / nblock; /* number of rows per block */
gsl_multilarge_linear_workspace * w =
gsl_multilarge_linear_alloc(T, p);
gsl_matrix *X = gsl_matrix_alloc(nrows, p);
gsl_vector *y = gsl_vector_alloc(nrows);
gsl_rng *r = gsl_rng_alloc(gsl_rng_default);
const size_t nlcurve = 200;
gsl_vector *reg_param = gsl_vector_alloc(nlcurve);
gsl_vector *rho = gsl_vector_alloc(nlcurve);
gsl_vector *eta = gsl_vector_alloc(nlcurve);
size_t rowidx = 0;
double rnorm, snorm, rcond;
double t = 0.0;
double dt = 1.0 / (n - 1.0);
while (rowidx < n)
{
size_t nleft = n - rowidx; /* number of rows left to accumulate */
size_t nr = GSL_MIN(nrows, nleft); /* number of rows in this block */
gsl_matrix_view Xv = gsl_matrix_submatrix(X, 0, 0, nr, p);
gsl_vector_view yv = gsl_vector_subvector(y, 0, nr);
size_t i;
/* build (X,y) block with 'nr' rows */
for (i = 0; i < nr; ++i)
{
gsl_vector_view row = gsl_matrix_row(&Xv.matrix, i);
double fi = func(t);
double ei = gsl_ran_gaussian (r, 0.1 * fi); /* noise */
double yi = fi + ei;
/* construct this row of LS matrix */
build_row(t, &row.vector);
/* set right hand side value with added noise */
gsl_vector_set(&yv.vector, i, yi);
if (print_data && (i % 100 == 0))
printf("%f %f\n", t, yi);
t += dt;
}
/* accumulate (X,y) block into LS system */
gsl_multilarge_linear_accumulate(&Xv.matrix, &yv.vector, w);
rowidx += nr;
}
if (print_data)
printf("\n\n");
/* compute L-curve */
gsl_multilarge_linear_lcurve(reg_param, rho, eta, w);
/* solve large LS system and store solution in c */
gsl_multilarge_linear_solve(lambda, c, &rnorm, &snorm, w);
/* compute reciprocal condition number */
gsl_multilarge_linear_rcond(&rcond, w);
fprintf(stderr, "=== Method %s ===\n", gsl_multilarge_linear_name(w));
fprintf(stderr, "condition number = %e\n", 1.0 / rcond);
fprintf(stderr, "residual norm = %e\n", rnorm);
fprintf(stderr, "solution norm = %e\n", snorm);
/* output L-curve */
{
size_t i;
for (i = 0; i < nlcurve; ++i)
{
printf("%.12e %.12e %.12e\n",
gsl_vector_get(reg_param, i),
gsl_vector_get(rho, i),
gsl_vector_get(eta, i));
}
printf("\n\n");
}
gsl_matrix_free(X);
gsl_vector_free(y);
gsl_multilarge_linear_free(w);
gsl_rng_free(r);
gsl_vector_free(reg_param);
gsl_vector_free(rho);
gsl_vector_free(eta);
return 0;
}
int
main(int argc, char *argv[])
{
const size_t n = 50000; /* number of observations */
const size_t p = 16; /* polynomial order + 1 */
double lambda = 0.0; /* regularization parameter */
gsl_vector *c_tsqr = gsl_vector_alloc(p);
gsl_vector *c_normal = gsl_vector_alloc(p);
if (argc > 1)
lambda = atof(argv[1]);
/* solve system with TSQR method */
solve_system(1, gsl_multilarge_linear_tsqr, lambda, n, p, c_tsqr);
/* solve system with Normal equations method */
solve_system(0, gsl_multilarge_linear_normal, lambda, n, p, c_normal);
/* output solutions */
{
gsl_vector *v = gsl_vector_alloc(p);
double t;
for (t = 0.0; t <= 1.0; t += 0.01)
{
double f_exact = func(t);
double f_tsqr, f_normal;
build_row(t, v);
gsl_blas_ddot(v, c_tsqr, &f_tsqr);
gsl_blas_ddot(v, c_normal, &f_normal);
printf("%f %e %e %e\n", t, f_exact, f_tsqr, f_normal);
}
gsl_vector_free(v);
}
gsl_vector_free(c_tsqr);
gsl_vector_free(c_normal);
return 0;
}
38.9 References and Further Reading
===================================
A summary of formulas and techniques for least squares fitting can be
found in the "Statistics" chapter of the Annual Review of Particle
Physics prepared by the Particle Data Group,
'Review of Particle Properties', R.M. Barnett et al., Physical
Review D54, 1 (1996)
The Review of Particle Physics is available online at the website given
above.
The tests used to prepare these routines are based on the NIST
Statistical Reference Datasets. The datasets and their documentation
are available from NIST at the following website,
.
More information on Tikhonov regularization can be found in
Hansen, P. C. (1998), Rank-Deficient and Discrete Ill-Posed
Problems: Numerical Aspects of Linear Inversion. SIAM Monogr. on
Mathematical Modeling and Computation, Society for Industrial and
Applied Mathematics
M. Rezghi and S. M. Hosseini (2009), A new variant of L-curve for
Tikhonov regularization, Journal of Computational and Applied
Mathematics, Volume 231, Issue 2, pages 914-924.
The GSL implementation of robust linear regression closely follows the
publications
DuMouchel, W. and F. O'Brien (1989), "Integrating a robust option
into a multiple regression computing environment," Computer Science
and Statistics: Proceedings of the 21st Symposium on the Interface,
American Statistical Association
Street, J.O., R.J. Carroll, and D. Ruppert (1988), "A note on
computing robust regression estimates via iteratively reweighted
least squares," The American Statistician, v. 42, pp. 152-154.
More information about the normal equations and TSQR approach for
solving large linear least squares systems can be found in the
publications
Trefethen, L. N. and Bau, D. (1997), "Numerical Linear Algebra",
SIAM.
Demmel, J., Grigori, L., Hoemmen, M. F., and Langou, J.
"Communication-optimal parallel and sequential QR and LU
factorizations", UCB Technical Report No. UCB/EECS-2008-89, 2008.
39 Nonlinear Least-Squares Fitting
**********************************
This chapter describes functions for multidimensional nonlinear
least-squares fitting. There are generally two classes of algorithms
for solving nonlinear least squares problems, which fall under line
search methods and trust region methods. GSL currently implements only
trust region methods and provides the user with full access to
intermediate steps of the iteration. The user also has the ability to
tune a number of parameters which affect low-level aspects of the
algorithm which can help to accelerate convergence for the specific
problem at hand. GSL provides two separate interfaces for nonlinear
least squares fitting. The first is designed for small to moderate
sized problems, and the second is designed for very large problems,
which may or may not have significant sparse structure.
The header file 'gsl_multifit_nlinear.h' contains prototypes for the
multidimensional nonlinear fitting functions and related declarations
relating to the small to moderate sized systems.
The header file 'gsl_multilarge_nlinear.h' contains prototypes for
the multidimensional nonlinear fitting functions and related
declarations relating to large systems.
39.1 Overview
=============
The problem of multidimensional nonlinear least-squares fitting requires
the minimization of the squared residuals of n functions, f_i, in p
parameters, x_i,
\Phi(x) = (1/2) || f(x) ||^2
= (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2
In trust region methods, the objective (or cost) function \Phi(x) is
approximated by a model function m_k(\delta) in the vicinity of some
point x_k. The model function is often simply a second order Taylor
series expansion around the point x_k, ie:
\Phi(x_k + \delta) ~=~ m_k(\delta) = \Phi(x_k) + g_k^T \delta + 1/2 \delta^T B_k \delta
where g_k = \nabla \Phi(x_k) = J^T f is the gradient vector at the
point x_k, B_k = \nabla^2 \Phi(x_k) is the Hessian matrix at x_k, or
some approximation to it, and J is the n-by-p Jacobian matrix J_{ij} = d
f_i / d x_j. In order to find the next step \delta, we minimize the
model function m_k(\delta), but search for solutions only within a
region where we trust that m_k(\delta) is a good approximation to the
objective function \Phi(x_k + \delta). In other words, we seek a
solution of the trust region subproblem (TRS)
\min_(\delta \in R^p) m_k(\delta), s.t. || D_k \delta || <= \Delta_k
where \Delta_k > 0 is the trust region radius and D_k is a scaling
matrix. If D_k = I, then the trust region is a ball of radius \Delta_k
centered at x_k. In some applications, the parameter vector x may have
widely different scales. For example, one parameter might be a
temperature on the order of 10^3 K, while another might be a length on
the order of 10^{-6} m. In such cases, a spherical trust region may not
be the best choice, since if \Phi changes rapidly along directions with
one scale, and more slowly along directions with a different scale, the
model function m_k may be a poor approximation to \Phi along the rapidly
changing directions. In such problems, it may be best to use an
elliptical trust region, by setting D_k to a diagonal matrix whose
entries are designed so that the scaled step D_k \delta has entries of
approximately the same order of magnitude.
The trust region subproblem above normally amounts to solving a
linear least squares system (or multiple systems) for the step \delta.
Once \delta is computed, it is checked whether or not it reduces the
objective function \Phi(x). A useful statistic for this is to look at
the ratio
\rho_k = ( \Phi(x_k) - \Phi(x_k + \delta_k) / ( m_k(0) - m_k(\delta_k) )
where the numerator is the actual reduction of the objective function
due to the step \delta_k, and the denominator is the predicted reduction
due to the model m_k. If \rho_k is negative, it means that the step
\delta_k increased the objective function and so it is rejected. If
\rho_k is positive, then we have found a step which reduced the
objective function and it is accepted. Furthermore, if \rho_k is close
to 1, then this indicates that the model function is a good
approximation to the objective function in the trust region, and so on
the next iteration the trust region is enlarged in order to take more
ambitious steps. When a step is rejected, the trust region is made
smaller and the TRS is solved again. An outline for the general trust
region method used by GSL can now be given.
Trust Region Algorithm
1. Initialize: given x_0, construct m_0(\delta), D_0 and \Delta_0 > 0
2. For k = 0, 1, 2, ...
a. If converged, then stop
b. Solve TRS for trial step \delta_k
c. Evaluate trial step by computing \rho_k
1. if step is accepted, set x_{k+1} = x_k + \delta_k and
increase radius, \Delta_{k+1} = \alpha \Delta_k
2. if step is rejected, set x_{k+1} = x_k and decrease
radius, \Delta_{k+1} = {\Delta_k \over \beta}; goto 2(b)
d. Construct m_{k+1}(\delta) and D_{k+1}
GSL offers the user a number of different algorithms for solving the
trust region subproblem in 2(b), as well as different choices of scaling
matrices D_k and different methods of updating the trust region radius
\Delta_k. Therefore, while reasonable default methods are provided, the
user has a lot of control to fine-tune the various steps of the
algorithm for their specific problem.
39.2 Solving the Trust Region Subproblem (TRS)
==============================================
Below we describe the methods available for solving the trust region
subproblem. The methods available provide either exact or approximate
solutions to the trust region subproblem. In all algorithms below, the
Hessian matrix B_k is approximated as B_k \approx J_k^T J_k, where J_k =
J(x_k). In all methods, the solution of the TRS involves solving a
linear least squares system involving the Jacobian matrix. For small to
moderate sized problems ('gsl_multifit_nlinear' interface), this is
accomplished by factoring the full Jacobian matrix, which is provided by
the user, with the Cholesky, QR, or SVD decompositions. For large
systems ('gsl_multilarge_nlinear' interface), the user has two choices.
One is to solve the system iteratively, without needing to store the
full Jacobian matrix in memory. With this method, the user must provide
a routine to calculate the matrix-vector products J u or J^T u for a
given vector u. This iterative method is particularly useful for
systems where the Jacobian has sparse structure, since forming
matrix-vector products can be done cheaply. The second option for large
systems involves forming the normal equations matrix J^T J and then
factoring it using a Cholesky decomposition. The normal equations
matrix is p-by-p, typically much smaller than the full n-by-p Jacobian,
and can usually be stored in memory even if the full Jacobian matrix
cannot. This option is useful for large, dense systems, or if the
iterative method has difficulty converging.
39.2.1 Levenberg-Marquardt
--------------------------
There is a theorem which states that if \delta_k is a solution to the
trust region subproblem given above, then there exists \mu_k \ge 0 such
that
( B_k + \mu_k D_k^T D_k ) \delta_k = -g_k
with \mu_k (\Delta_k - ||D_k \delta_k||) = 0. This forms the basis
of the Levenberg-Marquardt algorithm, which controls the trust region
size by adjusting the parameter \mu_k rather than the radius \Delta_k
directly. For each radius \Delta_k, there is a unique parameter \mu_k
which solves the TRS, and they have an inverse relationship, so that
large values of \mu_k correspond to smaller trust regions, while small
values of \mu_k correspond to larger trust regions.
With the approximation B_k \approx J_k^T J_k, on each iteration, in
order to calculate the step \delta_k, the following linear least squares
problem is solved:
[J_k; sqrt(mu_k) D_k] \delta_k = - [f_k; 0]
If the step \delta_k is accepted, then \mu_k is decreased on the next
iteration in order to take a larger step, otherwise it is increased to
take a smaller step. The Levenberg-Marquardt algorithm provides an
exact solution of the trust region subproblem, but typically has a
higher computational cost per iteration than the approximate methods
discussed below, since it may need to solve the least squares system
above several times for different values of \mu_k.
39.2.2 Levenberg-Marquardt with Geodesic Acceleration
-----------------------------------------------------
This method applies a so-called geodesic acceleration correction to the
standard Levenberg-Marquardt step \delta_k (Transtrum et al, 2011). By
interpreting \delta_k as a first order step along a geodesic in the
model parameter space (ie: a velocity \delta_k = v_k), the geodesic
acceleration a_k is a second order correction along the geodesic which
is determined by solving the linear least squares system
[J_k; sqrt(mu_k) D_k] a_k = - [f_vv(x_k); 0]
where f_{vv} is the second directional derivative of the residual vector
in the velocity direction v, f_{vv}(x) = D_v^2 f = \sum_{\alpha\beta}
v_{\alpha} v_{\beta} \partial_{\alpha} \partial_{\beta} f(x), where
\alpha and \beta are summed over the p parameters. The new total step
is then \delta_k' = v_k + {1 \over 2}a_k. The second order correction
a_k can be calculated with a modest additional cost, and has been shown
to dramatically reduce the number of iterations (and expensive Jacobian
evaluations) required to reach convergence on a variety of different
problems. In order to utilize the geodesic acceleration, the user must
supply a function which provides the second directional derivative
vector f_{vv}(x), or alternatively the library can use a finite
difference method to estimate this vector with one additional function
evaluation of f(x + h v) where h is a tunable step size (see the 'h_fvv'
parameter description).
39.2.3 Dogleg
-------------
This is Powell's dogleg method, which finds an approximate solution to
the trust region subproblem, by restricting its search to a piecewise
linear "dogleg" path, composed of the origin, the Cauchy point which
represents the model minimizer along the steepest descent direction, and
the Gauss-Newton point, which is the overall minimizer of the
unconstrained model. The Gauss-Newton step is calculated by solving
J_k \delta_gn = -f_k
which is the main computational task for each iteration, but only
needs to be performed once per iteration. If the Gauss-Newton point is
inside the trust region, it is selected as the step. If it is outside,
the method then calculates the Cauchy point, which is located along the
gradient direction. If the Cauchy point is also outside the trust
region, the method assumes that it is still far from the minimum and so
proceeds along the gradient direction, truncating the step at the trust
region boundary. If the Cauchy point is inside the trust region, with
the Gauss-Newton point outside, the method uses a dogleg step, which is
a linear combination of the gradient direction and the Gauss-Newton
direction, stopping at the trust region boundary.
39.2.4 Double Dogleg
--------------------
This method is an improvement over the classical dogleg algorithm, which
attempts to include information about the Gauss-Newton step while the
iteration is still far from the minimum. When the Cauchy point is
inside the trust region and the Gauss-Newton point is outside, the
method computes a scaled Gauss-Newton point and then takes a dogleg step
between the Cauchy point and the scaled Gauss-Newton point. The scaling
is calculated to ensure that the reduction in the model m_k is about the
same as the reduction provided by the Cauchy point.
39.2.5 Two Dimensional Subspace
-------------------------------
The dogleg methods restrict the search for the TRS solution to a 1D
curve defined by the Cauchy and Gauss-Newton points. An improvement to
this is to search for a solution using the full two dimensional subspace
spanned by the Cauchy and Gauss-Newton directions. The dogleg path is
of course inside this subspace, and so this method solves the TRS at
least as accurately as the dogleg methods. Since this method searches a
larger subspace for a solution, it can converge more quickly than dogleg
on some problems. Because the subspace is only two dimensional, this
method is very efficient and the main computation per iteration is to
determine the Gauss-Newton point.
39.2.6 Steihaug-Toint Conjugate Gradient
----------------------------------------
One difficulty of the dogleg methods is calculating the Gauss-Newton
step when the Jacobian matrix is singular. The Steihaug-Toint method
also computes a generalized dogleg step, but avoids solving for the
Gauss-Newton step directly, instead using an iterative conjugate
gradient algorithm. This method performs well at points where the
Jacobian is singular, and is also suitable for large-scale problems
where factoring the Jacobian matrix could be prohibitively expensive.
39.3 Weighted Nonlinear Least-Squares
=====================================
Weighted nonlinear least-squares fitting minimizes the function
\Phi(x) = (1/2) || f(x) ||_W^2
= (1/2) \sum_{i=1}^{n} f_i(x_1, ..., x_p)^2
where W = diag(w_1,w_2,...,w_n) is the weighting matrix, and
||f||_W^2 = f^T W f. The weights w_i are commonly defined as w_i =
1/\sigma_i^2, where \sigma_i is the error in the ith measurement. A
simple change of variables \tilde{f} = W^{1 \over 2} f yields \Phi(x) =
{1 \over 2} ||\tilde{f}||^2, which is in the same form as the unweighted
case. The user can either perform this transform directly on their
function residuals and Jacobian, or use the 'gsl_multifit_nlinear_winit'
interface which automatically performs the correct scaling. To manually
perform this transformation, the residuals and Jacobian should be
modified according to
f~_i = f_i / \sigma_i
J~_ij = 1 / \sigma_i df_i/dx_j
For large systems, the user must perform their own weighting.
39.4 Tunable Parameters
=======================
The user can tune nearly all aspects of the iteration at allocation
time. For the 'gsl_multifit_nlinear' interface, the user may modify the
'gsl_multifit_nlinear_parameters' structure, which is defined as
follows:
typedef struct
{
const gsl_multifit_nlinear_trs *trs; /* trust region subproblem method */
const gsl_multifit_nlinear_scale *scale; /* scaling method */
const gsl_multifit_nlinear_solver *solver; /* solver method */
gsl_multifit_nlinear_fdtype fdtype; /* finite difference method */
double factor_up; /* factor for increasing trust radius */
double factor_down; /* factor for decreasing trust radius */
double avmax; /* max allowed |a|/|v| */
double h_df; /* step size for finite difference Jacobian */
double h_fvv; /* step size for finite difference fvv */
} gsl_multifit_nlinear_parameters;
For the 'gsl_multilarge_nlinear' interface, the user may modify the
'gsl_multilarge_nlinear_parameters' structure, which is defined as
follows:
typedef struct
{
const gsl_multilarge_nlinear_trs *trs; /* trust region subproblem method */
const gsl_multilarge_nlinear_scale *scale; /* scaling method */
const gsl_multilarge_nlinear_solver *solver; /* solver method */
gsl_multilarge_nlinear_fdtype fdtype; /* finite difference method */
double factor_up; /* factor for increasing trust radius */
double factor_down; /* factor for decreasing trust radius */
double avmax; /* max allowed |a|/|v| */
double h_df; /* step size for finite difference Jacobian */
double h_fvv; /* step size for finite difference fvv */
size_t max_iter; /* maximum iterations for trs method */
double tol; /* tolerance for solving trs */
} gsl_multilarge_nlinear_parameters;
Each of these parameters is discussed in further detail below.
-- Parameter: const gsl_multifit_nlinear_trs * trs
-- Parameter: const gsl_multilarge_nlinear_trs * trs
This parameter determines the method used to solve the trust region
subproblem, and may be selected from the following choices,
-- Default: gsl_multifit_nlinear_trs_lm
-- Default: gsl_multilarge_nlinear_trs_lm
This selects the Levenberg-Marquardt algorithm.
-- Option: gsl_multifit_nlinear_trs_lmaccel
-- Option: gsl_multilarge_nlinear_trs_lmaccel
This selects the Levenberg-Marquardt algorithm with geodesic
acceleration.
-- Option: gsl_multifit_nlinear_trs_dogleg
-- Option: gsl_multilarge_nlinear_trs_dogleg
This selects the dogleg algorithm.
-- Option: gsl_multifit_nlinear_trs_ddogleg
-- Option: gsl_multilarge_nlinear_trs_ddogleg
This selects the double dogleg algorithm.
-- Option: gsl_multifit_nlinear_trs_subspace2D
-- Option: gsl_multilarge_nlinear_trs_subspace2D
This selects the 2D subspace algorithm.
-- Option: gsl_multilarge_nlinear_trs_cgst
This selects the Steihaug-Toint conjugate gradient algorithm.
This method is available only for large systems.
-- Parameter: const gsl_multifit_nlinear_scale * scale
-- Parameter: const gsl_multilarge_nlinear_scale * scale
This parameter determines the diagonal scaling matrix D and may be
selected from the following choices,
-- Default: gsl_multifit_nlinear_scale_more
-- Default: gsl_multilarge_nlinear_scale_more
This damping strategy was suggested by More', and corresponds
to D^T D = max(diag(J^T J)), in other words the maximum
elements of diag(J^T J) encountered thus far in the iteration.
This choice of D makes the problem scale-invariant, so that if
the model parameters x_i are each scaled by an arbitrary
constant, \tilde{x}_i = a_i x_i, then the sequence of iterates
produced by the algorithm would be unchanged. This method can
work very well in cases where the model parameters have widely
different scales (ie: if some parameters are measured in
nanometers, while others are measured in degrees Kelvin).
This strategy has been proven effective on a large class of
problems and so it is the library default, but it may not be
the best choice for all problems.
-- Option: gsl_multifit_nlinear_scale_levenberg
-- Option: gsl_multilarge_nlinear_scale_levenberg
This damping strategy was originally suggested by Levenberg,
and corresponds to D^T D = I. This method has also proven
effective on a large class of problems, but is not
scale-invariant. However, some authors (e.g. Transtrum and
Sethna 2012) argue that this choice is better for problems
which are susceptible to parameter evaporation (ie: parameters
go to infinity)
-- Option: gsl_multifit_nlinear_scale_marquardt
-- Option: gsl_multilarge_nlinear_scale_marquardt
This damping strategy was suggested by Marquardt, and
corresponds to D^T D = diag(J^T J). This method is
scale-invariant, but it is generally considered inferior to
both the Levenberg and More' strategies, though may work well
on certain classes of problems.
-- Parameter: const gsl_multifit_nlinear_solver * solver
-- Parameter: const gsl_multilarge_nlinear_solver * solver