This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 1.8 of the library. Please report any errors in this manual to bug-gsl@gnu.org.
More information about GSL can be found at the project homepage, http://www.gnu.org/software/gsl/.
Printed copies of this manual can be purchased from Network Theory Ltd at http://www.network-theory.co.uk/gsl/manual/. The money raised from sales of the manual helps support the development of GSL.
Copyright © 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006 The GSL Team.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 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 freedom to copy and modify this GNU Manual, like GNU software.”
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.
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
| |
| 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.
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 (see 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 http://www.fsf.org/).
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. Precompiled versions of the library can be purchased from commercial redistributors listed on the website above.
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.
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 (see GNU General Public License).
A list of known bugs can be found in the BUGS file included in the GSL distribution. 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 bug-gsl@gnu.org.
All bug reports should include:
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.
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 (2nd Ed.), ISBN 0954161734.
If you want to give a url, use “http://www.gnu.org/software/gsl/”.
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).
This chapter describes how to compile programs that use GSL, and introduces its conventions.
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 <stdio.h>
#include <gsl/gsl_sf_bessel.h>
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,
J0(5) = -1.775967713143382920e-01
The steps needed to compile this program are described in the following sections.
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 <gsl/gsl_math.h>
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.
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 following command line shows how you would link the same application with an alternative cblas library called libcblas,
$ 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
For more information see BLAS Support.
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, 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:$LD_LIBRARY_PATH
$ 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:$LD_LIBRARY_PATH
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 should 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
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_.
The inline keyword is not part of the original ANSI C standard
(C89) and the library does not export any inline function definitions by
default. However, the library provides optional inline versions of
performance-critical functions by conditional compilation. 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.
Note that the actual usage of the inline keyword is extern
inline, which eliminates unnecessary function definitions in gcc.
If the form extern inline causes problems with other compilers a
stricter autoconf test can be used, see Autoconf Macros.
The extended numerical type long double is part of the ANSI C
standard and should be available in every modern compiler. However, the
precision of long double is platform dependent, and this should
be considered when using it. The IEEE standard only specifies the
minimum precision of extended precision numbers, while the precision of
double is the same on all platforms.
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.
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 <config.h> 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 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.
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.
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 <gsl/gsl_foo.h> All types
#include <gsl/gsl_foo_double.h> double
#include <gsl/gsl_foo_long_double.h> long double
#include <gsl/gsl_foo_float.h> float
#include <gsl/gsl_foo_long.h> long
#include <gsl/gsl_foo_ulong.h> unsigned long
#include <gsl/gsl_foo_int.h> int
#include <gsl/gsl_foo_uint.h> unsigned int
#include <gsl/gsl_foo_short.h> short
#include <gsl/gsl_foo_ushort.h> unsigned short
#include <gsl/gsl_foo_char.h> char
#include <gsl/gsl_foo_uchar.h> unsigned char
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.
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.
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.
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.
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.
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.
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.
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. 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,
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)
Range error; used by mathematical functions when the result value is not representable because of overflow or underflow (like ERANGE in the C library)
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.
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.
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 errorfor a status value ofGSL_ERANGE.
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,
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 typevoid. 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,
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 (
aborton error) set the error handler toNULL,old_handler = gsl_set_error_handler (NULL);
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.
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:
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);
}
This macro is the same as
GSL_ERRORbut 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);
}
Here is an example of some code which checks the return value of a function where an error might be reported,
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
...
int status;
size_t n = 37;
gsl_set_error_handler_off();
status = gsl_fft_complex_radix2_forward (data, 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.
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.
The library ensures that the standard bsd mathematical constants are defined. For reference, here is a list of the constants:
M_EM_LOG2EM_LOG10EM_SQRT2M_SQRT1_2M_SQRT3M_PIM_PI_2M_PI_4M_SQRTPIM_2_SQRTPIM_1_PIM_2_PIM_LN10M_LN2M_LNPIM_EULERThis macro contains the IEEE representation of positive infinity, +\infty. It is computed from the expression
+1.0/0.0.
This macro contains the IEEE representation of negative infinity, -\infty. It is computed from the expression
-1.0/0.0.
This macro contains the IEEE representation of the Not-a-Number symbol,
NaN. It is computed from the ratio0.0/0.0.
This function returns +1 if x is positive infinity, -1 if x is negative infinity and 0 otherwise.
This function returns 1 if x is a real number, and 0 if it is infinite or not-a-number.
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 (see Portability functions).
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).
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).
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).
This function computes the value of \arccosh(x). It provides an alternative to the standard math function
acosh(x).
This function computes the value of \arcsinh(x). It provides an alternative to the standard math function
asinh(x).
This function computes the value of \arctanh(x). It provides an alternative to the standard math function
atanh(x).
This function computes the value of x * 2^e. It provides an alternative to the standard math function
ldexp(x,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).
A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides a simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.
This routine 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.
These functions can be used to compute small integer powers x^2, x^3, etc. efficiently. The functions will be inlined when possible so that use of these functions should be as efficient as explicitly writing the corresponding product expression.
#include <gsl/gsl_math.h>
double y = gsl_pow_4 (3.141) /* compute 3.141**4 */
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).
This macro evaluates to 1 if n is odd and 0 if n is even. The argument n must be of integer type.
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.
This macro returns the maximum of a and b. It is defined as
((a) > (b) ? (a):(b)).
This macro returns the minimum of a and b. It is defined as
((a) < (b) ? (a):(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_MAXwill be automatically substituted.
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_MINwill be automatically substituted.
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_MAXorGSL_MINwill be automatically substituted.
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_MAXorGSL_MINwill be automatically substituted.
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).
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.
The implementation is based on the package
fcmpby T.C. Belding.
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.
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.
This function uses the rectangular cartesian components (x,y) to return the complex number z = x + i y.
This function returns the complex number z = r \exp(i \theta) = r (\cos(\theta) + i \sin(\theta)) from the polar representation (r,theta).
These macros return the real and imaginary parts of the complex number z.
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.
These macros allow the real and imaginary parts of the complex number pointed to by zp to be set independently.
This function returns the argument of the complex number z, \arg(z), where -\pi < \arg(z) <= \pi.
This function returns the magnitude of the complex number z, |z|.
This function returns the squared magnitude of the complex number z, |z|^2.
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.
This function returns the sum of the complex numbers a and b, z=a+b.
This function returns the difference of the complex numbers a and b, z=a-b.
This function returns the product of the complex numbers a and b, z=ab.
This function returns the quotient of the complex numbers a and b, z=a/b.
This function returns the sum of the complex number a and the real number x, z=a+x.
This function returns the difference of the complex number a and the real number x, z=a-x.
This function returns the product of the complex number a and the real number x, z=ax.
This function returns the quotient of the complex number a and the real number x, z=a/x.
This function returns the sum of the complex number a and the imaginary number iy, z=a+iy.
This function returns the difference of the complex number a and the imaginary number iy, z=a-iy.
This function returns the product of the complex number a and the imaginary number iy, z=a*(iy).
This function returns the quotient of the complex number a and the imaginary number iy, z=a/(iy).
This function returns the complex conjugate of the complex number z, z^* = x - i y.
This function returns the inverse, or reciprocal, of the complex number z, 1/z = (x - i y)/(x^2 + y^2).
This function returns the negative of the complex number z, -z = (-x) + i(-y).
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.
This function returns the complex square root of the real number x, where x may be negative.
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.
This function returns the complex number z raised to the real power x, z^x.
This function returns the complex exponential of the complex number z, \exp(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.
This function returns the complex base-10 logarithm of the complex number z, \log_10 (z).
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).
This function returns the complex sine of the complex number z, \sin(z) = (\exp(iz) - \exp(-iz))/(2i).
This function returns the complex cosine of the complex number z, \cos(z) = (\exp(iz) + \exp(-iz))/2.
This function returns the complex tangent of the complex number z, \tan(z) = \sin(z)/\cos(z).
This function returns the complex secant of the complex number z, \sec(z) = 1/\cos(z).
This function returns the complex cosecant of the complex number z, \csc(z) = 1/\sin(z).
This function returns the complex cotangent of the complex number z, \cot(z) = 1/\tan(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.
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.
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.
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.
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.
This function returns the complex arcsecant of the complex number z, \arcsec(z) = \arccos(1/z).
This function returns the complex arcsecant of the real number z, \arcsec(z) = \arccos(1/z).
This function returns the complex arccosecant of the complex number z, \arccsc(z) = \arcsin(1/z).
This function returns the complex arccosecant of the real number z, \arccsc(z) = \arcsin(1/z).
This function returns the complex arccotangent of the complex number z, \arccot(z) = \arctan(1/z).
This function returns the complex hyperbolic sine of the complex number z, \sinh(z) = (\exp(z) - \exp(-z))/2.
This function returns the complex hyperbolic cosine of the complex number z, \cosh(z) = (\exp(z) + \exp(-z))/2.
This function returns the complex hyperbolic tangent of the complex number z, \tanh(z) = \sinh(z)/\cosh(z).
This function returns the complex hyperbolic secant of the complex number z, \sech(z) = 1/\cosh(z).
This function returns the complex hyperbolic cosecant of the complex number z, \csch(z) = 1/\sinh(z).
This function returns the complex hyperbolic cotangent of the complex number z, \coth(z) = 1/\tanh(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.
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.
This function returns the complex hyperbolic arccosine of the real number z, \arccosh(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.
This function returns the complex hyperbolic arctangent of the real number z, \arctanh(z).
This function returns the complex hyperbolic arcsecant of the complex number z, \arcsech(z) = \arccosh(1/z).
This function returns the complex hyperbolic arccosecant of the complex number z, \arccsch(z) = \arcsin(1/z).
This function returns the complex hyperbolic arccotangent of the complex number z, \arccoth(z) = \arctanh(1/z).
The implementations of the elementary and trigonometric functions are based on the following papers,
The general formulas and details of branch cuts can be found in the following books,
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.
This function evaluates the polynomial c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using Horner's method for stability. The function is inlined when possible.
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.
This function computes a divided-difference representation of the interpolating polynomial for the points (xa, ya) 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.
This function evaluates the polynomial stored in divided-difference form in the arrays dd and xa of length size at the point x.
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.
This function finds the real roots of the quadratic equation,
a x^2 + b x + c = 0The 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.
This function finds the complex roots of the quadratic equation,
a z^2 + b z + c = 0The 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.
This function finds the real roots of the cubic equation,
x^3 + a x^2 + b x + c = 0with 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.
This function finds the complex roots of the cubic equation,
z^3 + a z^2 + b z + c = 0The 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.
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.
This function allocates space for a
gsl_poly_complex_workspacestruct and a workspace suitable for solving a polynomial with n coefficients using the routinegsl_poly_complex_solve.The function returns a pointer to the newly allocated
gsl_poly_complex_workspaceif no errors were detected, and a null pointer in the case of error.
This function frees all the memory associated with the workspace w.
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_SUCCESSif all the roots are found andGSL_EFAILEDif the QR reduction does not converge. 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).
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 <stdio.h>
#include <gsl/gsl_poly.h>
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 = -0.809016994374947451 +0.587785252292473137
z1 = -0.809016994374947451 -0.587785252292473137
z2 = +0.309016994374947451 +0.951056516295153642
z3 = +0.309016994374947451 -0.951056516295153642
z4 = +1.000000000000000000 +0.000000000000000000
which agrees with the analytic result, z_n = \exp(2 \pi n i/5).
The balanced-QR method and its error analysis are described in the following papers,
The formulas for divided differences are given in Abramowitz and Stegun,
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.
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.
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;
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_DOUBLEGSL_PREC_SINGLEGSL_PREC_APPROXThe approximate mode provides the fastest evaluation at the lowest accuracy.
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) + \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.
These routines compute the Airy function Ai(x) with an accuracy specified by mode.
These routines compute the Airy function Bi(x) with an accuracy specified by mode.
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.
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.
These routines compute the Airy function derivative Ai'(x) with an accuracy specified by mode.
These routines compute the Airy function derivative Bi'(x) with an accuracy specified by mode.
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.
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.
These routines compute the location of the s-th zero of the Airy function Ai(x).
These routines compute the location of the s-th zero of the Airy function Bi(x).
These routines compute the location of the s-th zero of the Airy function derivative Ai'(x).
These routines compute the location of the s-th zero of the Airy function derivative Bi'(x).
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.
These routines compute the regular cylindrical Bessel function of zeroth order, J_0(x).
These routines compute the regular cylindrical Bessel function of first order, J_1(x).
These routines compute the regular cylindrical Bessel function of order n, J_n(x).
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.
These routines compute the irregular cylindrical Bessel function of zeroth order, Y_0(x), for x>0.
These routines compute the irregular cylindrical Bessel function of first order, Y_1(x), for x>0.
These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.
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.
These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x).
These routines compute the regular modified cylindrical Bessel function of first order, I_1(x).
These routines compute the regular modified cylindrical Bessel function of order n, I_n(x).
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.
These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x).
These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x).
These routines compute the scaled regular modified cylindrical Bessel function of order n, \exp(-|x|) I_n(x)
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.
These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.
These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.
These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.
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.
These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0.
These routines compute the scaled irregular modified cylindrical Bessel function of first order \exp(x) K_1(x) for x>0.
These routines compute the scaled irregular modified cylindrical Bessel function of order n, \exp(x) K_n(x), for x>0.
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.
These routines compute the regular spherical Bessel function of zeroth order, j_0(x) = \sin(x)/x.
These routines compute the regular spherical Bessel function of first order, j_1(x) = (\sin(x)/x - \cos(x))/x.
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.
These routines compute the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0.
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.
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.
These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x.
These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.
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).
These routines compute the irregular spherical Bessel function of order l, y_l(x), for l >= 0.
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.
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)
These routines compute the scaled regular modified spherical Bessel function of zeroth order, \exp(-|x|) i_0(x).
These routines compute the scaled regular modified spherical Bessel function of first order, \exp(-|x|) i_1(x).
These routines compute the scaled regular modified spherical Bessel function of second order, \exp(-|x|) i_2(x)
These routines compute the scaled regular modified spherical Bessel function of order l, \exp(-|x|) i_l(x)
This routine computes the values of the scaled regular modified cylindrical 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.
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).
These routines compute the scaled irregular modified spherical Bessel function of zeroth order, \exp(x) k_0(x), for x>0.
These routines compute the scaled irregular modified spherical Bessel function of first order, \exp(x) k_1(x), for x>0.
These routines compute the scaled irregular modified spherical Bessel function of second order, \exp(x) k_2(x), for x>0.
These routines compute the scaled irregular modified spherical Bessel function of order l, \exp(x) k_l(x), for x>0.
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.
These routines compute the regular cylindrical Bessel function of fractional order \nu, J_\nu(x).
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).
These routines compute the irregular cylindrical Bessel function of fractional order \nu, Y_\nu(x).
These routines compute the regular modified Bessel function of fractional order \nu, I_\nu(x) for x>0, \nu>0.
These routines compute the scaled regular modified Bessel function of fractional order \nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.
These routines compute the irregular modified Bessel function of fractional order \nu, K_\nu(x) for x>0, \nu>0.
These routines compute the logarithm of the irregular modified Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0, \nu>0.
These routines compute the scaled irregular modified Bessel function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.
These routines compute the location of the s-th positive zero of the Bessel function J_0(x).
These routines compute the location of the s-th positive zero of the Bessel function J_1(x).
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.
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.
These routines compute the Clausen integral Cl_2(x).
The prototypes of the Coulomb functions are declared in the header file gsl_sf_coulomb.h. Both bound state and scattering solutions are available.
These routines compute the lowest-order normalized hydrogenic bound state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r).
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 (see Laguerre Functions). The normalization is chosen such that the wavefunction \psi is given by \psi(n,l,r) = R_n Y_{lm}.
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)
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_EOVRFLWis returned and scaling exponents are stored in the modifiable parameters exp_F, exp_G.
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.
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.
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.
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.
The Coulomb wave function normalization constant is defined in Abramowitz 14.1.7.
This function computes the Coulomb wave function normalization constant C_L(\eta) for L > -1.
This function computes the Coulomb wave function normalization constant C_L(\eta) for L = Lmin \dots Lmin + kmax, Lmin > -1.
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.
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.
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.
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.
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.
These routines compute the value of Dawson's integral for x.
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.
These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)).
These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)).
These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)).
These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)).
These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)).
These routines compute the fourth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).
The functions described in this section are declared in the header file gsl_sf_dilog.h.
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.
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.
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.
This function multiplies x and y storing the product and its associated error in 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.
The functions described in this section are declared in the header file gsl_sf_ellint.h.
The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and P(\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)))
P(\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). Further information on the Legendre forms of elliptic integrals can be found in Abramowitz & Stegun, Chapter 17. The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun.
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)
These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode.
These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral P(\phi,k,n) to the accuracy specified by the mode variable mode.
These functions compute the incomplete elliptic integral D(\phi,k,n) which is defined through the Carlson form RD(x,y,z) by the following relation,
D(\phi,k,n) = RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
These routines compute the incomplete elliptic integral RC(x,y) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral RD(x,y,z) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral RF(x,y,z) to the accuracy specified by the mode variable mode.
These routines compute the incomplete elliptic integral RJ(x,y,z,p) to the accuracy specified by the mode variable mode.
The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16. The functions are declared in the header file gsl_sf_elljac.h.
This function computes the Jacobian elliptic functions sn(u|m), cn(u|m), dn(u|m) by descending Landen transformations.
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.
These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).
These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).
These routines compute the logarithm of the complementary error function \log(\erfc(x)).
The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.
These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).
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 Mill's 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.
These routines compute the hazard function for the normal distribution.
The functions described in this section are declared in the header file gsl_sf_exp.h.
These routines provide an exponential function \exp(x) using GSL semantics and error checking.
This function computes the exponential \exp(x) using the
gsl_sf_result_e10type to return a result with extended range. This function may be useful if the value of \exp(x) would overflow the numeric range ofdouble.
These routines exponentiate x and multiply by the factor y to return the product y \exp(x).
This function computes the product y \exp(x) using the
gsl_sf_result_e10type to return a result with extended numeric range.
These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x.
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.
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.
These routines compute the N-relative exponential, which is the n-th generalization of the functions
gsl_sf_exprelandgsl_sf_exprel2. 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)
This function exponentiates x with an associated absolute error dx.
This function exponentiates a quantity x with an associated absolute error dx using the
gsl_sf_result_e10type to return a result with extended range.
This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy.
This routine computes the product y \exp(x) for the quantities x, y with associated absolute errors dx, dy using the
gsl_sf_result_e10type to return a result with extended range.
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.
These routines compute the exponential integral E_1(x),
E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.
These routines compute the second-order exponential integral E_2(x),
E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.
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.
These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.
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).
These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.
These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.
These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.
These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.
The functions described in this section are declared in the header file gsl_sf_fermi_dirac.h.
The complete Fermi-Dirac integral F_j(x) is given by,
F_j(x) := (1/r\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
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).
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).
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)).
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)).
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)).
These routines compute the complete Fermi-Dirac integral F_{-1/2}(x).
These routines compute the complete Fermi-Dirac integral F_{1/2}(x).
These routines compute the complete Fermi-Dirac integral F_{3/2}(x).
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))
These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).
The functions described in this section are declared in the header file gsl_sf_gamma.h.
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. The functions described in this section are declared in the header file gsl_sf_gamma.h.
These routines compute the Gamma function \Gamma(x), subject to x not being a negative integer. 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_XMAXand is 171.0.
These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not a being negative integer. 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.
This routine computes the sign of the gamma function and the logarithm of its magnitude, subject to x not being a negative integer. The function is computed using the real Lanczos method. The value of the gamma function can be reconstructed using the relation \Gamma(x) = sgn * \exp(resultlg).
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 \inftyand is a useful suggestion of Temme.
These routines compute the reciprocal of the gamma function, 1/\Gamma(x) using the real Lanczos method.
This routine computes \log(\Gamma(z)) for complex z=z_r+i z_i and z not a negative integer, 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_ELOSSerror when it occurs. The absolute value part (lnr), however, never suffers from loss of precision.
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.
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_NMAXand is 170.
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_NMAXand is 297.
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_lngammafor n < 170, but defers for larger n.
These routines compute the logarithm of the double factorial of n, \log(n!!).
These routines compute the combinatorial factor
n choose m= n!/(m!(n-m)!)
These routines compute the logarithm of
n choose m. This is equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).
These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.
These routines compute the Pochhammer symbol (a)_x = \Gamma(a + x)/\Gamma(a), subject to a and a+x not being negative integers. The Pochhammer symbol is also known as the Apell symbol and sometimes written as (a,x).
These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0.
These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed parameters are result = \log(|(a)_x|) and sgn = \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a), subject to a, a+x not being negative integers.
These routines compute the relative Pochhammer symbol ((a)_x - 1)/x where (a)_x = \Gamma(a + x)/\Gamma(a).
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.
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.
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).
These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) for a > 0, b > 0.
These routines compute the logarithm of the Beta Function, \log(B(a,b)) for a > 0, b > 0.
These routines compute the normalized incomplete Beta function B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt for a > 0, b > 0, and 0 <= x <= 1.
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.
These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) using explicit representations for n =1, 2, 3.
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.
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.
Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15. These functions are declared in the header file gsl_sf_hyperg.h.
These routines compute the hypergeometric function 0F1(c,x).
These routines compute the confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n.
These routines compute the confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters a, b.
These routines compute the confluent hypergeometric function U(m,n,x) for integer parameters m, n.
This routine computes the confluent hypergeometric function U(m,n,x) for integer parameters m, n using the
gsl_sf_result_e10type to return a result with extended range.
These routines compute the confluent hypergeometric function U(a,b,x).
This routine computes the confluent hypergeometric function U(a,b,x) using the
gsl_sf_result_e10type to return a result with extended range.
These routines compute the Gauss hypergeometric function 2F1(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_EMAXITERwhen the series approximation converges too slowly. This occurs in the region of x=1, c - a - b = m for integer m.
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. exceptions:
These routines compute the renormalized Gauss hypergeometric function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.
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.
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)
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.
These routines evaluate the generalized Laguerre polynomials L^a_1(x), L^a_2(x), L^a_3(x) using explicit representations.
These routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.
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.
These compute the principal branch of the Lambert W function, W_0(x).
These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).
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.
These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3.
These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1
These functions compute an array of Legendre polynomials P_l(x), and optionally their derivatives dP_l(x)/dx, for l = 0, \dots, lmax, |x| <= 1
These routines compute the Legendre function Q_0(x) for x > -1, x != 1.
These routines compute the Legendre function Q_1(x) for x > -1, x != 1.
These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.
The following functions compute the associated Legendre Polynomials
P_l^m(x). Note that this function grows combinatorially with
l and can overflow for l larger than about 150. There is
no trouble for small m, but overflow occurs when m and
l are both large. Rather than allow overflows, these functions
refuse to calculate P_l^m(x) and return GSL_EOVRFLW when
they can sense that l and m are too big.
If you want to calculate a spherical harmonic, then do not use
these functions. Instead use gsl_sf_legendre_sphPlm() below,
which uses a similar recursion, but with the normalized functions.
These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.
These functions compute an array of Legendre polynomials P_l^m(x), and optionally their derivatives dP_l^m(x)/dx, for m >= 0, l = |m|, ..., lmax, |x| <= 1.
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).
These functions compute an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$, and optionally their derivatives, for m >= 0, l = |m|, ..., lmax, |x| <= 1.0
This function returns the size of result_array[] needed for the array versions of P_l^m(x), lmax - m + 1.
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.
These routines compute the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the conical function P^0_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the conical function P^1_{-1/2 + i \lambda}(x) for x > -1.
These routines compute the Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1.
These routines compute the Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1.
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.
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).
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).
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).
This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for 0 <= l <= lmax.
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.
These routines compute the logarithm of x, \log(x), for x > 0.
These routines compute the logarithm of the magnitude of x, \log(|x|), for x \ne 0.
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].
These routines compute \log(1 + x) for x > -1 using an algorithm that is accurate for small x.
These routines compute \log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.
The following functions are equivalent to the function gsl_pow_int
(see Small integer powers) with an error estimate. These functions are
declared in the header file gsl_sf_pow_int.h.
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 <gsl/gsl_sf_pow_int.h>
/* compute 3.0**12 */
double y = gsl_sf_pow_int(3.0, 12);
The polygamma functions of order m are defined by
\psi^{(m)}(x) = (d/dx)^m \psi(x) = (d/dx)^{m+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.
These routines compute the digamma function \psi(n) for positive integer n. The digamma function is also called the Psi function.
These routines compute the digamma function \psi(x) for general x, x \ne 0.
These routines compute the real part of the digamma function on the line 1+i y, \Re[\psi(1 + i y)].
These routines compute the Trigamma function \psi'(n) for positive integer n.
These routines compute the Trigamma function \psi'(x) for general x.
These routines compute the polygamma function \psi^{(m)}(x) for m >= 0, x > 0.
The functions described in this section are declared in the header file gsl_sf_synchrotron.h.
These routines compute the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0.
These routines compute the second synchrotron function x K_{2/3}(x) for x >= 0.
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.
These routines compute the transport function J(2,x).
These routines compute the transport function J(3,x).
These routines compute the transport function J(4,x).
These routines compute the transport function J(5,x).
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.
These routines compute the hypotenuse function \sqrt{x^2 + y^2} avoiding overflow and underflow.
These routines compute \sinc(x) = \sin(\pi x) / (\pi x) for any value of x.
This function computes the complex sine, \sin(z_r + i z_i) storing the real and imaginary parts in szr, szi.
This function computes the complex cosine, \cos(z_r + i z_i) storing the real and imaginary parts in szr, szi.
This function computes the logarithm of the complex sine, \log(\sin(z_r + i z_i)) storing the real and imaginary parts in szr, szi.
This function converts the polar coordinates (r,theta) to rectilinear coordinates (x,y), x = r\cos(\theta), y = r\sin(\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].
These routines force the angle theta to lie in the range (-\pi,\pi].
These routines force the angle theta to lie in the range [0, 2\pi).
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.
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.
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.
The Riemann zeta function is defined by the infinite sum \zeta(s) = \sum_{k=1}^\infty k^{-s}.
These routines compute the Riemann zeta function \zeta(n) for integer n, n \ne 1.
These routines compute the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.
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.
These routines compute \zeta(n) - 1 for integer n, n \ne 1.
These routines compute \zeta(s) - 1 for arbitrary s, s \ne 1.
The Hurwitz zeta function is defined by \zeta(s,q) = \sum_0^\infty (k+q)^{-s}.
These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.
The eta function is defined by \eta(s) = (1-2^{1-s}) \zeta(s).
These routines compute the eta function \eta(n) for integer n.
These routines compute the eta function \eta(s) for arbitrary s.
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 <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_sf_bessel.h>
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) = -0.177596771314338292
+/- 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 <stdio.h>
#include <gsl/gsl_sf_bessel.h>
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) = -0.177596771314338292
exact = -0.177596771314338292
The library follows the conventions of Abramowitz & Stegun where possible,
The following papers contain information on the algorithms used to compute the special functions,
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.
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.
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
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.
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_callocif 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.
This function allocates memory for a block and initializes all the elements of the block to zero.
This function frees the memory used by a block b previously allocated with
gsl_block_allocorgsl_block_calloc.
The library provides functions for reading and writing blocks to a file as binary data or formatted text.
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_EFAILEDif 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.
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_EFAILEDif 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.
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,%eor%fformats for floating point numbers and%dfor integers. The function returns 0 for success andGSL_EFAILEDif there was a problem writing to the file.
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_EFAILEDif there was a problem reading from the file.
The following program shows how to allocate a block,
#include <stdio.h>
#include <gsl/gsl_block.h>
int
main (void)
{
gsl_block * b = gsl_block_alloc (100);
printf ("length of block = %u\n", b->size);
printf ("block data address = %#x\n", b->data);
gsl_block_free (b);
return 0;
}
Here is the output from the program,
length of block = 100
block data address = 0x804b0d8
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
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.
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.
This function allocates memory for a vector of length n and initializes all the elements of the vector to zero.
This function frees a previously allocated vector v. If the vector was created using
gsl_vector_allocthen 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.
Unlike fortran compilers, C compilers do not usually provide
support for range checking of vectors and matrices. Range checking is
available in the GNU C Compiler bounds-checking extension, but it is not
part of the default installation of GCC. 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 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.
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.
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.
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.
This function sets all the elements of the vector v to the value x.
This function sets all the elements of the vector v to zero.
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.
The library provides functions for reading and writing vectors to a file as binary data or formatted text.
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_EFAILEDif 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.
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_EFAILEDif 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.
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,%eor%fformats for floating point numbers and%dfor integers. The function returns 0 for success andGSL_EFAILEDif there was a problem writing to the file.
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_EFAILEDif there was a problem reading from the file.
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.
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
datapointer 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_subvectoris equivalent togsl_vector_subvectorbut can be used for vectors which are declaredconst.
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_subvectorbut 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
vof lengthn, 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 ofvusing 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_strideis equivalent togsl_vector_subvector_with_stridebut can be used for vectors which are declaredconst.
These functions return a vector view of the real parts of the complex vector v.
The function
gsl_vector_complex_const_realis equivalent togsl_vector_complex_realbut can be used for vectors which are declaredconst.
These functions return a vector view of the imaginary parts of the complex vector v.
The function
gsl_vector_complex_const_imagis equivalent togsl_vector_complex_imagbut can be used for vectors which are declaredconst.
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_arrayis equivalent togsl_vector_view_arraybut can be used for arrays which are declaredconst.
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_arraybut 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_strideis equivalent togsl_vector_view_array_with_stridebut can be used for arrays which are declaredconst.
Common operations on vectors such as addition and multiplication are available in the blas part of the library (see 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.
This function copies the elements of the vector src into the vector dest. The two vectors must have the same length.
This function exchanges the elements of the vectors v and w by copying. The two vectors must have the same length.
The following function can be used to exchange, or permute, the elements of a vector.
This function exchanges the i-th and j-th elements of the vector v in-place.
This function reverses the order of the elements of the vector v.
The following operations are only defined for real vectors.
This function adds the elements of vector b to the elements of vector a, a'_i = a_i + b_i. The two vectors must have the same length.
This function subtracts the elements of vector b from the elements of vector a, a'_i = a_i - b_i. The two vectors must have the same length.
This function multiplies the elements of vector a by the elements of vector b, a'_i = a_i * b_i. The two vectors must have the same length.
This function divides the elements of vector a by the elements of vector b, a'_i = a_i / b_i. The two vectors must have the same length.
This function multiplies the elements of vector a by the constant factor x, a'_i = x a_i.
This function adds the constant value x to the elements of the vector a, a'_i = a_i + x.
This function returns the maximum value in the vector v.
This function returns the minimum value in the vector v.
This function returns the minimum and maximum values in the vector v, storing them in min_out and max_out.
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.
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.
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.
This function returns 1 if all the elements of the vector v are zero, and 0 otherwise.
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 <stdio.h>
#include <gsl/gsl_vector.h>
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++)
{
printf ("v_%d = %g\n", i, gsl_vector_get (v, i));
}
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 <stdio.h>
#include <gsl/gsl_vector.h>
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);
}
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 <stdio.h>
#include <gsl/gsl_vector.h>
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));
}
return 0;
}
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
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 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.
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.
This function allocates memory for a matrix of size n1 rows by n2 columns and initializes all the elements of the matrix to zero.
This function frees a previously allocated matrix m. If the matrix was created using
gsl_matrix_allocthen 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.
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.
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.
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.
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.
This function sets all the elements of the matrix m to the value x.
This function sets all the elements of the matrix m to zero.
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.
The library provides functions for reading and writing matrices to a file as binary data or formatted text.
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_EFAILEDif 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.
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_EFAILEDif 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.
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,%eor%fformats for floating point numbers and%dfor integers. The function returns 0 for success andGSL_EFAILEDif there was a problem writing to the file.
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_EFAILEDif there was a problem reading from the file.
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.
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
n1-1and the index j runs from 0 ton2-1.The
datapointer 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_submatrixis equivalent togsl_matrix_submatrixbut can be used for matrices which are declaredconst.
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
n1-1and the index j runs from 0 ton2-1.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_arrayis equivalent togsl_matrix_view_arraybut can be used for matrices which are declaredconst.
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
n1-1and the index j runs from 0 ton2-1.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_tdais equivalent togsl_matrix_view_array_with_tdabut can be used for matrices which are declaredconst.
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
n1-1and the index j runs from 0 ton2-1.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_vectoris equivalent togsl_matrix_view_vectorbut can be used for matrices which are declaredconst.
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
n1-1and the index j runs from 0 ton2-1.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_tdais equivalent togsl_matrix_view_vector_with_tdabut can be used for matrices which are declaredconst.
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.
These functions return a vector view of the i-th row of the matrix m. The
datapointer of the new vector is set to null if i is out of range.The function
gsl_vector_const_rowis equivalent togsl_matrix_rowbut can be used for matrices which are declaredconst.
These functions return a vector view of the j-th column of the matrix m. The
datapointer of the new vector is set to null if j is out of range.The function
gsl_vector_const_columnis equivalent togsl_matrix_columnbut can be used for matrices which are declaredconst.
These functions returns 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_diagonalis equivalent togsl_matrix_diagonalbut can be used for matrices which are declaredconst.
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_subdiagonalis equivalent togsl_matrix_subdiagonalbut can be used for matrices which are declaredconst.
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_superdiagonalis equivalent togsl_matrix_superdiagonalbut can be used for matrices which are declaredconst.
This function copies the elements of the matrix src into the matrix dest. The two matrices must have the same size.
This function exchanges the elements of the matrices m1 and m2 by copying. The two matrices must have the same size.
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.
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.
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.
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.
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.
The following functions can be used to exchange the rows and columns of a matrix.
This function exchanges the i-th and j-th rows of the matrix m in-place.
This function exchanges the i-th and j-th columns of the matrix m in-place.
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.
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.
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.
The following operations are defined for real and complex matrices.
This function adds the elements of matrix b to the elements of matrix a, a'(i,j) = a(i,j) + b(i,j). The two matrices must have the same dimensions.
This function subtracts the elements of matrix b from the elements of matrix a, a'(i,j) = a(i,j) - b(i,j). The two matrices must have the same dimensions.
This function multiplies the elements of matrix a by the elements of matrix b, a'(i,j) = a(i,j) * b(i,j). The two matrices must have the same dimensions.
This function divides the elements of matrix a by the elements of matrix b, a'(i,j) = a(i,j) / b(i,j). The two matrices must have the same dimensions.
This function multiplies the elements of matrix a by the constant factor x, a'(i,j) = x a(i,j).
This function adds the constant value x to the elements of the matrix a, a'(i,j) = a(i,j) + x.
The following operations are only defined for real matrices.
This function returns the maximum value in the matrix m.
This function returns the minimum value in the matrix m.
This function returns the minimum and maximum values in the matrix m, storing them in min_out and max_out.
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.
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.
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.
This function returns 1 if all the elements of the matrix m are zero, and 0 otherwise.
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 <stdio.h>
#include <gsl/gsl_matrix.h>
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++)
for (j = 0; j < 3; j++)
printf ("m(%d,%d) = %g\n", i, j,
gsl_matrix_get (m, i, j));
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 <stdio.h>
#include <gsl/gsl_matrix.h>
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++;
}
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 <math.h>
#include <stdio.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_blas.h>
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 %d, 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
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,
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.
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;
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_callocif 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.
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.
This function initializes the permutation p to the identity, i.e. (0,1,2,...,n-1).
This function frees all the memory used by the permutation p.
This function copies the elements of the permutation src into the permutation dest. The two permutations must have the same size.
The following functions can be used to access and manipulate permutations.
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.
This function exchanges the i-th and j-th elements of the permutation p.
This function returns the size of the permutation p.
This function returns a pointer to the array of elements in the 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.
This function computes the inverse of the permutation p, storing the result in inv.
This function advances the permutation p to the next permutation in lexicographic order and returns
GSL_SUCCESS. If no further permutations are available it returnsGSL_FAILUREand leaves p unmodified. Starting with the identity permutation and repeatedly applying this function will iterate through all possible permutations of a given order.
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 returnsGSL_FAILUREand leaves p unmodified.
This function applies the permutation p to the array data of size n with stride stride.
This function applies the inverse of the permutation p to the array data of size n with stride stride.
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.
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.
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.
The library provides functions for reading and writing permutations to a file as binary data or formatted text.
This function writes the elements of the permutation p to the stream stream in binary format. The function returns
GSL_EFAILEDif 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.
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_EFAILEDif 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.
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. On a GNU system the type modifier
Zrepresentssize_t, so"%Zu\n"is a suitable format. The function returnsGSL_EFAILEDif there was a problem writing to the file.
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_EFAILEDif there was a problem reading from the file.
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,
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.
This function computes the canonical form of the permutation p and stores it in the output argument q.
This function converts a permutation q in canonical form back into linear form storing it in the output argument 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.
This function counts the number of cycles in the permutation p, given in linear form.
This function counts the number of cycles in the permutation q, given in canonical form.
The example program below creates a random permutation (by shuffling the elements of the identity) and finds its inverse.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_permutation.h>
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");
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 <stdio.h>
#include <gsl/gsl_permutation.h>
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);
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.
The subject of permutations is covered extensively in Knuth's Sorting and Searching,
For the definition of the canonical form see,
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.
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;
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_callocif 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.
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.
This function initializes the combination c to the lexicographically first combination, i.e. (0,1,2,...,k-1).
This function initializes the combination c to the lexicographically last combination, i.e. (n-k,n-k+1,...,n-1).
This function frees all the memory used by the combination c.
This function copies the elements of the combination src into the combination dest. The two combinations must have the same size.
The following function can be used to access the elements of a combination.
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.
This function returns the range (n) of the combination c.
This function returns the number of elements (k) in the combination c.
This function returns a pointer to the array of elements in the 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.
This function advances the combination c to the next combination in lexicographic order and returns
GSL_SUCCESS. If no further combinations are available it returnsGSL_FAILUREand leaves c unmodified. Starting with the first combination and repeatedly applying this function will iterate through all possible combinations of a given order.
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 returnsGSL_FAILUREand leaves c unmodified.
The library provides functions for reading and writing combinations to a file as binary data or formatted text.
This function writes the elements of the combination c to the stream stream in binary format. The function returns
GSL_EFAILEDif 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.
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_EFAILEDif 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.
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. On a GNU system the type modifier
Zrepresentssize_t, so"%Zu\n"is a suitable format. The function returnsGSL_EFAILEDif there was a problem writing to the file.
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_EFAILEDif there was a problem reading from the file.
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 <stdio.h>
#include <gsl/gsl_combination.h>
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.
Further information on combinations can be found in,
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.
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.
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,
0if 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
qsortthe 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.
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.
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.
This function sorts the n elements of the array data with stride stride into ascending numerical order.
This function sorts the elements of the vector v into ascending numerical order.
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.
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.
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.
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.
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.
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,
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.
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.
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.
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);
}
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 <gsl/gsl_rng.h>
#include <gsl/gsl_sort_double.h>
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 ("%d smallest values from %d\n", k, N);
for (i = 0; i < k; i++)
{
printf ("%d: %.18f\n", i, small[i]);
}
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
The subject of sorting is covered extensively in Knuth's Sorting and Searching,
The Heapsort algorithm is described in the following book,
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, which is declared in the file
gsl_blas.h. This 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.
The interface for the gsl_cblas layer is specified in the file
gsl_cblas.h. This interface corresponds to the blas Technical
Forum's draft 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 draft cblas standard and can
be obtained from Netlib. The complete set of cblas functions is
listed in an appendix (see GSL CBLAS Library).
There are three levels of blas operations,
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,
The types of matrices are,
Each operation is defined for four precisions,
Thus, for example, the name sgemm stands for “single-precision general matrix-matrix multiply” and zgemm stands for “double-precision complex matrix-matrix multiply”.
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.
This function computes the sum \alpha + x^T y for the vectors x and y, returning the result in result.
These functions compute the scalar product x^T y for the vectors x and y, returning the result in result.
These functions compute the complex scalar product x^T y for the vectors x and y, returning the result in result
These functions compute the complex conjugate scalar product x^H y for the vectors x and y, returning the result in result
These functions compute the Euclidean norm ||x||_2 = \sqrt {\sum x_i^2} of the vector x.
These functions compute the Euclidean norm of the complex vector x,
||x||_2 = \sqrt {\sum (\Re(x_i)^2 + \Im(x_i)^2)}.
These functions compute the absolute sum \sum |x_i| of the elements of the vector 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)|.
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.
These functions exchange the elements of the vectors x and y.
These functions copy the elements of the vector x into the vector y.
These functions compute the sum y = \alpha x + y for the vectors x and y.
These functions rescale the vector x by the multiplicative factor alpha.
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.
These functions apply a Givens rotation (x', y') = (c x + s y, -s x + c y) to the vectors x, y.
These functions compute a modified Givens transformation. The modified Givens transformation is defined in the original Level-1 blas specification, given in the references.
These functions apply a modified Givens transformation.
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.
These functions compute the matrix-vector product x =\alpha op(A) x for the triangular matrix A, where op(A) = A, A^T, A^H for TransA =
CblasNoTrans,CblasTrans,CblasConjTrans. When Uplo isCblasUpperthen the upper triangle of A is used, and when Uplo isCblasLowerthen the lower triangle of A is used. If Diag isCblasNonUnitthen the diagonal of the matrix is used, but if Diag isCblasUnitthen the diagonal elements of the matrix A are taken as unity and are not referenced.
These functions compute inv(op(A)) x for x, where op(A) = A, A^T, A^H for TransA =
CblasNoTrans,CblasTrans,CblasConjTrans. When Uplo isCblasUpperthen the upper triangle of A is used, and when Uplo isCblasLowerthen the lower triangle of A is used. If Diag isCblasNonUnitthen the diagonal of the matrix is used, but if Diag isCblasUnitthen the diagonal elements of the matrix A are taken as unity and are not referenced.
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
CblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used.
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
CblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen 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.
These functions compute the rank-1 update A = \alpha x y^T + A of the matrix A.
These functions compute the conjugate rank-1 update A = \alpha x y^H + A of the 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
CblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used.
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
CblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.
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
CblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used.
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
CblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.
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,CblasConjTransand similarly for the parameter TransB.
These functions compute the matrix-matrix product and sum C = \alpha A B + \beta C for Side is
CblasLeftand C = \alpha B A + \beta C for Side isCblasRight, where the matrix A is symmetric. When Uplo isCblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used.
These functions compute the matrix-matrix product and sum C = \alpha A B + \beta C for Side is
CblasLeftand C = \alpha B A + \beta C for Side isCblasRight, where the matrix A is hermitian. When Uplo isCblasUpperthen the upper triangle and diagonal of A are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of A are used. The imaginary elements of the diagonal are automatically set to zero.
These functions compute the matrix-matrix product B = \alpha op(A) B for Side is
CblasLeftand B = \alpha B op(A) for Side isCblasRight. The matrix A is triangular and op(A) = A, A^T, A^H for TransA =CblasNoTrans,CblasTrans,CblasConjTransWhen Uplo isCblasUpperthen the upper triangle of A is used, and when Uplo isCblasLowerthen the lower triangle of A is used. If Diag isCblasNonUnitthen the diagonal of A is used, but if Diag isCblasUnitthen the diagonal elements of the matrix A are taken as unity and are not referenced.
These functions compute the inverse-matrix matrix product B = \alpha op(inv(A))B for Side is
CblasLeftand B = \alpha B op(inv(A)) for Side isCblasRight. The matrix A is triangular and op(A) = A, A^T, A^H for TransA =CblasNoTrans,CblasTrans,CblasConjTransWhen Uplo isCblasUpperthen the upper triangle of A is used, and when Uplo isCblasLowerthen the lower triangle of A is used. If Diag isCblasNonUnitthen the diagonal of A is used, but if Diag isCblasUnitthen the diagonal elements of the matrix A are taken as unity and are not referenced.
These functions compute a rank-k update of the symmetric matrix C, C = \alpha A A^T + \beta C when Trans is
CblasNoTransand C = \alpha A^T A + \beta C when Trans isCblasTrans. Since the matrix C is symmetric only its upper half or lower half need to be stored. When Uplo isCblasUpperthen the upper triangle and diagonal of C are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of C are used.
These functions compute a rank-k update of the hermitian matrix C, C = \alpha A A^H + \beta C when Trans is
CblasNoTransand C = \alpha A^H A + \beta C when Trans isCblasTrans. Since the matrix C is hermitian only its upper half or lower half need to be stored. When Uplo isCblasUpperthen the upper triangle and diagonal of C are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of C are used. The imaginary elements of the diagonal are automatically set to zero.
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
CblasNoTransand C = \alpha A^T B + \alpha B^T A + \beta C when Trans isCblasTrans. Since the matrix C is symmetric only its upper half or lower half need to be stored. When Uplo isCblasUpperthen the upper triangle and diagonal of C are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of C are used.
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
CblasNoTransand C = \alpha A^H B + \alpha^* B^H A + \beta C when Trans isCblasConjTrans. Since the matrix C is hermitian only its upper half or lower half need to be stored. When Uplo isCblasUpperthen the upper triangle and diagonal of C are used, and when Uplo isCblasLowerthen the lower triangle and diagonal of C are used. The imaginary elements of the diagonal are automatically set to zero.
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 <stdio.h>
#include <gsl/gsl_blas.h>
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 ]
Information on the blas standards, including both the legacy and draft interface standards, is available online from the blas Homepage and blas Technical Forum web-site.
The following papers contain the specifications for Level 1, Level 2 and Level 3 blas.
Postscript versions of the latter two papers are available from http://www.netlib.org/blas/. A cblas wrapper for Fortran blas libraries is available from the same location.
This chapter describes functions for solving linear systems. The
library provides simple linear algebra operations which operate directly
on the gsl_vector and gsl_matrix objects. These are
intended for use with “small” systems where simple algorithms are
acceptable.
Anyone interested in large systems will want to use the sophisticated routines found in lapack. The Fortran version of lapack is recommended as the standard package for large-scale linear algebra. It supports blocked algorithms, specialized data representations and other optimizations.
The functions described in this chapter are declared in the header file gsl_linalg.h.
A general 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.
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. 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).
These functions solve the square system A x = b using the LU decomposition of A into (LU, p) given by
gsl_linalg_LU_decomporgsl_linalg_complex_LU_decomp.
These functions solve the square system A x = b in-place using the 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.
These functions apply an iterative improvement to x, the solution of A x = b, using the LU decomposition of A into (LU,p). The initial residual r = A x - b is also computed and stored in residual.
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).
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.
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.
These functions compute the sign or phase factor of the determinant of a matrix A, \det(A)/|\det(A)|, from its LU decomposition, LU.
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.
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).
This function solves the square system A x = b using the QR decomposition of A into (QR, tau) given by
gsl_linalg_QR_decomp. The least-squares solution for rectangular systems can be found usinggsl_linalg_QR_lssolve.
This function solves the square system A x = b in-place using the QR decomposition of A into (QR,tau) given by
gsl_linalg_QR_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.
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 uses 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.
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.
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.
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.
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.
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.
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).
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.
This function solves the triangular system R x = b for the N-by-N matrix R.
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.
The QR decomposition 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.
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).
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.
This function solves the square system A x = b using the QRP^T decomposition of A into (QR, tau, p) given by
gsl_linalg_QRPT_decomp.
This function solves the square system A x = b in-place using the QRP^T decomposition of A into (QR,tau,p). On input x should contain the right-hand side b, which is replaced by the solution on output.
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).
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 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.
This function solves the triangular system R P^T x = b for the N-by-N matrix R contained in QR.
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.
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.
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.
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.
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).
This function solves the system A x = b using the singular value decomposition (U, S, V) of A given by
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.
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.
This function factorizes the positive-definite symmetric square matrix A into the Cholesky decomposition A = L L^T. On output the diagonal and lower triangular part of the input matrix A contain the matrix L. The upper triangular part of the input matrix contains 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.
This function solves the system A x = b using the Cholesky decomposition of A into the matrix cholesky given by
gsl_linalg_cholesky_decomp.
This function solves the system A x = b in-place using the Cholesky decomposition of A into the matrix cholesky given by
gsl_linalg_cholesky_decomp. On input x should contain the right-hand side b, which is replaced by the solution on output.
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.
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.
This function unpacks the encoded symmetric tridiagonal decomposition (A, tau) obtained from
gsl_linalg_symmtd_decompinto the orthogonal matrix Q, the vector of diagonal elements diag and the vector of subdiagonal elements subdiag.
This function unpacks the diagonal and subdiagonal of the encoded symmetric tridiagonal decomposition (A, tau) obtained from
gsl_linalg_symmtd_decompinto the vectors diag and subdiag.
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.
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 orthogonal matrix Q. 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.
This function unpacks the encoded tridiagonal decomposition (A, tau) obtained from
gsl_linalg_hermtd_decompinto the unitary matrix U, the real vector of diagonal elements diag and the real vector of subdiagonal elements subdiag.
This function unpacks the diagonal and subdiagonal of the encoded tridiagonal decomposition (A, tau) obtained from the
gsl_linalg_hermtd_decompinto the real vectors diag and subdiag.
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.
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.
This function unpacks the bidiagonal decomposition of A given 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.
This function unpacks the bidiagonal decomposition of A given 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.
This function unpacks the diagonal and superdiagonal of the bidiagonal decomposition of A given by
gsl_linalg_bidiag_decomp, into the diagonal vector diag and superdiagonal vector superdiag.
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.
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 except the first. On output the transformation is stored in the vector v and the scalar \tau is returned.
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.
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.
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.
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.
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.
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 )
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 )
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 )
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 )
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 <stdio.h>
#include <gsl/gsl_linalg.h>
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);
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.
Further information on the algorithms described in this section can be found in the following book,
The lapack library is described in the following manual,
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,
The Jacobi algorithm for singular value decomposition is described in the following papers,
lawns or
lawnspdf directories.
This chapter describes functions for computing eigenvalues and eigenvectors of matrices. There are routines for real symmetric and complex hermitian matrices, and eigenvalues can be computed with or without eigenvectors. The algorithms used are symmetric bidiagonalization followed by QR reduction.
These routines are intended for “small” systems where simple algorithms are acceptable. Anyone interested in finding eigenvalues and eigenvectors of large matrices will want to use the sophisticated routines found in lapack. The Fortran version of lapack is recommended as the standard package for large-scale linear algebra.
The functions described in this chapter are declared in the header file gsl_eigen.h.
This function allocates a workspace for computing eigenvalues of n-by-n real symmetric matrices. The size of the workspace is O(2n).
This function frees the memory associated with the 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.
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).
This function frees the memory associated with the 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.
This function allocates a workspace for computing eigenvalues of n-by-n complex hermitian matrices. The size of the workspace is O(3n).
This function frees the memory associated with the 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.
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).
This function frees the memory associated with the 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.
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
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.
The following program computes the eigenvalues and eigenvectors of the 4-th order Hilbert matrix, H(i,j) = 1/(i + j + 1).
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_eigen.h>
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");
}
}
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.
Further information on the algorithms described in this section can be found in the following book,
The lapack library is described in,
The lapack source code can be found at the website above along with an online copy of the users guide.
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 (see FFT References and Further Reading)
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_n 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.
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_complex_vector
* 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.
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.
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
transformversion of the function the sign argument can be eitherforward(-1) orbackward(+1).The functions return a value of
GSL_SUCCESSif no errors were detected, orGSL_EDOMif the length of the data n is not a power of two.
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 <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#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.
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.
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_wavetableif 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 tosinandcos, 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.
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.
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
nwas decomposed into.size_t factor[64]- This is the array of factors. Only the first
nfelements are used.gsl_complex * trig- This is a pointer to a preallocated trigonometric lookup table of
ncomplex 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.
This function allocates a workspace for a complex transform of length n.
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,
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
transformversion of the function the sign argument can be eitherforward(-1) orbackward(+1).The functions return a value of
0if no errors were detected. The followinggsl_errnoconditions 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 <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#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 < wavetable->nf; i++)
{
printf ("# factor %d: %d\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.
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).
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
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,
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]
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.
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.
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
sinandcos, 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.
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,
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.
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,
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_transformdata is an array of time-ordered real data. Forgsl_fft_halfcomplex_transformdata 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.
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_complexroutines. The algorithm for the conversion is simply,for (i = 0; i < n; i++) { complex_coefficient[i].real = real_coefficient[i]; complex_coefficient[i].imag = 0.0; }
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]; double hc_imag = halfcomplex_coefficient[2 * i]; complex_coefficient[i].real = hc_real; complex_coefficient[i].imag = hc_imag; complex_coefficient[n - i].real = hc_real; complex_coefficient[n - i].imag = -hc_imag; } if (i == n - i) { complex_coefficient[i].real = halfcomplex_coefficient[n - 1]; complex_coefficient[i].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 <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_real.h>
#include <gsl/gsl_fft_halfcomplex.h>
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;
}
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,
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,
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,
The derivation of FFTs for real-valued data is explained in the following two articles,
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,
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.
The source code for fftpack is available from Netlib,
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, Doncker-Kapenga, Uberhuber and Kahaner. Fortran code for quadpack is available on Netlib.
The functions described in this chapter are declared in the header file gsl_integration.h.
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|)
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 routineN- non-adaptive integratorA- adaptive integratorG- general integrand (user-defined)W- weight function with integrandS- singularities can be more readily integratedP- points of special difficulty can be suppliedI- infinite range of integrationO- oscillatory weight function, cos or sinF- Fourier integralC- 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.
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.
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.
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.
The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.
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.
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.
This function allocates a workspace sufficient to hold n double precision intervals, their integration results and error estimates.
This function frees the memory associated with the workspace w.
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.
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.
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.
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] = bwith npts = 5.
If you know the locations of the singular points in the integration region then this routine will be faster than
QAGS.
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.
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^2and then integrated using the QAGS algorithm.
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^2and then integrated using the QAGS algorithm.
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.
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.
This function allocates space for a
gsl_integration_qaws_tablestruct and associated workspace 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
gsl_integration_qaws_tableif no errors were detected, and 0 in the case of error.
This function modifies the parameters (\alpha, \beta, \mu, \nu) of an existing
gsl_integration_qaws_tablestruct t.
This function frees all the memory associated with the
gsl_integration_qaws_tablestruct t.
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.
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.
This function allocates space for a
gsl_integration_qawo_tablestruct 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_SINEThe
gsl_integration_qawo_tableis 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 routinegsl_integration_qaworeturns the errorGSL_ETABLEif the number of levels is insufficient for the requested accuracy.
This function changes the parameters omega, L and sine of the existing workspace t.
This function allows the length parameter L of the workspace t to be changed.
This function frees all the memory associated with the workspace t.
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.
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 abserrwhere 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_{i<k}{E_i})where E_k is the estimated error on the interval C_k.
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. The integration over each subinterval uses the memory provided by cycle_workspace as workspace for the QAWO algorithm.
In addition to the standard error codes for invalid arguments the functions can return the following values,
GSL_EMAXITERGSL_EROUNDGSL_ESINGGSL_EDIVERGEThe integrator QAGS will handle a large class of definite
integrals. For example, consider the following integral, which has a
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 <stdio.h>
#include <math.h>
#include <gsl/gsl_integration.h>
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 = %d\n", w->size);
return 0;
}
The results below show that the desired accuracy is achieved after 8 subdivisions.
$ ./a.out
result = -3.999999999999973799
exact result = -4.000000000000000000
estimated error = 0.000000000000246025
actual error = 0.000000000000026201
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.
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.
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.
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”.
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.
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 variableGSL_RNG_SEED(see Random number environment variables).The details of the available generator types are described later in this chapter.
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 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
ranluxgenerator used a seed of 314159265, and so choosing s equal to zero reproduces this when usinggsl_rng_ranlux.
This function frees all the memory associated with the generator r.
The following functions return uniformly distributed random numbers, either as integers or double precision floating point numbers. To obtain non-uniform distributions see Random Number Distributions.
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 determined using the auxiliary functions
gsl_rng_max (r)andgsl_rng_min (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)bygsl_rng_max(r) + 1.0in 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 singleunsigned long int).
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_uniformuntil a non-zero value is obtained. You can use this function if you need to avoid a singularity at 0.0.
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_EINVALand 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 mimimum value, such as
gsl_rng_ranlxd1,gsl_rng_mt19937orgsl_rng_taus, and sample it directly usinggsl_rng_get(). The range of each generator can be found using the auxiliary functions described in the next section.
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.
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.
gsl_rng_maxreturns the largest value thatgsl_rng_getcan return.
gsl_rng_minreturns the smallest value thatgsl_rng_getcan 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.
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);
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); }
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.
This function reads the environment variables
GSL_RNG_TYPEandGSL_RNG_SEEDand uses their values to set the corresponding library variablesgsl_rng_defaultandgsl_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_seedThe environment variable
GSL_RNG_TYPEshould be the name of a generator, such astausormt19937. The environment variableGSL_RNG_SEEDshould contain the desired seed value. It is converted to anunsigned long intusing the C library functionstrtoul.If you don't specify a generator for
GSL_RNG_TYPEthengsl_rng_mt19937is used as the default. The initial value ofgsl_rng_default_seedis 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 <stdio.h>
#include <gsl/gsl_rng.h>
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));
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
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:
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.
This function returns a pointer to a newly created generator which is an exact copy of the generator r.
The library provides functions for reading and writing the random number state to a file as binary data or formatted text.
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_EFAILEDif 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.
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_EFAILEDif 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.
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.
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_setreproduces this.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_mt19937uses 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 generatorsgsl_rng_mt19937_1999andgsl_rng_mt19937_1998.
The generator
ranlxs0is a second-generation version of the ranlux algorithm of Lüscher, which produces “luxury random numbers”. This generator provides single precision output (24 bits) at three luxury levelsranlxs0,ranlxs1andranlxs2. It uses double-precision floating point arithmetic internally and can be significantly faster than the integer version ofranlux, 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.
These generators produce double precision output (48 bits) from the ranlxs generator. The library provides two luxury levels
ranlxd1andranlxd2.
The
ranluxgenerator is an implementation of the original algorithm developed by Lüscher. 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 Lüscher is provided bygsl_rng_ranlux, whilegsl_rng_ranlux389gives 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. Lüscher, “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 Lüscher”, Computer Physics Communications, 79 (1994) 111–114
This is a combined multiple recursive generator by L'Ecuyer. Its sequence is,
z_n = (x_n - y_n) mod m_1where 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_2with 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.
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 mwith 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).
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
0xFFFFFFFFto 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_taus2uses the same algorithm asgsl_rng_tausbut 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_taus2should now be used in preference togsl_rng_taus.
The
gfsr4generator is like a lagged-fibonacci generator, and produces each number as anxor'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.
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.
This is the BSD
rand()generator. Its sequence isx_{n+1} = (a x_n + c) mod mwith 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.
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 ofrandom()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_bsdwhere the numeric suffix indicates the buffer length. The original BSD
randomfunction used a 128-byte default buffer and sogsl_rng_random_bsdhas been made equivalent togsl_rng_random128_bsd. Corresponding versions of thelibc5andglibc2generators are also available, with the namesgsl_rng_random8_libc5,gsl_rng_random8_glibc2, etc.
This is the Unix
rand48generator. Its sequence isx_{n+1} = (a x_n + c) mod mdefined 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 functiongsl_rng_getreturns the upper 32 bits from each term of the sequence. This does not have a direct parallel in the originalrand48functions, but forcing the result to typelong intreproduces the output ofmrand48. The functiongsl_rng_uniformuses the full 48 bits of internal state to return the double precision number x_n/m, which is equivalent to the functiondrand48. Note that some versions of the GNU C Library contained a bug inmrand48function which caused it to produce different results (only the lower 16-bits of the return value were set).
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.
This is the CRAY random number generator
RANF. Its sequence isx_{n+1} = (a x_n) mod mdefined 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_getreturns the upper 32 bits from each term of the sequence. The functiongsl_rng_uniformuses the full 48 bits to return the double precision number x_n/m.The period of this generator is 2^46.
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.
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)
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.
This is the VAX generator
MTH$RANDOM. Its sequence is,x_{n+1} = (a x_n + c) mod mwith 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.
This is the random number generator from the INMOS Transputer Development system. Its sequence is,
x_{n+1} = (a x_n) mod mwith a = 1664525 and m = 2^32. The seed specifies the initial value, x_1.
This is the IBM
RANDUgenerator. Its sequence isx_{n+1} = (a x_n) mod mwith 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.
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 mwith 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 is used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function). 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.
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.
This is the SLATEC random number generator RAND. It is ancient. The original source code is available from NETLIB.
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).
This is the Borosh-Niederreiter random number generator. It is taken from Knuth's Seminumerical Algorithms, 3rd Ed., pages 106–108. Its sequence is,
x_{n+1} = (a x_n) mod mwith a = 1812433253 and m = 2^32. The seed specifies the initial value, x_1.
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 mwith m = 2^32. The seed specifies the initial value, x_1.
This is the Fishman, Moore III random number generator. It is taken from Knuth's Seminumerical Algorithms, 3rd Ed., pages 106–108. Its sequence is,
x_{n+1} = (a x_n) mod mwith a = 62089911 and m = 2^31 - 1. The seed specifies the initial value, x_1.
This is the Fishman random number generator. It is taken from Knuth's Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
x_{n+1} = (a x_n) mod mwith a = 48271 and m = 2^31 - 1. The seed specifies the initial value, x_1.
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 mwith m = 2^31 - 1. x_n and y_n are given by the
fishman20andlecuyer21algorithms. The seed specifies the initial value, x_1.
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 mwith a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.
This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth provides its C code.
This is the L'Ecuyer random number generator. It is taken from Knuth's Seminumerical Algorithms, 3rd Ed., page 106–108. Its sequence is,
x_{n+1} = (a x_n) mod mwith a = 40692 and m = 2^31 - 249. The seed specifies the initial value, x_1.
This is the Waterman random number generator. It is taken from Knuth's Seminumerical Algorithms, 3rd Ed., page 106–108. Its sequence is,
x_{n+1} = (a x_n) mod mwith a = 1566083941 and m = 2^32. The seed specifies the initial value, x_1.
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
1852 k ints/sec, 935 k doubles/sec, ran3
813 k ints/sec, 575 k doubles/sec, ran0
787 k ints/sec, 476 k doubles/sec, ran1
379 k ints/sec, 292 k doubles/sec, ran2
The following program demonstrates the use of a random number generator to produce uniform random numbers in the range [0.0, 1.0),
#include <stdio.h>
#include <gsl/gsl_rng.h>
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
GSL_RNG_TYPE=mrg
GSL_RNG_SEED=123
0.33050
0.86631
0.32982
0.67620
0.53391
0.06457
0.16847
0.70229
0.04371
0.86374
The subject of random number generation and testing is reviewed extensively in Knuth's Seminumerical Algorithms.
Further information is available in the review paper written by Pierre L'Ecuyer,
http://www.iro.umontreal.ca/~lecuyer/papers.html in the file handsim.ps.
The source code for the diehard random number generator tests is also available online,
A comprehensive set of random number generator tests is available from nist,
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 Lüscher for providing notes and source code for the ranlxs and ranlxd generators.
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.
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.
This function frees all the memory associated with the generator 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.
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.
This function returns a pointer to the name of the generator.
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);
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.
This function returns a pointer to a newly created generator which is an exact copy of the generator q.
The following quasi-random sequence algorithms are available,
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.
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.
The following program prints the first 1024 points of the 2-dimensional Sobol sequence.
#include <stdio.h>
#include <gsl/gsl_qrng.h>
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.
The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,
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.
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).
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) dxfor x in the range -\infty to +\infty. Use the transformation z = \mu + x on the numbers returned by
gsl_ran_gaussianto obtain a Gaussian distribution with mean \mu. This function uses the Box-Mueller algorithm which requires two calls to the random number generator r.
This function computes the probability density p(x) at x for a Gaussian distribution with standard deviation sigma, using the formula given above.
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.
These functions compute results for the unit Gaussian distribution. They are equivalent to the functions above with a standard deviation of one, sigma = 1.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Gaussian distribution with standard deviation sigma.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the unit Gaussian distribution.
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)) dxfor x > a where N(a;\sigma) is the normalization constant,
N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
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.
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.
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 dyfor x,y in the range -\infty to +\infty. The correlation coefficient rho should lie between 1 and -1.
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.
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) dxfor x >= 0.
This function computes the probability density p(x) at x for an exponential distribution with mean mu, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the exponential distribution with mean mu.
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|) dxfor -\infty < x < \infty.
This function computes the probability density p(x) at x for a Laplace distribution with width a, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Laplace distribution with width a.
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) dxfor 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.
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.
These functions compute the cumulative distribution functions P(x), Q(x) for the exponential power distribution with parameters a and b.
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) } dxfor x in the range -\infty to +\infty. The Cauchy distribution is also known as the Lorentz distribution.
This function computes the probability density p(x) at x for a Cauchy distribution with scale parameter a, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Cauchy distribution with scale parameter a.
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)) dxfor x > 0.
This function computes the probability density p(x) at x for a Rayleigh distribution with scale parameter sigma, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Rayleigh distribution with scale parameter 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)) dxfor x > a.
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.
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).
This function computes the probability density p(x) at x for the Landau distribution using an approximation to the formula given above.
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
The algorithm only works for 0 < alpha <= 2.
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
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).
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} dxfor x > 0.
The gamma distribution with an integer parameter a is known as the Erlang distribution. The variates are computed using the algorithms from Knuth (vol 2).
This function returns a gamma variate using the Marsaglia-Tsang fast gamma method.
This function computes the probability density p(x) at x for a gamma distribution with parameters a and b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the gamma distribution with parameters a and 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)} dxif a <= x < b and 0 otherwise.
This function computes the probability density p(x) at x for a uniform distribution from a to b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for a uniform distribution from a to b.
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) dxfor x > 0.
This function computes the probability density p(x) at x for a lognormal distribution with parameters zeta and sigma, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the lognormal distribution with parameters zeta and sigma.
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.
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) dxfor x >= 0.
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.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the chi-squared distribution with nu degrees of freedom.
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).
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.
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.
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.
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.
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} dxfor -\infty < x < +\infty.
This function computes the probability density p(x) at x for a t-distribution with nu degrees of freedom, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the t-distribution with nu degrees of freedom.
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} dxfor 0 <= x <= 1.
This function computes the probability density p(x) at x for a beta distribution with parameters a and b, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the beta distribution with parameters a and b.
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 } dxfor -\infty < x < +\infty.
This function computes the probability density p(x) at x for a logistic distribution with scale parameter a, using the formula given above.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the logistic distribution with scale parameter a.
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} dxfor x >= 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.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Pareto distribution with exponent a and scale b.
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.
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).
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).
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).
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) dxfor x >= 0.
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.
These functions compute the cumulative distribution functions P(x), Q(x) and their inverses for the Weibull distribution with scale a and exponent 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)) dxfor -\infty < x < \infty.
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.
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.
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}) dxfor 0 < x < \infty.
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.
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.
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_Kfor 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).
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.
This function computes the logarithm of the probability density p(\theta_1, ... , \theta_K) for a Dirichlet distribution with parameters alpha[K].
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 numbers 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.
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_discretefunction below.
After the preprocessor, above, has been called, you use this function to get the discrete random numbers.
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.
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.
This function computes the probability p(k) of obtaining k from a Poisson distribution with mean mu, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the Poisson distribution with parameter mu.
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
This function computes the probability p(k) of obtaining k from a Bernoulli distribution with probability parameter p, using the formula given above.
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.
This function computes the probability p(k) of obtaining k from a binomial distribution with parameters p and n, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the binomial distribution with parameters p and n.
This function returns an array of K random variates from a multinomial distribution. The distribution function 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_Kwhere (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.
Random variates are generated using the conditional binomial method (see C.S. David, The computer generation of multinomial random variates, Comp. Stat. Data Anal. 16 (1993) 205–217 for details).
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.
This function returns the logarithm of the probability for the multinomial distribution P(n_1, n_2, ..., n_K) with parameters p[K].
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)^kNote that n is not required to be an integer.
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.
These functions compute the cumulative distribution functions P(k), Q(k) for the negative binomial distribution with parameters p and 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)^kfor k >= 0
This function computes the probability p(k) of obtaining k from a Pascal distribution with parameters p and n, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the Pascal distribution with parameters p and n.
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.
This function computes the probability p(k) of obtaining k from a geometric distribution with probability parameter p, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the geometric distribution with parameter p.
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.
This function computes the probability p(k) of obtaining k from a hypergeometric distribution with parameters n1, n2, t, using the formula given above.
These functions compute the cumulative distribution functions P(k), Q(k) for the hypergeometric distribution with parameters n1, n2 and t.
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.
This function computes the probability p(k) of obtaining k from a logarithmic distribution with probability parameter p, using the formula given above.
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”.
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));
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 callgsl_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));
This function is like
gsl_ran_choosebut 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.
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 <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
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");
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.out2 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
GSL_RNG_SEED=123
4 5 6 3 3 1 4 2 5 5
The following program generates a random walk in two dimensions.
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
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);
}
return 0;
}
Here is the output from the program, three 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 <stdio.h>
#include <gsl/gsl_cdf.h>
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
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.
The subject of random variate generation is also reviewed by Knuth, who describes algorithms for all the major distributions.
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.
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.
The cumulative distribution functions for the Gaussian distribution are based on the following papers,
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.
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_iwhere 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.
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)^2where 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 togsl_stats_variance_m.
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
The standard deviation is defined as the square root of the variance. These functions return the square root of the corresponding variance functions above.
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
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.
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.
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).
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)^3where 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_meanandgsl_stats_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)^3These functions are useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.
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) - 3The kurtosis measures how sharply peaked a distribution is, relative to its width. The kurtosis is normalized to zero for a gaussian distribution.
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) - 3This function is useful if you have already computed the mean and standard deviation of data and want to avoid recomputing them.
This function computes the lag-1 autocorrelation of the dataset data.
a_1 = {\sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i-1} - \Hat\mu) \over \sum_{i = 1}^{n} (x_{i} - \Hat\mu) (x_{i} - \Hat\mu)}
This function computes the lag-1 autocorrelation of the dataset data using the given value of the mean mean.
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)
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.
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.
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)
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 defined as,
\Hat\sigma^2 = ((\sum w_i)/((\sum w_i)^2 - \sum (w_i^2))) \sum w_i (x_i - \Hat\mu)^2Note that this expression reduces to an unweighted variance with the familiar 1/(N-1) factor when there are N equal non-zero weights.
This function returns the estimated variance of the weighted dataset data using the given weighted mean wmean.
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_wvarianceabove.
This function returns the square root of the corresponding variance function
gsl_stats_wvariance_mabove.
This function computes an unbiased estimate of the variance of 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)
The standard deviation is defined as the square root of the variance. This function returns the square root of the corresponding variance function above.
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)
This function computes the absolute deviation of the weighted dataset data about the given weighted mean wmean.
This function computes the weighted skewness of the dataset data.
skew = (\sum w_i ((x_i - xbar)/\sigma)^3) / (\sum w_i)
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.
This function computes the weighted kurtosis of the dataset data.
kurtosis = ((\sum w_i ((x_i - xbar)/sigma)^4) / (\sum w_i)) - 3
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.
The following functions find the maximum and minimum values of a
dataset (or their indices). If the data contains NaNs 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.
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
fabsorabsto your data before calling this function.
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
fabsorabsto your data before calling this function.
This function finds both the minimum and maximum values min, max in data in a single pass.
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.
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.
This function returns the indexes min_index, max_index of the minimum and maximum values in data in a single pass.
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).
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_sortshould 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.
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_sortshould 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.
Here is a basic example of how to use the statistical functions:
#include <stdio.h>
#include <gsl/gsl_statistics.h>
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 4.2984
The largest value is 18.3
The smallest value is 12.6
Here is an example using sorted data,
#include <stdio.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_statistics.h>
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
The standard reference for almost any topic in statistics is the multi-volume Advanced Theory of Statistics by Kendall and Stuart.
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.
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.
The Review of Particle Physics is available online at the website http://pdg.lbl.gov/.
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.
A histogram is defined by the following struct,
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.
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.
This function allocates memory for a histogram with n bins, and returns a pointer to a newly created
gsl_histogramstruct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code ofGSL_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.
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
rangearray 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.
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 < xmaxwhere d is the bin spacing, d = (xmax-xmin)/n.
This function frees the histogram h and all of the memory associated with it.
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.
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.
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.
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 returnsGSL_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.
This function is similar to
gsl_histogram_incrementbut increases the value of the appropriate bin in the histogram h by the floating-point number weight.
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_EDOMand the function returns 0.
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.
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_histogramstruct directly.
This function resets all the bins in the histogram h to zero.
The following functions are used by the access and update routines to locate the bin which corresponds to a given x coordinate.
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 returnsGSL_EDOMand the error handler is invoked.
This function returns the maximum value contained in the histogram bins.
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.
This function returns the minimum value contained in the histogram bins.
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.
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.
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.
This function returns the sum of all bin values. Negative bin values are included in the sum.
This function returns 1 if the all of the individual bin ranges of the two histograms are identical, and 0 otherwise.
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.
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.
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.
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.
This function multiplies the contents of the bins of histogram h by the constant scale, i.e. h'_1(i) = h_1(i) * scale.
This function shifts the contents of the bins of histogram h by the constant offset, i.e. h'_1(i) = h_1(i) + offset.
The library provides functions for reading and writing histograms to a file as binary data or formatted text.
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_EFAILEDif 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.
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_EFAILEDif 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.
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,%eor%fformats for floating point numbers. The function returns 0 for success andGSL_EFAILEDif 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.
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 andGSL_EFAILEDif there was a problem reading from the file.
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.
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.
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.
This function allocates memory for a probability distribution with n bins and returns a pointer to a newly initialized
gsl_histogram_pdfstruct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code ofGSL_ENOMEM.
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_EDOMbecause a probability distribution cannot contain negative values.
This function frees the probability distribution function p and all of the memory associated with it.
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]).
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 <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_histogram.h>
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
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).
Two dimensional histograms are defined by the following struct,
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.
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.
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_histogram2dstruct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code ofGSL_ENOMEM. The bins and ranges must be initialized with one of the functions below before the histogram is ready for use.
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.
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.
This function frees the 2D histogram h and all of the memory associated with it.
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.
This function returns a pointer to a newly created histogram which is an exact copy of the histogram src.
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.
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.
This function is similar to
gsl_histogram2d_incrementbut increases the value of the appropriate bin in the histogram h by the floating-point number weight.
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_EDOMand the function returns 0.
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.
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_histogram2dstruct directly.
This function resets all the bins of the histogram h to zero.
The following functions are used by the access and update routines to locate the bin which corresponds to a given (x,y) coordinate.
This function finds and sets the indices i and j to the 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 returnsGSL_EDOMand the error handler is invoked.
This function returns the maximum value contained in the histogram bins.
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.
This function returns the minimum value contained in the histogram bins.
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.
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.
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.
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.
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.
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.
This function returns the sum of all bin values. Negative bin values are included in the sum.
This function returns 1 if all the individual bin ranges of the two histograms are identical, and 0 otherwise.
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.
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.
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.
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.
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.
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.
The library provides functions for reading and writing two dimensional histograms to a file as binary data or formatted text.
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_EFAILEDif 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.
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_EFAILEDif 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.
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,%eor%fformats for floating point numbers. The function returns 0 for success andGSL_EFAILEDif 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.
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_histogram_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 andGSL_EFAILEDif there was a problem reading from the file.
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.
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.
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_pdfstruct. If insufficient memory is available a null pointer is returned and the error handler is invoked with an error code ofGSL_ENOMEM.
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_EDOMbecause a probability distribution cannot contain negative values.
This function frees the two-dimensional probability distribution function p and all of the memory associated with it.
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.
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 <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_histogram2d.h>
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);
}
}
return 0;
}
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
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;
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.
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.
This function writes the current ntuple ntuple->ntuple_data of size ntuple->size to the corresponding file.
This function is a synonym for
gsl_ntuple_write.
This function reads the current row of the ntuple file for ntuple and stores the values in ntuple->data.
This function closes the ntuple file ntuple and frees its associated allocated memory.
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.
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.
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 <gsl/gsl_ntuple.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
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);
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 <math.h>
#include <gsl/gsl_ntuple.h>
#include <gsl/gsl_histogram.h>
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)
{
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.
Further information on the use of ntuples can be found in the documentation for the cern packages paw and hbook (available online).
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.
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.
This data type defines a general function with parameters for Monte Carlo integration.
double (* f) (double *x, size_tdim, 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)
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 <f> = (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 / N) \sum_i^N (f(x_i) - <f>)^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.
This function allocates and initializes a workspace for Monte Carlo integration in dim dimensions.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
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.
This function frees the memory associated with the integrator state s.
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.
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.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
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.
This function frees the memory associated with the integrator state s.
The miser algorithm has several configurable parameters. The
following variables can be accessed through the
gsl_monte_miser_state struct,
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.
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.
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.
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.
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.
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).
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.
This function initializes a previously allocated integration state. This allows an existing workspace to be reused for different integrations.
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.
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,
The vegas algorithm is highly configurable. The following variables
can be accessed through the gsl_monte_vegas_state struct,
These parameters contain the raw value of the integral result and its error sigma from the last iteration of the algorithm.
This parameter gives the chi-squared per degree of freedom for the weighted estimate of the integral. The value of chisq should be close to 1. A value of chisq 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.
The parameter
alphacontrols 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.
The number of iterations to perform for each call to the routine. The default value is 5 iterations.
Setting this determines the stage of the calculation. Normally,
stage = 0which begins with a new uniform grid and empty weighted average. Calling vegas withstage = 1retains 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 withstage = 1on the optimized grid. Settingstage = 2keeps 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. Choosingstage = 3enters at the main loop, so that nothing is changed, and is equivalent to performing additional iterations in a previous call.
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).
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 of0prints summary information about the weighted average and final result, while a value of1also displays the grid coordinates. A value of2prints information from the rebinning procedure for each iteration.
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 <stdlib.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_monte.h>
#include <gsl/gsl_monte_plain.h>
#include <gsl/gsl_monte_miser.h>
#include <gsl/gsl_monte_vegas.h>
/* 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)
{
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 = %.1g 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, s->chisq);
}
while (fabs (s->chisq - 1.0) > 0.5);
display_results ("vegas final", res, err);
gsl_monte_vegas_free (s);
}
return 0;
}
With 500,000 function calls the plain Monte Carlo algorithm achieves a
fractional error of 0.6%. The estimated error sigma is
consistent with the actual error, and the computed result differs from
the true result by about one standard deviation,
plain ==================
result = 1.385867
sigma = 0.007938
exact = 1.393204
error = -0.007337 = 0.9 sigma
The miser algorithm reduces the error by a factor of two, and also correctly estimates the error,
miser ==================
result = 1.390656
sigma = 0.003743
exact = 1.393204
error = -0.002548 = 0.7 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 = 1.386925
sigma = 0.002651
exact = 1.393204
error = -0.006278 = 2 sigma
converging...
result = 1.392957 sigma = 0.000452 chisq/dof = 1.1
vegas final ==================
result = 1.392957
sigma = 0.000452
exact = 1.393204
error = -0.000247 = 0.5 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.
The miser algorithm is described in the following article by Press and Farrar,
The vegas algorithm is described in the following papers,
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.
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.
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,memcpyandfree. 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
stdoutwith the following columns:number_of_iterations temperature x x-(*x0_p) Ef(x)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.
This function type should return the energy of a configuration xp.
double (*gsl_siman_Efunc_t) (void *xp)
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)
This function type should return the distance between two configurations xp and yp.
double (*gsl_siman_metric_t) (void *xp, void *yp)
This function type should print the contents of the configuration xp.
void (*gsl_siman_print_t) (void *xp)
This function type should copy the configuration source into dest.
void (*gsl_siman_copy_t) (void *source, void *dest)
This function type should create a new copy of the configuration xp.
void * (*gsl_siman_copy_construct_t) (void *xp)
This function type should destroy the configuration xp, freeing its memory.
void (*gsl_siman_destroy_t) (void *xp)
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.
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.
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 <math.h>
#include <stdlib.h>
#include <gsl/gsl_siman.h>
/* 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 10
/* max step size in random walk */
#define STEP_SIZE 10
/* Boltzmann constant */
#define K 1.0
/* initial temperature */
#define T_INITIAL 0.002
/* damping factor for temperature */
#define MU_T 1.005
#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(int argc, char *argv[])
{
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);
return 0;
}
Here are a couple of plots that are generated by running
siman_test in the following way:
$ ./siman_test | grep -v "^#"
| xyplot -xyil -y -0.88 -0.83 -d "x...y"
| xyps -d > siman-test.eps
$ ./siman_test | grep -v "^#"
| xyplot -xyil -xl "generation" -yl "energy" -d "x..y"
| xyps -d > siman-energy.eps
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
| xyplot -xyil -d "x................y"
-lx "generation" -ly "distance"
-lt "TSP -- 12 southwest cities"
| xyps -d > 12-cities.eps
$ grep initial_city_coord tsp.output
| awk '{print $2, $3, $4, $5}'
| xyplot -xyil -lb0 -cs 0.8
-lx "longitude (- means west)" -ly "latitude"
-lt "TSP -- initial-order"
| xyps -d > initial-route.eps
$ grep final_city_coord tsp.output
| awk '{print $2, $3, $4, $5}'
| xyplot -xyil -lb0 -cs 0.8
-lx "longitude (- means west)" -ly "latitude"
-lt "TSP -- final-order"
| xyps -d > 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: -106.28 35.89 Los Alamos
###final_city_coord: -106.62 35.12 Albuquerque
###final_city_coord: -107.84 35.15 Grants
###final_city_coord: -107.87 37.29 Durango
###final_city_coord: -108.58 37.35 Cortez
###final_city_coord: -108.74 35.52 Gallup
###final_city_coord: -112.07 33.54 Phoenix
###final_city_coord: -106.76 32.34 Las Cruces
###final_city_coord: -96.77 32.79 Dallas
###final_city_coord: -103.2 34.41 Clovis
###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:
Further information is available in the following book,
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.
These functions are declared in the header file gsl_odeiv.h.
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_odeiv_system
datatype.
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_SUCCESSif the calculation was completed successfully. Any other return value indicates an error.int (* jacobian) (double t, const double y[], double * dfdy, double dfdt[], void * params);- This function should store the vector of derivative elements df_i(t,y,params)/dt 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]wheredimensionis the dimension of the system. The function should returnGSL_SUCCESSif the calculation was completed successfully. Any other return value indicates an error.Some of the simpler solver algorithms do not make use of the Jacobian matrix, so it is not always strictly necessary to provide it (the
jacobianelement of the struct can be replaced by a null pointer for those algorithms). However, it is useful to provide the Jacobian to allow the solver algorithms to be interchanged—the best algorithms make use of the Jacobian.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.
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.
This function returns a pointer to a newly allocated instance of a stepping function of type T for a system of dim dimensions.
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.
This function frees all the memory associated with the stepping function s.
This function returns a pointer to the name of the stepping function. For example,
printf ("step method is '%s'\n", gsl_odeiv_step_name (s));would print something like
step method is 'rk4'.
This function returns the order of the stepping function on the previous step. This order can vary if the stepping function itself is adaptive.
This function applies the stepping function s to the system of equations defined by dydt, 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.
If the user-supplied functions defined in the system dydt return a status other than
GSL_SUCCESSthe step will be aborted. In this 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. To distinguish between error codes from the user-supplied functions and those fromgsl_odeiv_step_applyitself, any user-defined return values should be distinct from the standard GSL error codes.
The following algorithms are available,
Embedded Runge-Kutta-Fehlberg (4, 5) method. This method is a good general-purpose integrator.
Implicit Bulirsch-Stoer method of Bader and Deuflhard. This algorithm requires the Jacobian.
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.
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'_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.
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.
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.
This function creates a new control object which uses the same algorithm as
gsl_odeiv_control_standard_newbut 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'_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.
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.
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).
This function frees all the memory associated with the control function c.
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 andGSL_ODEIV_HADJ_INCis returned. The function returnsGSL_ODEIV_HADJ_NILif 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.
This function returns a pointer to the name of the control function. For example,
printf ("control method is '%s'\n", gsl_odeiv_control_name (c));would print something like
control method is 'standard'
The highest level of the system is the evolution function which combines the results of a stepping function and control function to reliably advance the solution forward over an interval (t_0, t_1). If the control function signals that the step-size should be decreased the evolution function backs out of the current step and tries the proposed smaller step-size. This process is continued until an acceptable step-size is found.
This function returns a pointer to a newly allocated instance of an evolution function for a system of dim dimensions.
This function advances the system (e, dydt) 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, but this will be modified using the control function c to achieve the appropriate error bound if necessary. The routine may make several calls to step in order to determine the optimum step-size. If the step-size has been changed the value of h will be modified on output. 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.
If the user-supplied functions defined in the system dydt return a status other than
GSL_SUCCESSthe step will be aborted. In this case, t and y will be restored to their pre-step values and the error code from the user-supplied function will be returned. To distinguish between error codes from the user-supplied functions and those fromgsl_odeiv_evolve_applyitself, any user-defined return values should be distinct from the standard GSL error codes.
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.
This function frees all the memory associated with the evolution function e.
The following program solves the second-order nonlinear Van der Pol oscillator equation,
x''(t) + \mu x'(t) (x(t)^2 - 1) + x(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, y = x'(t),
x' = y
y' = -x + \mu y (1-x^2)
The program begins by defining functions for these derivatives and their Jacobian,
#include <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_matrix.h>
#include <gsl/gsl_odeiv.h>
int
func (double t, const double y[], double f[],
void *params)
{
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)
{
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)
{
const gsl_odeiv_step_type * T
= gsl_odeiv_step_rk8pd;
gsl_odeiv_step * s
= gsl_odeiv_step_alloc (T, 2);
gsl_odeiv_control * c
= gsl_odeiv_control_y_new (1e-6, 0.0);
gsl_odeiv_evolve * e
= gsl_odeiv_evolve_alloc (2);
double mu = 10;
gsl_odeiv_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_odeiv_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_odeiv_evolve_free (e);
gsl_odeiv_control_free (c);
gsl_odeiv_step_free (s);
return 0;
}
For functions with multiple parameters, the appropriate information can be passed in through the params argument using a pointer to a struct.
The main loop of the program evolves the solution from (y, y') = (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 y.
To obtain the values at regular intervals, rather than the variable spacings chosen by the control function, the main loop can be modified to advance the solution from one point to the next. For example, the following main loop prints the solution at the fixed points t = 0, 1, 2, \dots, 100,
for (i = 1; i <= 100; i++)
{
double ti = i * t1 / 100.0;
while (t < ti)
{
gsl_odeiv_evolve_apply (e, c, s,
&sys,
&t, ti, &h,
y);
}
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
It is also possible to work with a non-adaptive integrator, using only
the stepping function itself. The following program uses the rk4
fourth-order Runge-Kutta stepping function with a fixed stepsize of
0.01,
int
main (void)
{
const gsl_odeiv_step_type * T
= gsl_odeiv_step_rk4;
gsl_odeiv_step * s
= gsl_odeiv_step_alloc (T, 2);
double mu = 10;
gsl_odeiv_system sys = {func, jac, 2, &mu};
double t = 0.0, t1 = 100.0;
double h = 1e-2;
double y[2] = { 1.0, 0.0 }, y_err[2];
double dydt_in[2], dydt_out[2];
/* initialise dydt_in from system parameters */
GSL_ODEIV_FN_EVAL(&sys, t, y, dydt_in);
while (t < t1)
{
int status = gsl_odeiv_step_apply (s, t, h,
y, y_err,
dydt_in,
dydt_out,
&sys);
if (status != GSL_SUCCESS)
break;
dydt_in[0] = dydt_out[0];
dydt_in[1] = dydt_out[1];
t += h;
printf ("%.5e %.5e %.5e\n", t, y[0], y[1]);
}
gsl_odeiv_step_free (s);
return 0;
}
The derivatives must be initialized for the starting point t=0 before the first step is taken. Subsequent steps use the output derivatives dydt_out as inputs to the next step by copying their values into dydt_in.
Many of the basic Runge-Kutta formulas can be found in the Handbook of Mathematical Functions,
The implicit Bulirsch-Stoer algorithm bsimp is described in the
following paper,
This chapter describes functions for performing interpolation. The library provides a variety of interpolation methods, including Cubic splines and Akima 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.
The functions described in this section are declared in the header files gsl_interp.h and gsl_spline.h.
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.
The interpolation function for a given dataset is stored in a
gsl_interp object. These are created by the following functions.
This function returns a pointer to a newly allocated interpolation object of type T for size data-points.
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; the behavior for other arrangements is not defined.
This function frees the interpolation object interp.
The interpolation library provides five interpolation types:
Linear interpolation. This interpolation method does not require any additional memory.
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.
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.
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.
Non-rounded Akima spline with natural boundary conditions. This method uses the non-rounded corner algorithm of Wodicka.
Non-rounded Akima spline with periodic boundary conditions. This method uses the non-rounded corner algorithm of Wodicka.
The following related functions are available:
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.
This function returns the minimum number of points required by the interpolation type of interp. For example, Akima spline interpolation requires a minimum of 5 points.
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.
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].
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.
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].
This function frees the accelerator object acc.
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.
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.
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.
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.
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.
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 <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_spline.h>
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 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 <stdlib.h>
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_spline.h>
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: first = last
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.
Descriptions of the interpolation algorithms and further references can be found in the following books:
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.
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, 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.
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.
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_forwardwith a negative step-size.
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 <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_deriv.h>
double f (double x, void * params)
{
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 +/- 0.0000004064
exact = 2.1213203436
x = 0.0
f'(x) = 0.0000000160 +/- 0.0000000339
exact = 0.0000000000
The algorithms used by these functions are described in the following sources:
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.
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.
This function allocates space for a Chebyshev series of order n and returns a pointer to a new
gsl_cheb_seriesstruct.
This function frees a previously allocated Chebyshev series cs.
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.
This function evaluates the Chebyshev series cs at a given point x.
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.
This function evaluates the Chebyshev series cs at a given point n, to (at most) the given order order.
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.
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.
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.
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.
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 <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_chebyshev.h>
double
f (double x, void *p)
{
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.
The following paper describes the use of Chebyshev series,
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.
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.
This function allocates a workspace for a Levin u-transform of n terms. The size of the workspace is O(2n^2 + 3n).
This function frees the memory associated with the workspace w.
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.
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.
This function allocates a workspace for a Levin u-transform of n terms, without error estimation. The size of the workspace is O(3n).
This function frees the memory associated with the workspace w.
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.
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 <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_sum.h>
#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 %d terms\n",
w->sum_plain, w->terms_used);
printf ("exact value = % .16f\n", zeta_2);
printf ("accelerated sum = % .16f using %d 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 = 1.6449340668166479 using 13 terms
estimated error = 0.0000000000508580
actual error = -0.0000000000315785
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.
The algorithms used by these functions are described in the following papers,
The theory of the u-transform was presented by Levin,
A review paper on the Levin Transform is available online,
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.
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.
The gsl_wavelet structure contains the filter coefficients
defining the wavelet and any associated offset parameters.
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:
The is the Daubechies wavelet family of maximum phase with k/2 vanishing moments. The implemented wavelets are k=4, 6, ..., 20, with k even.
This is the Haar wavelet. The only valid choice of k for the Haar wavelet is k=2.
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.
This function returns a pointer to the name of the wavelet family for 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.
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.
This function frees the allocated workspace work.
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.
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
transformversion of the function the argument dir can be eitherforward(+1) orbackward(-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_SUCCESSupon successful completion.GSL_EINVALis returned if n is not an integer power of 2 or if insufficient workspace is provided.
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.
These functions compute two-dimensional in-place forward and inverse discrete wavelet transforms in standard and non-standard forms 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
transformversion of the function the argument dir can be eitherforward(+1) orbackward(-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_SUCCESSupon successful completion.GSL_EINVALis returned if size1 and size2 are not equal and integer powers of 2, or if insufficient workspace is provided.
These functions compute the two-dimensional in-place wavelet transform on a matrix a.
These functions compute the two-dimensional wavelet transform in non-standard form.
These functions compute the non-standard form of the two-dimensional in-place wavelet transform on a matrix a.
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 <stdio.h>
#include <math.h>
#include <gsl/gsl_sort.h>
#include <gsl/gsl_wavelet.h>
int
main (int argc, char **argv)
{
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));
gsl_wavelet *w;
gsl_wavelet_workspace *work;
w = gsl_wavelet_alloc (gsl_wavelet_daubechies, 4);
work = gsl_wavelet_workspace_alloc (n);
FILE *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]);
}
}
The output can be used with the gnu plotutils graph program,
$ ./a.out ecg.dat > dwt.dat
$ graph -T ps -x 0 256 32 -h 0.3 -a dwt.dat > dwt.ps
The mathematical background to wavelet transforms is covered in the original lectures by Daubechies,
An easy to read introduction to the subject with an emphasis on the application of the wavelet transform in various branches of science is,
For extensive coverage of signal analysis by wavelets, wavelet packets and local cosine bases see,
The concept of multiresolution analysis underlying the wavelet transform is described in,
The coefficients for the individual wavelet families implemented by the library can be found in the following papers,
The PhysioNet archive of physiological datasets can be found online at http://www.physionet.org/ and is described in the following paper,
This chapter describes functions for performing Discrete Hankel Transforms (DHTs). The functions are declared in the header file gsl_dht.h.
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 zeroes 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.
Specifically, let f(t) be a function on the unit interval. Then the finite \nu-Hankel transform of f(t) is defined to be the set of numbers g_m given by,
g_m = \int_0^1 t dt J_\nu(j_(\nu,m)t) f(t),
so that,
f(t) = \sum_{m=1}^\infty (2 J_\nu(j_(\nu,m)x) / J_(\nu+1)(j_(\nu,m))^2) g_m.
Suppose that f is band-limited in the sense that g_m=0 for m > M. Then we have the following fundamental sampling theorem.
g_m = (2 / j_(\nu,M)^2)
\sum_{k=1}^{M-1} f(j_(\nu,k)/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. 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 a scaling of
the fundamental interval, for convenience, so that one can assume the
function is defined on the interval [0,X], rather than the unit
interval.
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.
This function allocates a Discrete Hankel transform object of size size.
This function initializes the transform t for the given values of nu and x.
This function allocates a Discrete Hankel transform object of size size and initializes it for the given values of nu and x.
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.
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.
This function returns the value of the n-th sample point in “k-space”, j_{\nu,n+1}/X.
The algorithms used by these functions are described in the following papers,
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.
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,
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.
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.
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.
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.
This function initializes, or reinitializes, an existing solver s to use the function f and the initial search interval [x_lower, x_upper].
This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial guess root.
These functions free all the memory associated with the solver 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.
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:
This data type defines a general function with parameters.
double (* function) (doublex, 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:
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 following macro,
#define GSL_FN_EVAL(F,x)
(*((F)->function))(x,(F)->params)
This data type defines a general function with parameters and its first derivative.
double (* f) (doublex, void *params)- this function should return the value of f(x,params) for argument x and parameters params
double (* df) (doublex, 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) (doublex, 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 following macro,
#define GSL_FN_FDF_EVAL_F(FDF,x)
(*((FDF)->f))(x,(FDF)->params)
The derivative f'(x) can be evaluated using the following macro,
#define GSL_FN_FDF_EVAL_DF(FDF,x)
(*((FDF)->df))(x,(FDF)->params)
and both the function y = f(x) and its derivative dy = f'(x) can be evaluated at the same time using the following macro,
#define GSL_FN_FDF_EVAL_F_DF(FDF,x,y,dy)
(*((FDF)->fdf))(x,(FDF)->params,(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.
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 a 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.
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.
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
InforNaN.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,
These functions return the current estimate of the root for the solver s.
These functions return the current bracketing interval for the solver s.
A root finding procedure should stop when one of the following conditions is true:
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.
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_SUCCESSif 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.
This function tests for the convergence of the sequence ..., x0, x1 with absolute error epsabs and relative error epsrel. The test returns
GSL_SUCCESSif the following condition is achieved,|x_1 - x_0| < epsabs + epsrel |x_1|and returns
GSL_CONTINUEotherwise.
This function tests the residual value f against the absolute error bound epsabs. The test returns
GSL_SUCCESSif the following condition is achieved,|f| < epsabsand returns
GSL_CONTINUEotherwise. 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.
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.
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.
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.
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.
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.
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.
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.
The Steffenson Method 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.
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;
double c = p->c;
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 <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>
#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);
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 <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_roots.h>
#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);
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.
For information on the Brent-Dekker algorithm see the following two papers,
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.
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,
The state for the minimizers is held in a gsl_min_fminimizer
struct. The updating procedure uses only function evaluations (not
derivatives).
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.
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.
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.
This function is equivalent to
gsl_min_fminimizer_setbut uses the values f_minimum, f_lower and f_upper instead of computingf(x_minimum),f(x_lower)andf(x_upper).
This function frees all the memory associated with the minimizer 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.
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
(see Providing the function to solve).
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.
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
InforNaN.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,
This function returns the current estimate of the position of the minimum for the minimizer s.
These functions return the current upper and lower bound of the interval for the minimizer 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.
A minimization procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The function below allows the user to test the precision of the current result.
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_SUCCESSif 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.
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.
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 = 0.3189660...) 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.
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.
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 <stdio.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_min.h>
double fn1 (double x, void * params)
{
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 %.7f\n",
iter, a, b,
m, m_expected, m - m_expected, b - a);
}
while (status == GSL_CONTINUE && iter < max_iter);
return status;
}
Here are the results of the minimization procedure.
$ ./a.out
0 [0.0000000, 6.0000000] 2.0000000 -1.1415927 6.0000000
1 [2.0000000, 6.0000000] 3.2758640 +0.1342713 4.0000000
2 [2.0000000, 3.2831929] 3.2758640 +0.1342713 1.2831929
3 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
4 [2.8689068, 3.2831929] 3.2758640 +0.1342713 0.4142862
5 [2.8689068, 3.2758640] 3.1460585 +0.0044658 0.4069572
6 [3.1346075, 3.2758640] 3.1460585 +0.0044658 0.1412565
7 [3.1346075, 3.1874620] 3.1460585 +0.0044658 0.0528545
8 [3.1346075, 3.1460585] 3.1460585 +0.0044658 0.0114510
9 [3.1346075, 3.1460585] 3.1424060 +0.0008133 0.0114510
10 [3.1346075, 3.1424060] 3.1415885 -0.0000041 0.0077985
Converged:
11 [3.1415885, 3.1424060] 3.1415927 -0.0000000 0.0008175
Further information on Brent's algorithm is available in the following book,
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.
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,
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.
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.
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.
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.
This function sets, or resets, an existing solver s to use the function f and the initial guess x.
This function sets, or resets, an existing solver s to use the function and derivative fdf and the initial guess x.
These functions free all the memory associated with the solver 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.
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:
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;
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.
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.
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
InforNaN.GSL_ENOPROG- the iteration is not making any progress, preventing the algorithm from continuing.
The solver maintains a current best estimate of the root at all times. This information can be accessed with the following auxiliary functions,
These functions return the current estimate of the root for the solver s.
These functions return the function value f(x) at the current estimate of the root for the solver s.
These functions return the last step dx taken by the solver s.
A root finding procedure should stop when one of the following conditions is true:
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.
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_SUCCESSif the following condition is achieved,|dx_i| < epsabs + epsrel |x_i|for each component of x and returns
GSL_CONTINUEotherwise.
This function tests the residual value f against the absolute error bound epsabs. The test returns
GSL_SUCCESSif the following condition is achieved,\sum_i |f_i| < epsabsand returns
GSL_CONTINUEotherwise. 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.
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.
This is a modified version of Powell's Hybrid method as implemented in the hybrj algorithm in minpack. Minpack was written by Jorge J. Moré, 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.
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 byhybridsjis inappropriate.
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.
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.
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.
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_fdjacwith a relative step size ofGSL_SQRT_DBL_EPSILON.
This is a finite difference version of the Hybrid algorithm without internal scaling.
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_jwhere \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.
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} dfwhere 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.
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 <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_multiroots.h>
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.
The original version of the Hybrid method is described in the following articles by Powell,
The following papers are also relevant to the algorithms described in this section,
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.
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.
The Nelder-Mead Simplex algorithm applies a different strategy. It maintains n+1 trial parameter vectors as the vertices of a n-dimensional simplex. In each iteration step it tries to improve the worst vertex by a simple geometrical transformation until the size of the simplex falls below a given tolerance.
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,
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.
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.
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.
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.
This function 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|. — Function: int gsl_multimin_fminimizer_set (gsl_multimin_fminimizer * s, gsl_multimin_function * f, const gsl_vector * x, const gsl_vector * step_size)
This function 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.
This function frees all the memory associated with the minimizer 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.
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:
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.
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.
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.
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 paraboloid with two parameters,
/* Paraboloid centered on (dp[0],dp[1]) */
double
my_f (const gsl_vector *v, void *params)
{
double x, y;
double *dp = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
return 10.0 * (x - dp[0]) * (x - dp[0]) +
20.0 * (y - dp[1]) * (y - dp[1]) + 30.0;
}
/* 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 *dp = (double *)params;
x = gsl_vector_get(v, 0);
y = gsl_vector_get(v, 1);
gsl_vector_set(df, 0, 20.0 * (x - dp[0]));
gsl_vector_set(df, 1, 40.0 * (y - dp[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;
double p[2] = { 1.0, 2.0 }; /* center at (1,2) */
my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = (void *)p;
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.
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 minimizer maintains a current best estimate of the minimum at all times. This information can be accessed with the following auxiliary functions,
These functions return the current best estimate of the location of the minimum, the value of the function at that point, its gradient, and minimizer specific characteristic size for the minimizer s.
This function resets the minimizer s to use the current point as a new starting point.
A minimization procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The functions below allow the user to test the precision of the current result.
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_SUCCESSif the following condition is achieved,|g| < epsabsand returns
GSL_CONTINUEotherwise. 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.
This function tests the minimizer specific characteristic size (if applicable to the used minimizer) against absolute tolerance epsabs. The test returns
GSL_SUCCESSif the size is smaller than tolerance, otherwiseGSL_CONTINUEis returned.
There are several minimization methods available. The best choice of algorithm depends on the problem. All of the algorithms use the value of the function and its gradient at each evaluation point, except for the Simplex algorithm which uses function values only.
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.
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.
This is the vector Broyden-Fletcher-Goldfarb-Shanno (BFGS) conjugate gradient algorithm. It 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 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.
This is the Simplex algorithm of Nelder and Mead. It constructs n vectors p_i from the starting vector x and the vector step_size as follows:
p_0 = (x_0, x_1, ... , x_n) p_1 = (x_0 + step_size_0, x_1, ... , x_n) p_2 = (x_0, x_1 + step_size_1, ... , x_n) ... = ... p_n = (x_0, x_1, ... , x_n+step_size_n)These vectors form the n+1 vertices of a simplex in n dimensions. On each iteration the algorithm tries to improve the parameter vector p_i corresponding to the highest function value by simple geometrical transformations. These are reflection, reflection followed by expansion, contraction and multiple contraction. Using these transformations the simplex moves through the parameter 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 routine calculates the minimizer specific characteristic size 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.
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). */
double par[2] = { 1.0, 2.0 };
gsl_vector *x;
gsl_multimin_function_fdf my_func;
my_func.f = &my_f;
my_func.df = &my_df;
my_func.fdf = &my_fdf;
my_func.n = 2;
my_func.params = ∥
/* 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)
{
size_t np = 2;
double par[2] = {1.0, 2.0};
const gsl_multimin_fminimizer_type *T =
gsl_multimin_fminimizer_nmsimplex;
gsl_multimin_fminimizer *s = NULL;
gsl_vector *ss, *x;
gsl_multimin_function minex_func;
size_t iter = 0, i;
int status;
double size;
/* Initial vertex size vector */
ss = gsl_vector_alloc (np);
/* Set all step sizes to 1 */
gsl_vector_set_all (ss, 1.0);
/* Starting point */
x = gsl_vector_alloc (np);
gsl_vector_set (x, 0, 5.0);
gsl_vector_set (x, 1, 7.0);
/* Initialize method and iterate */
minex_func.f = &my_f;
minex_func.n = np;
minex_func.params = (void *)∥
s = gsl_multimin_fminimizer_alloc (T, np);
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 ", iter);
for (i = 0; i < np; i++)
{
printf ("%10.3e ", gsl_vector_get (s->x, i));
}
printf ("f() = %7.3f size = %.3f\n", 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 = 1.082
2 5.250e+00 4.000e+00 f() = 290.625 size = 1.372
3 5.250e+00 4.000e+00 f() = 290.625 size = 1.372
4 5.500e+00 1.000e+00 f() = 252.500 size = 1.372
5 2.625e+00 3.500e+00 f() = 101.406 size = 1.823
6 3.469e+00 1.375e+00 f() = 98.760 size = 1.526
7 1.820e+00 3.156e+00 f() = 63.467 size = 1.105
8 1.820e+00 3.156e+00 f() = 63.467 size = 1.105
9 1.016e+00 2.812e+00 f() = 43.206 size = 1.105
10 2.041e+00 2.008e+00 f() = 40.838 size = 0.645
11 1.236e+00 1.664e+00 f() = 32.816 size = 0.645
12 1.236e+00 1.664e+00 f() = 32.816 size = 0.447
13 5.225e-01 1.980e+00 f() = 32.288 size = 0.447
14 1.103e+00 2.073e+00 f() = 30.214 size = 0.345
15 1.103e+00 2.073e+00 f() = 30.214 size = 0.264
16 1.103e+00 2.073e+00 f() = 30.214 size = 0.160
17 9.864e-01 1.934e+00 f() = 30.090 size = 0.132
18 9.190e-01 1.987e+00 f() = 30.069 size = 0.092
19 1.028e+00 2.017e+00 f() = 30.013 size = 0.056
20 1.028e+00 2.017e+00 f() = 30.013 size = 0.046
21 1.028e+00 2.017e+00 f() = 30.013 size = 0.033
22 9.874e-01 1.985e+00 f() = 30.006 size = 0.028
23 9.846e-01 1.995e+00 f() = 30.003 size = 0.023
24 1.007e+00 2.003e+00 f() = 30.001 size = 0.012
converged to minimum at
25 1.007e+00 2.003e+00 f() = 30.001 size = 0.010
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.
A brief description of multidimensional minimization algorithms and further references can be found in the following book,
The simplex algorithm is described in the following paper,
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. The functions are declared in the header file gsl_fit.h.
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 \langle \, \rangle 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
allowing the covariance matrix to be written in terms of the errors on the data,
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
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 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.
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.
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.
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.
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.
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.
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.
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.
The functions described in this section perform least-squares fits to a general linear model, y = X c where y is a vector of n observations, X is an n by p matrix of predictor variables, and the elements of the vector c are the p unknown best-fit parameters which are to be estimated.
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 functions described in this section are declared in the header file gsl_multifit.h.
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.
This function allocates a workspace for fitting a model to n observations using p parameters.
This function frees the memory associated with the workspace w.
These functions compute the best-fit parameters c of the model y = X c for the observations y and the matrix of predictor variables X. The variance-covariance matrix of the model parameters cov is estimated from the scatter of the observations about the best-fit. The sum of squares of the residuals from the best-fit, \chi^2, is returned in chisq.
The best-fit is found by singular value decomposition of the matrix X using the preallocated workspace provided in work. The modified Golub-Reinsch SVD algorithm is used, 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. In the second form of the function the components are discarded if the ratio of singular values s_i/s_0 falls below the user-specified tolerance tol, and the effective rank is returned in rank.
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. The covariance matrix of the model parameters cov is computed with the given weights. The weighted sum of squares of the residuals from the best-fit, \chi^2, is returned in chisq.
The best-fit is found by singular value decomposition of the matrix X using the preallocated workspace provided in work. Any components which have zero singular value (to machine precision) are discarded from the fit. In the second form of the function the components are discarded if the ratio of singular values s_i/s_0 falls below the user-specified tolerance tol, and the effective rank is returned in rank.
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.
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 <stdio.h>
#include <gsl/gsl_fit.h>
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
The next 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 <stdio.h>
#include <gsl/gsl_multifit.h>
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);
}
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 <stdio.h>
#include <math.h>
#include <gsl/gsl_randist.h>
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);
}
return 0;
}
The data can be prepared by running the resulting executable program,
$ ./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.
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,
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,
This chapter describes functions for multidimensional nonlinear least-squares fitting. 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_multifit_nlin.h contains prototypes for the multidimensional nonlinear fitting functions and related declarations.
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
All algorithms proceed from an initial guess using the linearization,
\psi(p) = || F(x+p) || ~=~ || F(x) + J p ||
where x is the initial point, p is the proposed step and J is the Jacobian matrix J_{ij} = d f_i / d x_j. Additional strategies are used to enlarge the region of convergence. These include requiring a decrease in the norm ||F|| on each step or using a trust region to avoid steps which fall outside the linear regime.
To perform a weighted least-squares fit of a nonlinear model Y(x,t) to data (t_i, y_i) with independent gaussian errors \sigma_i, use function components of the following form,
f_i = (Y(x, t_i) - y_i) / \sigma_i
Note that the model parameters are denoted by x in this chapter since the non-linear least-squares algorithms are described geometrically (i.e. finding the minimum of a surface). The independent variable of any data to be fitted is denoted by t.
With the definition above the Jacobian is J_{ij} =(1 / \sigma_i) d Y_i / d x_j, where Y_i = Y(x,t_i).
This function returns a pointer to a newly allocated instance of a solver of type T for n observations and p parameters. The number of observations n must be greater than or equal to parameters p.
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.
This function returns a pointer to a newly allocated instance of a derivative solver of type T for n observations and p parameters. For example, the following code creates an instance of a Levenberg-Marquardt solver for 100 data points and 3 parameters,
const gsl_multifit_fdfsolver_type * T = gsl_multifit_fdfsolver_lmder; gsl_multifit_fdfsolver * s = gsl_multifit_fdfsolver_alloc (T, 100, 3);The number of observations n must be greater than or equal to parameters p.
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.
This function initializes, or reinitializes, an existing solver s to use the function f and the initial guess x.
This function initializes, or reinitializes, an existing solver s to use the function and derivative fdf and the initial guess x.
These functions free all the memory associated with the solver s.
These functions return a pointer to the name of the solver. For example,
printf ("s is a '%s' solver\n", gsl_multifit_fdfsolver_name (s));would print something like
s is a 'lmder' solver.
You must provide n functions of p variables for the minimization algorithms to operate on. In order to allow for arbitrary parameters the functions are defined by the following data types:
This data type defines a general system of functions with arbitrary 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 arbitrary parameters params, returning an appropriate error code if the function cannot be computed.
size_t n- the number of functions, i.e. the number of components of the vector f.
size_t p- the number of independent variables, i.e. the number of components of the vector x.
void * params- a pointer to the arbitrary parameters of the function.
This data type defines a general system of functions with arbitrary 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 arbitrary 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-p matrix result J_ij = d f_i(x,params) / d x_j in J for argument x and arbitrary 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 arbitrary 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 number of functions, i.e. the number of components of the vector f.
size_t p- the number of independent variables, i.e. the number of components of the vector x.
void * params- a pointer to the arbitrary parameters of the function.
Note that when fitting a non-linear model against experimental data, the data is passed to the functions above using the params argument and the trial best-fit parameters through the x argument.
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.
These functions perform a single iteration of the solver s. If the iteration encounters an unexpected problem then an error code will be returned. The solver maintains a current estimate of the best-fit parameters at all times.
The solver struct s contains the following entries, which can be used to track the progress of the solution:
gsl_vector * xgsl_vector * fgsl_vector * dxgsl_matrix * Jgsl_multifit_fdfsolver struct only)
The best-fit information also can be accessed with the following auxiliary functions,
These functions return the current position (i.e. best-fit parameters)
s->xof the solver s.
A minimization procedure should stop when one of the following conditions is true:
The handling of these conditions is under user control. The functions below allow the user to test the current estimate of the best-fit parameters in several standard ways.
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_SUCCESSif the following condition is achieved,|dx_i| < epsabs + epsrel |x_i|for each component of x and returns
GSL_CONTINUEotherwise.
This function tests the residual gradient g against the absolute error bound epsabs. Mathematically, the gradient should be exactly zero at the minimum. The test returns
GSL_SUCCESSif the following condition is achieved,\sum_i |g_i| < epsabsand returns
GSL_CONTINUEotherwise. This criterion is suitable for situations where the precise location of the minimum, x, is unimportant provided a value can be found where the gradient is small enough.
This function computes the gradient g of \Phi(x) = (1/2) ||F(x)||^2 from the Jacobian matrix J and the function values f, using the formula g = J^T f.
The minimization algorithms described in this section make use of both the function and its derivative. They require an initial guess for the location of the minimum. 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 minimum for it to work.
This is a robust and efficient version of the Levenberg-Marquardt algorithm as implemented in the scaled lmder routine in minpack. Minpack was written by Jorge J. Moré, Burton S. Garbow and Kenneth E. Hillstrom.
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 attempts to minimize the linear system |F + J p| subject to the constraint |D p| < \Delta. The solution to this constrained linear system is found using the Levenberg-Marquardt method.
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, and follows the predicted behavior of the function within the trust region, then it is accepted and the size of the trust region is increased. If the proposed step fails to improve the solution, or differs significantly from the expected behavior within the trust region, then the size of the trust region is decreased and another trial step is computed.
The algorithm also monitors the progress of the solution and returns an error if the changes in the solution are smaller than the machine precision. The possible error codes are,
GSL_ETOLF- the decrease in the function falls below machine precision
GSL_ETOLX- the change in the position vector falls below machine precision
GSL_ETOLG- the norm of the gradient, relative to the norm of the function, falls below machine precision
These error codes indicate that further iterations will be unlikely to change the solution from its current value.
This is an unscaled version of the lmder algorithm. The elements of the diagonal scaling matrix D are set to 1. This algorithm may be useful in circumstances where the scaled version of lmder converges too slowly, or the function is already scaled appropriately.
There are no algorithms implemented in this section at the moment.
This function uses the Jacobian matrix J to compute the covariance matrix of the best-fit parameters, covar. The parameter epsrel is used to remove linear-dependent columns when J is rank deficient.
The covariance matrix is given by,
covar = (J^T J)^{-1}and is computed by QR decomposition of J with column-pivoting. Any columns of R which satisfy
|R_{kk}| <= epsrel |R_{11}|are considered linearly-dependent and are excluded from the covariance matrix (the corresponding rows and columns of the covariance matrix are set to zero).
If the minimisation uses the weighted least-squares function f_i = (Y(x, t_i) - y_i) / \sigma_i then the covariance matrix above gives the statistical error on the best-fit parameters resulting from the gaussian errors \sigma_i on the underlying data y_i. This can be verified from the relation \delta f = J \delta c and the fact that the fluctuations in f from the data y_i are normalised by \sigma_i and so satisfy <\delta f \delta f^T> = I.
For an unweighted least-squares function f_i = (Y(x, t_i) - y_i) the covariance matrix above should be multiplied by the variance of the residuals about the best-fit \sigma^2 = \sum (y_i - Y(x,t_i))^2 / (n-p) to give the variance-covariance matrix \sigma^2 C. This estimates the statistical error on the best-fit parameters from the scatter of the underlying data.
For more information about covariance matrices see Fitting Overview.
The following example program fits a weighted exponential model with
background to experimental data, Y = A \exp(-\lambda t) + b. The
first part of the program sets up the functions expb_f and
expb_df to calculate the model and its Jacobian. The appropriate
fitting function is given by,
f_i = ((A \exp(-\lambda t_i) + b) - y_i)/\sigma_i
where we have chosen t_i = i. The Jacobian matrix J is the derivative of these functions with respect to the three parameters (A, \lambda, b). It is given by,
J_{ij} = d f_i / d x_j
where x_0 = A, x_1 = \lambda and x_2 = b.
/* expfit.c -- model functions for exponential + background */
struct data {
size_t n;
double * y;
double * sigma;
};
int
expb_f (const gsl_vector * x, void *data,
gsl_vector * f)
{
size_t n = ((struct data *)data)->n;
double *y = ((struct data *)data)->y;
double *sigma = ((struct data *) data)->sigma;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
double b = gsl_vector_get (x, 2);
size_t i;
for (i = 0; i < n; i++)
{
/* Model Yi = A * exp(-lambda * i) + b */
double t = i;
double Yi = A * exp (-lambda * t) + b;
gsl_vector_set (f, i, (Yi - y[i])/sigma[i]);
}
return GSL_SUCCESS;
}
int
expb_df (const gsl_vector * x, void *data,
gsl_matrix * J)
{
size_t n = ((struct data *)data)->n;
double *sigma = ((struct data *) data)->sigma;
double A = gsl_vector_get (x, 0);
double lambda = gsl_vector_get (x, 1);
size_t i;
for (i = 0; i < n; i++)
{
/* Jacobian matrix J(i,j) = dfi / dxj, */
/* where fi = (Yi - yi)/sigma[i], */
/* Yi = A * exp(-lambda * i) + b */
/* and the xj are the parameters (A,lambda,b) */
double t = i;
double s = sigma[i];
double e = exp(-lambda * t);
gsl_matrix_set (J, i, 0, e/s);
gsl_matrix_set (J, i, 1, -t * A * e/s);
gsl_matrix_set (J, i, 2, 1/s);
}
return GSL_SUCCESS;
}
int
expb_fdf (const gsl_vector * x, void *data,
gsl_vector * f, gsl_matrix * J)
{
expb_f (x, data, f);
expb_df (x, data, J);
return GSL_SUCCESS;
}
The main part of the program sets up a Levenberg-Marquardt solver and some simulated random data. The data uses the known parameters (1.0,5.0,0.1) combined with gaussian noise (standard deviation = 0.1) over a range of 40 timesteps. The initial guess for the parameters is chosen as (0.0, 1.0, 0.0).
#include <stdlib.h>
#include <stdio.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_vector.h>
#include <gsl/gsl_blas.h>
#include <gsl/gsl_multifit_nlin.h>
#include "expfit.c"
#define N 40
void print_state (size_t iter, gsl_multifit_fdfsolver * s);
int
main (void)
{
const gsl_multifit_fdfsolver_type *T;
gsl_multifit_fdfsolver *s;
int status;
size_t i, iter = 0;
const size_t n = N;
const size_t p = 3;
gsl_matrix *covar = gsl_matrix_alloc (p, p);
double y[N], sigma[N];
struct data d = { n, y, sigma};
gsl_multifit_function_fdf f;
double x_init[3] = { 1.0, 0.0, 0.0 };
gsl_vector_view x = gsl_vector_view_array (x_init, p);
const gsl_rng_type * type;
gsl_rng * r;
gsl_rng_env_setup();
type = gsl_rng_default;
r = gsl_rng_alloc (type);
f.f = &expb_f;
f.df = &expb_df;
f.fdf = &expb_fdf;
f.n = n;
f.p = p;
f.params = &d;
/* This is the data to be fitted */
for (i = 0; i < n; i++)
{
double t = i;
y[i] = 1.0 + 5 * exp (-0.1 * t)
+ gsl_ran_gaussian (r, 0.1);
sigma[i] = 0.1;
printf ("data: %d %g %g\n", i, y[i], sigma[i]);
};
T = gsl_multifit_fdfsolver_lmsder;
s = gsl_multifit_fdfsolver_alloc (T, n, p);
gsl_multifit_fdfsolver_set (s, &f, &x.vector);
print_state (iter, s);
do
{
iter++;
status = gsl_multifit_fdfsolver_iterate (s);
printf ("status = %s\n", gsl_strerror (status));
print_state (iter, s);
if (status)
break;
status = gsl_multifit_test_delta (s->dx, s->x,
1e-4, 1e-4);
}
while (status == GSL_CONTINUE && iter < 500);
gsl_multifit_covar (s->J, 0.0, covar);
#define FIT(i) gsl_vector_get(s->x, i)
#define ERR(i) sqrt(gsl_matrix_get(covar,i,i))
{
double chi = gsl_blas_dnrm2(s->f);
double dof = n - p;
double c = GSL_MAX_DBL(1, chi / sqrt(dof));
printf("chisq/dof = %g\n", pow(chi, 2.0) / dof);
printf ("A = %.5f +/- %.5f\n", FIT(0), c*ERR(0));
printf ("lambda = %.5f +/- %.5f\n", FIT(1), c*ERR(1));
printf ("b = %.5f +/- %.5f\n", FIT(2), c*ERR(2));
}
printf ("status = %s\n", gsl_strerror (status));
gsl_multifit_fdfsolver_free (s);
return 0;
}
void
print_state (size_t iter, gsl_multifit_fdfsolver * s)
{
printf ("iter: %3u x = % 15.8f % 15.8f % 15.8f "
"|f(x)| = %g\n",
iter,
gsl_vector_get (s->x, 0),
gsl_vector_get (s->x, 1),
gsl_vector_get (s->x, 2),
gsl_blas_dnrm2 (s->f));
}
The iteration terminates when the change in x is smaller than 0.0001, as both an absolute and relative change. Here are the results of running the program:
iter: 0 x=1.00000000 0.00000000 0.00000000 |f(x)|=117.349
status=success
iter: 1 x=1.64659312 0.01814772 0.64659312 |f(x)|=76.4578
status=success
iter: 2 x=2.85876037 0.08092095 1.44796363 |f(x)|=37.6838
status=success
iter: 3 x=4.94899512 0.11942928 1.09457665 |f(x)|=9.58079
status=success
iter: 4 x=5.02175572 0.10287787 1.03388354 |f(x)|=5.63049
status=success
iter: 5 x=5.04520433 0.10405523 1.01941607 |f(x)|=5.44398
status=success
iter: 6 x=5.04535782 0.10404906 1.01924871 |f(x)|=5.44397
chisq/dof = 0.800996
A = 5.04536 +/- 0.06028
lambda = 0.10405 +/- 0.00316
b = 1.01925 +/- 0.03782
status = success
The approximate values of the parameters are found correctly, and the chi-squared value indicates a good fit (the chi-squared per degree of freedom is approximately 1). In this case the errors on the parameters can be estimated from the square roots of the diagonal elements of the covariance matrix.
If the chi-squared value shows a poor fit (i.e. chi^2/dof >> 1) then the error estimates obtained from the covariance matrix will be too small. In the example program the error estimates are multiplied by \sqrt{\chi^2/dof} in this case, a common way of increasing the errors for a poor fit. Note that a poor fit will result from the use an inappropriate model, and the scaled error estimates may then be outside the range of validity for gaussian errors.
The minpack algorithm is described in the following article,
The following paper is also relevant to the algorithms described in this section,
This chapter describes macros for the values of physical constants, such as the speed of light, c, and gravitational constant, G. The values are available in different unit systems, including the standard MKSA system (meters, kilograms, seconds, amperes) and the CGSM system (centimeters, grams, seconds, gauss), which is commonly used in Astronomy.
The definitions of constants in the MKSA system are available in the file gsl_const_mksa.h. The constants in the CGSM system are defined in gsl_const_cgsm.h. Dimensionless constants, such as the fine structure constant, which are pure numbers are defined in gsl_const_num.h.
The full list of constants is described briefly below. Consult the header files themselves for the values of the constants used in the library.
GSL_CONST_MKSA_SPEED_OF_LIGHTGSL_CONST_MKSA_VACUUM_PERMEABILITYGSL_CONST_MKSA_VACUUM_PERMITTIVITYGSL_CONST_MKSA_PLANCKS_CONSTANT_HGSL_CONST_MKSA_PLANCKS_CONSTANT_HBARGSL_CONST_NUM_AVOGADROGSL_CONST_MKSA_FARADAYGSL_CONST_MKSA_BOLTZMANNGSL_CONST_MKSA_MOLAR_GASGSL_CONST_MKSA_STANDARD_GAS_VOLUMEGSL_CONST_MKSA_STEFAN_BOLTZMANN_CONSTANTGSL_CONST_MKSA_GAUSSGSL_CONST_MKSA_ASTRONOMICAL_UNITGSL_CONST_MKSA_GRAVITATIONAL_CONSTANTGSL_CONST_MKSA_LIGHT_YEARGSL_CONST_MKSA_PARSECGSL_CONST_MKSA_GRAV_ACCELGSL_CONST_MKSA_SOLAR_MASSGSL_CONST_MKSA_ELECTRON_CHARGEGSL_CONST_MKSA_ELECTRON_VOLTGSL_CONST_MKSA_UNIFIED_ATOMIC_MASSGSL_CONST_MKSA_MASS_ELECTRONGSL_CONST_MKSA_MASS_MUONGSL_CONST_MKSA_MASS_PROTONGSL_CONST_MKSA_MASS_NEUTRONGSL_CONST_NUM_FINE_STRUCTUREGSL_CONST_MKSA_RYDBERGGSL_CONST_MKSA_BOHR_RADIUSGSL_CONST_MKSA_ANGSTROMGSL_CONST_MKSA_BARNGSL_CONST_MKSA_BOHR_MAGNETONGSL_CONST_MKSA_NUCLEAR_MAGNETONGSL_CONST_MKSA_ELECTRON_MAGNETIC_MOMENTGSL_CONST_MKSA_PROTON_MAGNETIC_MOMENTGSL_CONST_MKSA_THOMSON_CROSS_SECTIONGSL_CONST_MKSA_DEBYEGSL_CONST_MKSA_MINUTEGSL_CONST_MKSA_HOURGSL_CONST_MKSA_DAYGSL_CONST_MKSA_WEEKGSL_CONST_MKSA_INCHGSL_CONST_MKSA_FOOTGSL_CONST_MKSA_YARDGSL_CONST_MKSA_MILEGSL_CONST_MKSA_MILGSL_CONST_MKSA_KILOMETERS_PER_HOURGSL_CONST_MKSA_MILES_PER_HOURGSL_CONST_MKSA_NAUTICAL_MILEGSL_CONST_MKSA_FATHOMGSL_CONST_MKSA_KNOTGSL_CONST_MKSA_POINTGSL_CONST_MKSA_TEXPOINTGSL_CONST_MKSA_MICRONGSL_CONST_MKSA_HECTAREGSL_CONST_MKSA_ACREGSL_CONST_MKSA_LITERGSL_CONST_MKSA_US_GALLONGSL_CONST_MKSA_CANADIAN_GALLONGSL_CONST_MKSA_UK_GALLONGSL_CONST_MKSA_QUARTGSL_CONST_MKSA_PINTGSL_CONST_MKSA_POUND_MASSGSL_CONST_MKSA_OUNCE_MASSGSL_CONST_MKSA_TONGSL_CONST_MKSA_METRIC_TONGSL_CONST_MKSA_UK_TONGSL_CONST_MKSA_TROY_OUNCEGSL_CONST_MKSA_CARATGSL_CONST_MKSA_GRAM_FORCEGSL_CONST_MKSA_POUND_FORCEGSL_CONST_MKSA_KILOPOUND_FORCEGSL_CONST_MKSA_POUNDALGSL_CONST_MKSA_CALORIEGSL_CONST_MKSA_BTUGSL_CONST_MKSA_THERMGSL_CONST_MKSA_HORSEPOWERGSL_CONST_MKSA_BARGSL_CONST_MKSA_STD_ATMOSPHEREGSL_CONST_MKSA_TORRGSL_CONST_MKSA_METER_OF_MERCURYGSL_CONST_MKSA_INCH_OF_MERCURYGSL_CONST_MKSA_INCH_OF_WATERGSL_CONST_MKSA_PSIGSL_CONST_MKSA_POISEGSL_CONST_MKSA_STOKESGSL_CONST_MKSA_STILBGSL_CONST_MKSA_LUMENGSL_CONST_MKSA_LUXGSL_CONST_MKSA_PHOTGSL_CONST_MKSA_FOOTCANDLEGSL_CONST_MKSA_LAMBERTGSL_CONST_MKSA_FOOTLAMBERTGSL_CONST_MKSA_CURIEGSL_CONST_MKSA_ROENTGENGSL_CONST_MKSA_RADGSL_CONST_MKSA_NEWTONGSL_CONST_MKSA_DYNEGSL_CONST_MKSA_JOULEGSL_CONST_MKSA_ERGThese constants are dimensionless scaling factors.
GSL_CONST_NUM_YOTTAGSL_CONST_NUM_ZETTAGSL_CONST_NUM_EXAGSL_CONST_NUM_PETAGSL_CONST_NUM_TERAGSL_CONST_NUM_GIGAGSL_CONST_NUM_MEGAGSL_CONST_NUM_KILOGSL_CONST_NUM_MILLIGSL_CONST_NUM_MICROGSL_CONST_NUM_NANOGSL_CONST_NUM_PICOGSL_CONST_NUM_FEMTOGSL_CONST_NUM_ATTOGSL_CONST_NUM_ZEPTOGSL_CONST_NUM_YOCTOThe following program demonstrates the use of the physical constants in a calculation. In this case, the goal is to calculate the range of light-travel times from Earth to Mars.
The required data is the average distance of each planet from the Sun in astronomical units (the eccentricities and inclinations of the orbits will be neglected for the purposes of this calculation). The average radius of the orbit of Mars is 1.52 astronomical units, and for the orbit of Earth it is 1 astronomical unit (by definition). These values are combined with the MKSA values of the constants for the speed of light and the length of an astronomical unit to produce a result for the shortest and longest light-travel times in seconds. The figures are converted into minutes before being displayed.
#include <stdio.h>
#include <gsl/gsl_const_mksa.h>
int
main (void)
{
double c = GSL_CONST_MKSA_SPEED_OF_LIGHT;
double au = GSL_CONST_MKSA_ASTRONOMICAL_UNIT;
double minutes = GSL_CONST_MKSA_MINUTE;
/* distance stored in meters */
double r_earth = 1.00 * au;
double r_mars = 1.52 * au;
double t_min, t_max;
t_min = (r_mars - r_earth) / c;
t_max = (r_mars + r_earth) / c;
printf ("light travel time from Earth to Mars:\n");
printf ("minimum = %.1f minutes\n", t_min / minutes);
printf ("maximum = %.1f minutes\n", t_max / minutes);
return 0;
}
Here is the output from the program,
light travel time from Earth to Mars:
minimum = 4.3 minutes
maximum = 21.0 minutes
The authoritative sources for physical constanst are the 2002 CODATA recommended values, published in the articles below. Further information on the values of physical constants is also available from the cited articles and the NIST website.
This chapter describes functions for examining the representation of floating point numbers and controlling the floating point environment of your program. The functions described in this chapter are declared in the header file gsl_ieee_utils.h.
The IEEE Standard for Binary Floating-Point Arithmetic defines binary formats for single and double precision numbers. Each number is composed of three parts: a sign bit (s), an exponent (E) and a fraction (f). The numerical value of the combination (s,E,f) is given by the following formula,
(-1)^s (1.fffff...) 2^E
The sign bit is either zero or one. The exponent ranges from a minimum value E_min to a maximum value E_max depending on the precision. The exponent is converted to an unsigned number e, known as the biased exponent, for storage by adding a bias parameter, e = E + bias. The sequence fffff... represents the digits of the binary fraction f. The binary digits are stored in normalized form, by adjusting the exponent to give a leading digit of 1. Since the leading digit is always 1 for normalized numbers it is assumed implicitly and does not have to be stored. Numbers smaller than 2^(E_min) are be stored in denormalized form with a leading zero,
(-1)^s (0.fffff...) 2^(E_min)
This allows gradual underflow down to 2^(E_min - p) for p bits of precision. A zero is encoded with the special exponent of 2^(E_min - 1) and infinities with the exponent of 2^(E_max + 1).
The format for single precision numbers uses 32 bits divided in the following way,
seeeeeeeefffffffffffffffffffffff
s = sign bit, 1 bit
e = exponent, 8 bits (E_min=-126, E_max=127, bias=127)
f = fraction, 23 bits
The format for double precision numbers uses 64 bits divided in the following way,
seeeeeeeeeeeffffffffffffffffffffffffffffffffffffffffffffffffffff
s = sign bit, 1 bit
e = exponent, 11 bits (E_min=-1022, E_max=1023, bias=1023)
f = fraction, 52 bits
It is often useful to be able to investigate the behavior of a calculation at the bit-level and the library provides functions for printing the IEEE representations in a human-readable form.
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream stream. A pointer is used to pass the number indirectly, to avoid any undesired promotion from
floattodouble. The output takes one of the following forms,
NaN- the Not-a-Number symbol
Inf, -Inf- positive or negative infinity
1.fffff...*2^E, -1.fffff...*2^E- a normalized floating point number
0.fffff...*2^E, -0.fffff...*2^E- a denormalized floating point number
0, -0- positive or negative zero
The output can be used directly in GNU Emacs Calc mode by preceding it with
2#to indicate binary.
These functions output a formatted version of the IEEE floating-point number pointed to by x to the stream
stdout.
The following program demonstrates the use of the functions by printing the single and double precision representations of the fraction 1/3. For comparison the representation of the value promoted from single to double precision is also printed.
#include <stdio.h>
#include <gsl/gsl_ieee_utils.h>
int
main (void)
{
float f = 1.0/3.0;
double d = 1.0/3.0;
double fd = f; /* promote from float to double */
printf (" f="); gsl_ieee_printf_float(&f);
printf ("\n");
printf ("fd="); gsl_ieee_printf_double(&fd);
printf ("\n");
printf (" d="); gsl_ieee_printf_double(&d);
printf ("\n");
return 0;
}
The binary representation of 1/3 is 0.01010101... . The output below shows that the IEEE format normalizes this fraction to give a leading digit of 1,
f= 1.01010101010101010101011*2^-2
fd= 1.0101010101010101010101100000000000000000000000000000*2^-2
d= 1.0101010101010101010101010101010101010101010101010101*2^-2
The output also shows that a single-precision number is promoted to double-precision by adding zeros in the binary representation.
The IEEE standard defines several modes for controlling the behavior of floating point operations. These modes specify the important properties of computer arithmetic: the direction used for rounding (e.g. whether numbers should be rounded up, down or to the nearest number), the rounding precision and how the program should handle arithmetic exceptions, such as division by zero.
Many of these features can now be controlled via standard functions such
as fpsetround, which should be used whenever they are available.
Unfortunately in the past there has been no universal API for
controlling their behavior—each system has had its own low-level way
of accessing them. To help you write portable programs GSL allows you
to specify modes in a platform-independent way using the environment
variable GSL_IEEE_MODE. The library then takes care of all the
necessary machine-specific initializations for you when you call the
function gsl_ieee_env_setup.
This function reads the environment variable
GSL_IEEE_MODEand attempts to set up the corresponding specified IEEE modes. The environment variable should be a list of keywords, separated by commas, like this,GSL_IEEE_MODE= "keyword,keyword,..."where keyword is one of the following mode-names,
single-precisiondouble-precisionextended-precisionround-to-nearestround-downround-upround-to-zeromask-allmask-invalidmask-denormalizedmask-division-by-zeromask-overflowmask-underflowtrap-inexacttrap-commonIf
GSL_IEEE_MODEis empty or undefined then the function returns immediately and no attempt is made to change the system's IEEE mode. When the modes fromGSL_IEEE_MODEare turned on the function prints a short message showing the new settings to remind you that the results of the program will be affected.If the requested modes are not supported by the platform being used then the function calls the error handler and returns an error code of
GSL_EUNSUP.When options are specified using this method, the resulting mode is based on a default setting of the highest available precision (double precision or extended precision, depending on the platform) in round-to-nearest mode, with all exceptions enabled apart from the inexact exception. The inexact exception is generated whenever rounding occurs, so it must generally be disabled in typical scientific calculations. All other floating-point exceptions are enabled by default, including underflows and the use of denormalized numbers, for safety. They can be disabled with the individual
mask-settings or together usingmask-all.The following adjusted combination of modes is convenient for many purposes,
GSL_IEEE_MODE="double-precision,"\ "mask-underflow,"\ "mask-denormalized"This choice ignores any errors relating to small numbers (either denormalized, or underflowing to zero) but traps overflows, division by zero and invalid operations.
To demonstrate the effects of different rounding modes consider the following program which computes e, the base of natural logarithms, by summing a rapidly-decreasing series,
e = 1 + 1/2! + 1/3! + 1/4! + ...
= 2.71828182846...
#include <stdio.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_ieee_utils.h>
int
main (void)
{
double x = 1, oldsum = 0, sum = 0;
int i = 0;
gsl_ieee_env_setup (); /* read GSL_IEEE_MODE */
do
{
i++;
oldsum = sum;
sum += x;
x = x / i;
printf ("i=%2d sum=%.18f error=%g\n",
i, sum, sum - M_E);
if (i > 30)
break;
}
while (sum != oldsum);
return 0;
}
Here are the results of running the program in round-to-nearest
mode. This is the IEEE default so it isn't really necessary to specify
it here,
$ GSL_IEEE_MODE="round-to-nearest" ./a.out
i= 1 sum=1.000000000000000000 error=-1.71828
i= 2 sum=2.000000000000000000 error=-0.718282
....
i=18 sum=2.718281828459045535 error=4.44089e-16
i=19 sum=2.718281828459045535 error=4.44089e-16
After nineteen terms the sum converges to within
4 \times 10^-16 of the correct value.
If we now change the rounding mode to
round-down the final result is less accurate,
$ GSL_IEEE_MODE="round-down" ./a.out
i= 1 sum=1.000000000000000000 error=-1.71828
....
i=19 sum=2.718281828459041094 error=-3.9968e-15
The result is about
4 \times 10^-15
below the correct value, an order of magnitude worse than the result
obtained in the round-to-nearest mode.
If we change to rounding mode to round-up then the series no
longer converges (the reason is that when we add each term to the sum
the final result is always rounded up. This is guaranteed to increase the sum
by at least one tick on each iteration). To avoid this problem we would
need to use a safer converge criterion, such as while (fabs(sum -
oldsum) > epsilon), with a suitably chosen value of epsilon.
Finally we can see the effect of computing the sum using
single-precision rounding, in the default round-to-nearest
mode. In this case the program thinks it is still using double precision
numbers but the CPU rounds the result of each floating point operation
to single-precision accuracy. This simulates the effect of writing the
program using single-precision float variables instead of
double variables. The iteration stops after about half the number
of iterations and the final result is much less accurate,
$ GSL_IEEE_MODE="single-precision" ./a.out
....
i=12 sum=2.718281984329223633 error=1.5587e-07
with an error of O(10^-7), which corresponds to single precision accuracy (about 1 part in 10^7). Continuing the iterations further does not decrease the error because all the subsequent results are rounded to the same value.
The reference for the IEEE standard is,
A more pedagogical introduction to the standard can be found in the following paper,
Corrigendum: ACM Computing Surveys, Vol. 23, No. 3 (September 1991), page 413. and see also the sections by B. A. Wichmann and Charles B. Dunham in Surveyor's Forum: “What Every Computer Scientist Should Know About Floating-Point Arithmetic”. ACM Computing Surveys, Vol. 24, No. 3 (September 1992), page 319.
A detailed textbook on IEEE arithmetic and its practical use is available from SIAM Press,
This chapter describes some tips and tricks for debugging numerical programs which use GSL.
Any errors reported by the library are passed to the function
gsl_error. By running your programs under gdb and setting a
breakpoint in this function you can automatically catch any library
errors. You can add a breakpoint for every session by putting
break gsl_error
into your .gdbinit file in the directory where your program is started.
If the breakpoint catches an error then you can use a backtrace
(bt) to see the call-tree, and the arguments which possibly
caused the error. By moving up into the calling function you can
investigate the values of variables at that point. Here is an example
from the program fft/test_trap, which contains the following
line,
status = gsl_fft_complex_wavetable_alloc (0, &complex_wavetable);
The function gsl_fft_complex_wavetable_alloc takes the length of
an FFT as its first argument. When this line is executed an error will
be generated because the length of an FFT is not allowed to be zero.
To debug this problem we start gdb, using the file
.gdbinit to define a breakpoint in gsl_error,
$ gdb test_trap
GDB is free software and you are welcome to distribute copies
of it under certain conditions; type "show copying" to see
the conditions. There is absolutely no warranty for GDB;
type "show warranty" for details. GDB 4.16 (i586-debian-linux),
Copyright 1996 Free Software Foundation, Inc.
Breakpoint 1 at 0x8050b1e: file error.c, line 14.
When we run the program this breakpoint catches the error and shows the reason for it.
(gdb) run
Starting program: test_trap
Breakpoint 1, gsl_error (reason=0x8052b0d
"length n must be positive integer",
file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
at error.c:14
14 if (gsl_error_handler)
The first argument of gsl_error is always a string describing the
error. Now we can look at the backtrace to see what caused the problem,
(gdb) bt
#0 gsl_error (reason=0x8052b0d
"length n must be positive integer",
file=0x8052b04 "c_init.c", line=108, gsl_errno=1)
at error.c:14
#1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
wavetable=0xbffff778) at c_init.c:108
#2 0x8048a00 in main (argc=1, argv=0xbffff9bc)
at test_trap.c:94
#3 0x80488be in ___crt_dummy__ ()
We can see that the error was generated in the function
gsl_fft_complex_wavetable_alloc when it was called with an
argument of n=0. The original call came from line 94 in the
file test_trap.c.
By moving up to the level of the original call we can find the line that caused the error,
(gdb) up
#1 0x8049376 in gsl_fft_complex_wavetable_alloc (n=0,
wavetable=0xbffff778) at c_init.c:108
108 GSL_ERROR ("length n must be positive integer", GSL_EDOM);
(gdb) up
#2 0x8048a00 in main (argc=1, argv=0xbffff9bc)
at test_trap.c:94
94 status = gsl_fft_complex_wavetable_alloc (0,
&complex_wavetable);
Thus we have found the line that caused the problem. From this point we
could also print out the values of other variables such as
complex_wavetable.
The contents of floating point registers can be examined using the
command info float (on supported platforms).
(gdb) info float
st0: 0xc4018b895aa17a945000 Valid Normal -7.838871e+308
st1: 0x3ff9ea3f50e4d7275000 Valid Normal 0.0285946
st2: 0x3fe790c64ce27dad4800 Valid Normal 6.7415931e-08
st3: 0x3ffaa3ef0df6607d7800 Spec Normal 0.0400229
st4: 0x3c028000000000000000 Valid Normal 4.4501477e-308
st5: 0x3ffef5412c22219d9000 Zero Normal 0.9580257
st6: 0x3fff8000000000000000 Valid Normal 1
st7: 0xc4028b65a1f6d243c800 Valid Normal -1.566206e+309
fctrl: 0x0272 53 bit; NEAR; mask DENOR UNDER LOS;
fstat: 0xb9ba flags 0001; top 7; excep DENOR OVERF UNDER LOS
ftag: 0x3fff
fip: 0x08048b5c
fcs: 0x051a0023
fopoff: 0x08086820
fopsel: 0x002b
Individual registers can be examined using the variables $reg, where reg is the register name.
(gdb) p $st1
$1 = 0.02859464454261210347719
It is possible to stop the program whenever a SIGFPE floating
point exception occurs. This can be useful for finding the cause of an
unexpected infinity or NaN. The current handler settings can be
shown with the command info signal SIGFPE.
(gdb) info signal SIGFPE
Signal Stop Print Pass to program Description
SIGFPE Yes Yes Yes Arithmetic exception
Unless the program uses a signal handler the default setting should be
changed so that SIGFPE is not passed to the program, as this would cause
it to exit. The command handle SIGFPE stop nopass prevents this.
(gdb) handle SIGFPE stop nopass
Signal Stop Print Pass to program Description
SIGFPE Yes Yes No Arithmetic exception
Depending on the platform it may be necessary to instruct the kernel to
generate signals for floating point exceptions. For programs using GSL
this can be achieved using the GSL_IEEE_MODE environment variable
in conjunction with the function gsl_ieee_env_setup() as described
in see IEEE floating-point arithmetic.
(gdb) set env GSL_IEEE_MODE=double-precision
Writing reliable numerical programs in C requires great care. The following GCC warning options are recommended when compiling numerical programs:
gcc -ansi -pedantic -Werror -Wall -W
-Wmissing-prototypes -Wstrict-prototypes
-Wtraditional -Wconversion -Wshadow
-Wpointer-arith -Wcast-qual -Wcast-align
-Wwrite-strings -Wnested-externs
-fshort-enums -fno-common -Dinline= -g -O4
For details of each option consult the manual Using and Porting GCC. The following table gives a brief explanation of what types of errors these options catch.
-ansi -pedantic-Werror-Wall-Wall, but it is not enough on its own.
-O2-Wall rely on the optimizer to analyze the code. If there is no
optimization then these warnings aren't generated.
-W-Wall, such as
missing return values and comparisons between signed and unsigned
integers.
-Wmissing-prototypes -Wstrict-prototypes-Wtraditional-Wconversionunsigned int x = -1. If you need
to perform such a conversion you can use an explicit cast.
-Wshadow-Wpointer-arith -Wcast-qual -Wcast-alignvoid, if you remove a const
cast from a pointer, or if you cast a pointer to a type which has a
different size, causing an invalid alignment.
-Wwrite-stringsconst qualifier so that it
will be a compile-time error to attempt to overwrite them.
-fshort-enumsenum as short as possible. Normally
this makes an enum different from an int. Consequently any
attempts to assign a pointer-to-int to a pointer-to-enum will generate a
cast-alignment warning.
-fno-commonextern declaration.
-Wnested-externsextern declaration is encountered within a
function.
-Dinline=inline keyword is not part of ANSI C. Thus if you want to use
-ansi with a program which uses inline functions you can use this
preprocessor definition to remove the inline keywords.
-ggdb. The only effect of debugging symbols
is to increase the size of the file, and you can use the strip
command to remove them later if necessary.
The following books are essential reading for anyone writing and debugging numerical programs with gcc and gdb.
For a tutorial introduction to the GNU C Compiler and related programs, see
(See the AUTHORS file in the distribution for up-to-date information.)
Thanks to Nigel Lowry for help in proofreading the manual.
For applications using autoconf the standard macro
AC_CHECK_LIB can be used to link with GSL automatically
from a configure script. The library itself depends on the
presence of a cblas and math library as well, so these must also be
located before linking with the main libgsl file. The following
commands should be placed in the configure.ac file to perform
these tests,
AC_CHECK_LIB(m,main)
AC_CHECK_LIB(gslcblas,main)
AC_CHECK_LIB(gsl,main)
It is important to check for libm and libgslcblas before
libgsl, otherwise the tests will fail. Assuming the libraries
are found the output during the configure stage looks like this,
checking for main in -lm... yes
checking for main in -lgslcblas... yes
checking for main in -lgsl... yes
If the library is found then the tests will define the macros
HAVE_LIBGSL, HAVE_LIBGSLCBLAS, HAVE_LIBM and add
the options -lgsl -lgslcblas -lm to the variable LIBS.
The tests above will find any version of the library. They are suitable for general use, where the versions of the functions are not important. An alternative macro is available in the file gsl.m4 to test for a specific version of the library. To use this macro simply add the following line to your configure.in file instead of the tests above:
AM_PATH_GSL(GSL_VERSION,
[action-if-found],
[action-if-not-found])
The argument GSL_VERSION should be the two or three digit
major.minor or major.minor.micro version number of the release
you require. A suitable choice for action-if-not-found is,
AC_MSG_ERROR(could not find required version of GSL)
Then you can add the variables GSL_LIBS and GSL_CFLAGS to
your Makefile.am files to obtain the correct compiler flags.
GSL_LIBS is equal to the output of the gsl-config --libs
command and GSL_CFLAGS is equal to gsl-config --cflags
command. For example,
libfoo_la_LDFLAGS = -lfoo $(GSL_LIBS) -lgslcblas
Note that the macro AM_PATH_GSL needs to use the C compiler so it
should appear in the configure.in file before the macro
AC_LANG_CPLUSPLUS for programs that use C++.
To test for inline the following test should be placed in your
configure.in file,
AC_C_INLINE
if test "$ac_cv_c_inline" != no ; then
AC_DEFINE(HAVE_INLINE,1)
AC_SUBST(HAVE_INLINE)
fi
and the macro will then be defined in the compilation flags or by including the file config.h before any library headers.
The following autoconf test will check for extern inline,
dnl Check for "extern inline", using a modified version
dnl of the test for AC_C_INLINE from acspecific.mt
dnl
AC_CACHE_CHECK([for extern inline], ac_cv_c_extern_inline,
[ac_cv_c_extern_inline=no
AC_TRY_COMPILE([extern $ac_cv_c_inline double foo(double x);
extern $ac_cv_c_inline double foo(double x) { return x+1.0; };
double foo (double x) { return x + 1.0; };],
[ foo(1.0) ],
[ac_cv_c_extern_inline="yes"])
])
if test "$ac_cv_c_extern_inline" != no ; then
AC_DEFINE(HAVE_INLINE,1)
AC_SUBST(HAVE_INLINE)
fi
The substitution of portability functions can be made automatically if
you use autoconf. For example, to test whether the BSD function
hypot is available you can include the following line in the
configure file configure.in for your application,
AC_CHECK_FUNCS(hypot)
and place the following macro definitions in the file config.h.in,
/* 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 <config.h> to substitute gsl_hypot for each
occurrence of hypot when hypot is not available.
The prototypes for the low-level cblas functions are declared in
the file gsl_cblas.h. For the definition of the functions
consult the documentation available from Netlib (see BLAS References and Further Reading).
The following program computes the product of two matrices using the Level-3 blas function sgemm,
[ 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 but could be stored in column
major order if the first argument of the call to cblas_sgemm was
changed to CblasColMajor.
#include <stdio.h>
#include <gsl/gsl_cblas.h>
int
main (void)
{
int lda = 3;
float A[] = { 0.11, 0.12, 0.13,
0.21, 0.22, 0.23 };
int ldb = 2;
float B[] = { 1011, 1012,
1021, 1022,
1031, 1032 };
int ldc = 2;
float C[] = { 0.00, 0.00,
0.00, 0.00 };
/* Compute C = A B */
cblas_sgemm (CblasRowMajor,
CblasNoTrans, CblasNoTrans, 2, 2, 3,
1.0, A, lda, B, ldb, 0.0, C, ldc);
printf ("[ %g, %g\n", C[0], C[1]);
printf (" %g, %g ]\n", C[2], C[3]);
return 0;
}
To compile the program use the following command line,
$ gcc -Wall demo.c -lgslcblas
There is no need to link with the main library -lgsl in this
case as the cblas library is an independent unit. Here is the output
from the program,
$ ./a.out
[ 367.76, 368.12
674.06, 674.72 ]
The following article was written by Richard Stallman, founder of the GNU Project.
The biggest deficiency in the free software community today is not in the software—it is the lack of good free documentation that we can include with the free software. Many of our most important programs do not come with free reference manuals and free introductory texts. Documentation is an essential part of any software package; when an important free software package does not come with a free manual and a free tutorial, that is a major gap. We have many such gaps today.
Consider Perl, for instance. The tutorial manuals that people normally use are non-free. How did this come about? Because the authors of those manuals published them with restrictive terms—no copying, no modification, source files not available—which exclude them from the free software world.
That wasn't the first time this sort of thing happened, and it was far from the last. Many times we have heard a GNU user eagerly describe a manual that he is writing, his intended contribution to the community, only to learn that he had ruined everything by signing a publication contract to make it non-free.
Free documentation, like free software, is a matter of freedom, not price. The problem with the non-free manual is not that publishers charge a price for printed copies—that in itself is fine. (The Free Software Foundation sells printed copies of manuals, too.) The problem is the restrictions on the use of the manual. Free manuals are available in source code form, and give you permission to copy and modify. Non-free manuals do not allow this.
The criteria of freedom for a free manual are roughly the same as for free software. Redistribution (including the normal kinds of commercial redistribution) must be permitted, so that the manual can accompany every copy of the program, both on-line and on paper.
Permission for modification of the technical content is crucial too. When people modify the software, adding or changing features, if they are conscientious they will change the manual too—so they can provide accurate and clear documentation for the modified program. A manual that leaves you no choice but to write a new manual to document a changed version of the program is not really available to our community.
Some kinds of limits on the way modification is handled are acceptable. For example, requirements to preserve the original author's copyright notice, the distribution terms, or the list of authors, are ok. It is also no problem to require modified versions to include notice that they were modified. Even entire sections that may not be deleted or changed are acceptable, as long as they deal with nontechnical topics (like this one). These kinds of restrictions are acceptable because they don't obstruct the community's normal use of the manual.
However, it must be possible to modify all the technical content of the manual, and then distribute the result in all the usual media, through all the usual channels. Otherwise, the restrictions obstruct the use of the manual, it is not free, and we need another manual to replace it.
Please spread the word about this issue. Our community continues to lose manuals to proprietary publishing. If we spread the word that free software needs free reference manuals and free tutorials, perhaps the next person who wants to contribute by writing documentation will realize, before it is too late, that only free manuals contribute to the free software community.
If you are writing documentation, please insist on publishing it under the GNU Free Documentation License or another free documentation license. Remember that this decision requires your approval—you don't have to let the publisher decide. Some commercial publishers will use a free license if you insist, but they will not propose the option; it is up to you to raise the issue and say firmly that this is what you want. If the publisher you are dealing with refuses, please try other publishers. If you're not sure whether a proposed license is free, write to licensing@gnu.org.
You can encourage commercial publishers to sell more free, copylefted manuals and tutorials by buying them, and particularly by buying copies from the publishers that paid for their writing or for major improvements. Meanwhile, try to avoid buying non-free documentation at all. Check the distribution terms of a manual before you buy it, and insist that whoever seeks your business must respect your freedom. Check the history of the book, and try reward the publishers that have paid or pay the authors to work on it.
The Free Software Foundation maintains a list of free documentation published by other publishers:
Copyright © 1989, 1991 Free Software Foundation, Inc.
59 Temple Place - Suite 330, Boston, MA 02111-1307, USA
Everyone is permitted to copy and distribute verbatim copies of this
license document, but changing it is not allowed.
The licenses for most software are designed to take away your freedom to share and change it. By contrast, the GNU General Public License is intended to guarantee your freedom to share and change free software—to make sure the software is free for all its users. This General Public License applies to most of the Free Software Foundation's software and to any other program whose authors commit to using it. (Some other Free Software Foundation software is covered by the GNU Library General Public License instead.) You can apply it to your programs, too.
When we speak of free software, we are referring to freedom, not price. Our General Public Licenses are designed to make sure that you have the freedom to distribute copies of free software (and charge for this service if you wish), that you receive source code or can get it if you want it, that you can change the software or use pieces of it in new free programs; and that you know you can do these things.
To protect your rights, we need to make restrictions that forbid anyone to deny you these rights or to ask you to surrender the rights. These restrictions translate to certain responsibilities for you if you distribute copies of the software, or if you modify it.
For example, if you distribute copies of such a program, whether gratis or for a fee, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must show them these terms so they know their rights.
We protect your rights with two steps: (1) copyright the software, and (2) offer you this license which gives you legal permission to copy, distribute and/or modify the software.
Also, for each author's protection and ours, we want to make certain that everyone understands that there is no warranty for this free software. If the software is modified by someone else and passed on, we want its recipients to know that what they have is not the original, so that any problems introduced by others will not reflect on the original authors' reputations.
Finally, any free program is threatened constantly by software patents. We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses, in effect making the program proprietary. To prevent this, we have made it clear that any patent must be licensed for everyone's free use or not licensed at all.
The precise terms and conditions for copying, distribution and modification follow.
Activities other than copying, distribution and modification are not covered by this License; they are outside its scope. The act of running the Program is not restricted, and the output from the Program is covered only if its contents constitute a work based on the Program (independent of having been made by running the Program). Whether that is true depends on what the Program does.
You may charge a fee for the physical act of transferring a copy, and you may at your option offer warranty protection in exchange for a fee.
These requirements apply to the modified work as a whole. If identifiable sections of that work are not derived from the Program, and can be reasonably considered independent and separate works in themselves, then this License, and its terms, do not apply to those sections when you distribute them as separate works. But when you distribute the same sections as part of a whole which is a work based on the Program, the distribution of the whole must be on the terms of this License, whose permissions for other licensees extend to the entire whole, and thus to each and every part regardless of who wrote it.
Thus, it is not the intent of this section to claim rights or contest your rights to work written entirely by you; rather, the intent is to exercise the right to control the distribution of derivative or collective works based on the Program.
In addition, mere aggregation of another work not based on the Program with the Program (or with a work based on the Program) on a volume of a storage or distribution medium does not bring the other work under the scope of this License.
The source code for a work means the preferred form of the work for making modifications to it. For an executable work, complete source code means all the source code for all modules it contains, plus any associated interface definition files, plus the scripts used to control compilation and installation of the executable. However, as a special exception, the source code distributed need not include anything that is normally distributed (in either source or binary form) with the major components (compiler, kernel, and so on) of the operating system on which the executable runs, unless that component itself accompanies the executable.
If distribution of executable or object code is made by offering access to copy from a designated place, then offering equivalent access to copy the source code from the same place counts as distribution of the source code, even though third parties are not compelled to copy the source along with the object code.
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It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims; this section has the sole purpose of protecting the integrity of the free software distribution system, which is implemented by public license practices. Many people have made generous contributions to the wide range of software distributed through that system in reliance on consistent application of that system; it is up to the author/donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice.
This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License.
Each version is given a distinguishing version number. If the Program specifies a version number of this License which applies to it and “any later version”, you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation. If the Program does not specify a version number of this License, you may choose any version ever published by the Free Software Foundation.
If you develop a new program, and you want it to be of the greatest possible use to the public, the best way to achieve this is to make it free software which everyone can redistribute and change under these terms.
To do so, attach the following notices to the program. It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty; and each file should have at least the “copyright” line and a pointer to where the full notice is found.
one line to give the program's name and a brief idea
of what it does. Copyright (C) yyyy name of author This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.
Also add information on how to contact you by electronic and paper mail.
If the program is interactive, make it output a short notice like this when it starts in an interactive mode:
Gnomovision version 69, Copyright (C) 19yy name of author
Gnomovision comes with ABSOLUTELY NO WARRANTY; for details
type `show w'. This is free software, and you are welcome
to redistribute it under certain conditions; type `show c'
for details.
The hypothetical commands `show w' and `show c' should show the appropriate parts of the General Public License. Of course, the commands you use may be called something other than `show w' and `show c'; they could even be mouse-clicks or menu items—whatever suits your program.
You should also get your employer (if you work as a programmer) or your school, if any, to sign a “copyright disclaimer” for the program, if necessary. Here is a sample; alter the names:
Yoyodyne, Inc., hereby disclaims all copyright interest in
the program `Gnomovision' (which makes passes at compilers)
written by James Hacker.
signature of Ty Coon, 1 April 1989
Ty Coon, President of Vice
This General Public License does not permit incorporating your program into proprietary programs. If your program is a subroutine library, you may consider it more useful to permit linking proprietary applications with the library. If this is what you want to do, use the GNU Library General Public License instead of this License.
Copyright © 2000,2001,2002 Free Software Foundation, Inc.
51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
Everyone is permitted to copy and distribute verbatim copies of this license
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The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.
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cblas_caxpy: Level 1 CBLAS Functionscblas_ccopy: Level 1 CBLAS Functionscblas_cdotc_sub: Level 1 CBLAS Functionscblas_cdotu_sub: Level 1 CBLAS Functionscblas_cgbmv: Level 2 CBLAS Functionscblas_cgemm: Level 3 CBLAS Functionscblas_cgemv: Level 2 CBLAS Functionscblas_cgerc: Level 2 CBLAS Functionscblas_cgeru: Level 2 CBLAS Functionscblas_chbmv: Level 2 CBLAS Functionscblas_chemm: Level 3 CBLAS Functionscblas_chemv: Level 2 CBLAS Functionscblas_cher: Level 2 CBLAS Functionscblas_cher2: Level 2 CBLAS Functionscblas_cher2k: Level 3 CBLAS Functionscblas_cherk: Level 3 CBLAS Functionscblas_chpmv: Level 2 CBLAS Functionscblas_chpr: Level 2 CBLAS Functionscblas_chpr2: Level 2 CBLAS Functionscblas_cscal: Level 1 CBLAS Functionscblas_csscal: Level 1 CBLAS Functionscblas_cswap: Level 1 CBLAS Functionscblas_csymm: Level 3 CBLAS Functionscblas_csyr2k: Level 3 CBLAS Functionscblas_csyrk: Level 3 CBLAS Functionscblas_ctbmv: Level 2 CBLAS Functionscblas_ctbsv: Level 2 CBLAS Functionscblas_ctpmv: Level 2 CBLAS Functionscblas_ctpsv: Level 2 CBLAS Functionscblas_ctrmm: Level 3 CBLAS Functionscblas_ctrmv: Level 2 CBLAS Functionscblas_ctrsm: Level 3 CBLAS Functionscblas_ctrsv: Level 2 CBLAS Functionscblas_dasum: Level 1 CBLAS Functionscblas_daxpy: Level 1 CBLAS Functionscblas_dcopy: Level 1 CBLAS Functionscblas_ddot: Level 1 CBLAS Functionscblas_dgbmv: Level 2 CBLAS Functionscblas_dgemm: Level 3 CBLAS Functionscblas_dgemv: Level 2 CBLAS Functionscblas_dger: Level 2 CBLAS Functionscblas_dnrm2: Level 1 CBLAS Functionscblas_drot: Level 1 CBLAS Functionscblas_drotg: Level 1 CBLAS Functionscblas_drotm: Level 1 CBLAS Functionscblas_drotmg: Level 1 CBLAS Functionscblas_dsbmv: Level 2 CBLAS Functionscblas_dscal: Level 1 CBLAS Functionscblas_dsdot: Level 1 CBLAS Functionscblas_dspmv: Level 2 CBLAS Functionscblas_dspr: Level 2 CBLAS Functionscblas_dspr2: Level 2 CBLAS Functionscblas_dswap: Level 1 CBLAS Functionscblas_dsymm: Level 3 CBLAS Functionscblas_dsymv: Level 2 CBLAS Functionscblas_dsyr: Level 2 CBLAS Functionscblas_dsyr2: Level 2 CBLAS Functionscblas_dsyr2k: Level 3 CBLAS Functionscblas_dsyrk: Level 3 CBLAS Functionscblas_dtbmv: Level 2 CBLAS Functionscblas_dtbsv: Level 2 CBLAS Functionscblas_dtpmv: Level 2 CBLAS Functionscblas_dtpsv: Level 2 CBLAS Functionscblas_dtrmm: Level 3 CBLAS Functionscblas_dtrmv: Level 2 CBLAS Functionscblas_dtrsm: Level 3 CBLAS Functionscblas_dtrsv: Level 2 CBLAS Functionscblas_dzasum: Level 1 CBLAS Functionscblas_dznrm2: Level 1 CBLAS Functionscblas_icamax: Level 1 CBLAS Functionscblas_idamax: Level 1 CBLAS Functionscblas_isamax: Level 1 CBLAS Functionscblas_izamax: Level 1 CBLAS Functionscblas_sasum: Level 1 CBLAS Functionscblas_saxpy: Level 1 CBLAS Functionscblas_scasum: Level 1 CBLAS Functionscblas_scnrm2: Level 1 CBLAS Functionscblas_scopy: Level 1 CBLAS Functionscblas_sdot: Level 1 CBLAS Functionscblas_sdsdot: Level 1 CBLAS Functionscblas_sgbmv: Level 2 CBLAS Functionscblas_sgemm: Level 3 CBLAS Functionscblas_sgemv: Level 2 CBLAS Functionscblas_sger: Level 2 CBLAS Functionscblas_snrm2: Level 1 CBLAS Functionscblas_srot: Level 1 CBLAS Functionscblas_srotg: Level 1 CBLAS Functionscblas_srotm: Level 1 CBLAS Functionscblas_srotmg: Level 1 CBLAS Functionscblas_ssbmv: Level 2 CBLAS Functionscblas_sscal: Level 1 CBLAS Functionscblas_sspmv: Level 2 CBLAS Functionscblas_sspr: Level 2 CBLAS Functionscblas_sspr2: Level 2 CBLAS Functionscblas_sswap: Level 1 CBLAS Functionscblas_ssymm: Level 3 CBLAS Functionscblas_ssymv: Level 2 CBLAS Functionscblas_ssyr: Level 2 CBLAS Functionscblas_ssyr2: Level 2 CBLAS Functionscblas_ssyr2k: Level 3 CBLAS Functionscblas_ssyrk: Level 3 CBLAS Functionscblas_stbmv: Level 2 CBLAS Functionscblas_stbsv: Level 2 CBLAS Functionscblas_stpmv: Level 2 CBLAS Functionscblas_stpsv: Level 2 CBLAS Functionscblas_strmm: Level 3 CBLAS Functionscblas_strmv: Level 2 CBLAS Functionscblas_strsm: Level 3 CBLAS Functionscblas_strsv: Level 2 CBLAS Functionscblas_xerbla: Level 3 CBLAS Functionscblas_zaxpy: Level 1 CBLAS Functionscblas_zcopy: Level 1 CBLAS Functionscblas_zdotc_sub: Level 1 CBLAS Functionscblas_zdotu_sub: Level 1 CBLAS Functionscblas_zdscal: Level 1 CBLAS Functionscblas_zgbmv: Level 2 CBLAS Functionscblas_zgemm: Level 3 CBLAS Functionscblas_zgemv: Level 2 CBLAS Functionscblas_zgerc: Level 2 CBLAS Functionscblas_zgeru: Level 2 CBLAS Functionscblas_zhbmv: Level 2 CBLAS Functionscblas_zhemm: Level 3 CBLAS Functionscblas_zhemv: Level 2 CBLAS Functionscblas_zher: Level 2 CBLAS Functionscblas_zher2: Level 2 CBLAS Functionscblas_zher2k: Level 3 CBLAS Functionscblas_zherk: Level 3 CBLAS Functionscblas_zhpmv: Level 2 CBLAS Functionscblas_zhpr: Level 2 CBLAS Functionscblas_zhpr2: Level 2 CBLAS Functionscblas_zscal: Level 1 CBLAS Functionscblas_zswap: Level 1 CBLAS Functionscblas_zsymm: Level 3 CBLAS Functionscblas_zsyr2k: Level 3 CBLAS Functionscblas_zsyrk: Level 3 CBLAS Functionscblas_ztbmv: Level 2 CBLAS Functionscblas_ztbsv: Level 2 CBLAS Functionscblas_ztpmv: Level 2 CBLAS Functionscblas_ztpsv: Level 2 CBLAS Functionscblas_ztrmm: Level 3 CBLAS Functionscblas_ztrmv: Level 2 CBLAS Functionscblas_ztrsm: Level 3 CBLAS Functionscblas_ztrsv: Level 2 CBLAS Functionsgsl_acosh: Elementary Functionsgsl_asinh: Elementary Functionsgsl_atanh: Elementary Functionsgsl_blas_caxpy: Level 1 GSL BLAS Interfacegsl_blas_ccopy: Level 1 GSL BLAS Interfacegsl_blas_cdotc: Level 1 GSL BLAS Interfacegsl_blas_cdotu: Level 1 GSL BLAS Interfacegsl_blas_cgemm: Level 3 GSL BLAS Interfacegsl_blas_cgemv: Level 2 GSL BLAS Interfacegsl_blas_cgerc: Level 2 GSL BLAS Interfacegsl_blas_cgeru: Level 2 GSL BLAS Interfacegsl_blas_chemm: Level 3 GSL BLAS Interfacegsl_blas_chemv: Level 2 GSL BLAS Interfacegsl_blas_cher: Level 2 GSL BLAS Interfacegsl_blas_cher2: Level 2 GSL BLAS Interfacegsl_blas_cher2k: Level 3 GSL BLAS Interfacegsl_blas_cherk: Level 3 GSL BLAS Interfacegsl_blas_cscal: Level 1 GSL BLAS Interfacegsl_blas_csscal: Level 1 GSL BLAS Interfacegsl_blas_cswap: Level 1 GSL BLAS Interfacegsl_blas_csymm: Level 3 GSL BLAS Interfacegsl_blas_csyr2k: Level 3 GSL BLAS Interfacegsl_blas_csyrk: Level 3 GSL BLAS Interfacegsl_blas_ctrmm: Level 3 GSL BLAS Interfacegsl_blas_ctrmv: Level 2 GSL BLAS Interfacegsl_blas_ctrsm: Level 3 GSL BLAS Interfacegsl_blas_ctrsv: Level 2 GSL BLAS Interfacegsl_blas_dasum: Level 1 GSL BLAS Interfacegsl_blas_daxpy: Level 1 GSL BLAS Interfacegsl_blas_dcopy: Level 1 GSL BLAS Interfacegsl_blas_ddot: Level 1 GSL BLAS Interfacegsl_blas_dgemm: Level 3 GSL BLAS Interfacegsl_blas_dgemv: Level 2 GSL BLAS Interfacegsl_blas_dger: Level 2 GSL BLAS Interfacegsl_blas_dnrm2: Level 1 GSL BLAS Interfacegsl_blas_drot: Level 1 GSL BLAS Interfacegsl_blas_drotg: Level 1 GSL BLAS Interfacegsl_blas_drotm: Level 1 GSL BLAS Interfacegsl_blas_drotmg: Level 1 GSL BLAS Interfacegsl_blas_dscal: Level 1 GSL BLAS Interfacegsl_blas_dsdot: Level 1 GSL BLAS Interfacegsl_blas_dswap: Level 1 GSL BLAS Interfacegsl_blas_dsymm: Level 3 GSL BLAS Interfacegsl_blas_dsymv: Level 2 GSL BLAS Interfacegsl_blas_dsyr: Level 2 GSL BLAS Interfacegsl_blas_dsyr2: Level 2 GSL BLAS Interfacegsl_blas_dsyr2k: Level 3 GSL BLAS Interfacegsl_blas_dsyrk: Level 3 GSL BLAS Interfacegsl_blas_dtrmm: Level 3 GSL BLAS Interfacegsl_blas_dtrmv: Level 2 GSL BLAS Interfacegsl_blas_dtrsm: Level 3 GSL BLAS Interfacegsl_blas_dtrsv: Level 2 GSL BLAS Interfacegsl_blas_dzasum: Level 1 GSL BLAS Interfacegsl_blas_dznrm2: Level 1 GSL BLAS Interfacegsl_blas_icamax: Level 1 GSL BLAS Interfacegsl_blas_idamax: Level 1 GSL BLAS Interfacegsl_blas_isamax: Level 1 GSL BLAS Interfacegsl_blas_izamax: Level 1 GSL BLAS Interfacegsl_blas_sasum: Level 1 GSL BLAS Interfacegsl_blas_saxpy: Level 1 GSL BLAS Interfacegsl_blas_scasum: Level 1 GSL BLAS Interfacegsl_blas_scnrm2: Level 1 GSL BLAS Interfacegsl_blas_scopy: Level 1 GSL BLAS Interfacegsl_blas_sdot: Level 1 GSL BLAS Interfacegsl_blas_sdsdot: Level 1 GSL BLAS Interfacegsl_blas_sgemm: Level 3 GSL BLAS Interfacegsl_blas_sgemv: Level 2 GSL BLAS Interfacegsl_blas_sger: Level 2 GSL BLAS Interfacegsl_blas_snrm2: Level 1 GSL BLAS Interfacegsl_blas_srot: Level 1 GSL BLAS Interfacegsl_blas_srotg: Level 1 GSL BLAS Interfacegsl_blas_srotm: Level 1 GSL BLAS Interfacegsl_blas_srotmg: Level 1 GSL BLAS Interfacegsl_blas_sscal: Level 1 GSL BLAS Interfacegsl_blas_sswap: Level 1 GSL BLAS Interfacegsl_blas_ssymm: Level 3 GSL BLAS Interfacegsl_blas_ssymv: Level 2 GSL BLAS Interfacegsl_blas_ssyr: Level 2 GSL BLAS Interfacegsl_blas_ssyr2: Level 2 GSL BLAS Interfacegsl_blas_ssyr2k: Level 3 GSL BLAS Interfacegsl_blas_ssyrk: Level 3 GSL BLAS Interfacegsl_blas_strmm: Level 3 GSL BLAS Interfacegsl_blas_strmv: Level 2 GSL BLAS Interfacegsl_blas_strsm: Level 3 GSL BLAS Interfacegsl_blas_strsv: Level 2 GSL BLAS Interfacegsl_blas_zaxpy: Level 1 GSL BLAS Interfacegsl_blas_zcopy: Level 1 GSL BLAS Interfacegsl_blas_zdotc: Level 1 GSL BLAS Interfacegsl_blas_zdotu: Level 1 GSL BLAS Interfacegsl_blas_zdscal: Level 1 GSL BLAS Interfacegsl_blas_zgemm: Level 3 GSL BLAS Interfacegsl_blas_zgemv: Level 2 GSL BLAS Interfacegsl_blas_zgerc: Level 2 GSL BLAS Interfacegsl_blas_zgeru: Level 2 GSL BLAS Interfacegsl_blas_zhemm: Level 3 GSL BLAS Interfacegsl_blas_zhemv: Level 2 GSL BLAS Interfacegsl_blas_zher: Level 2 GSL BLAS Interfacegsl_blas_zher2: Level 2 GSL BLAS Interfacegsl_blas_zher2k: Level 3 GSL BLAS Interfacegsl_blas_zherk: Level 3 GSL BLAS Interfacegsl_blas_zscal: Level 1 GSL BLAS Interfacegsl_blas_zswap: Level 1 GSL BLAS Interfacegsl_blas_zsymm: Level 3 GSL BLAS Interfacegsl_blas_zsyr2k: Level 3 GSL BLAS Interfacegsl_blas_zsyrk: Level 3 GSL BLAS Interfacegsl_blas_ztrmm: Level 3 GSL BLAS Interfacegsl_blas_ztrmv: Level 2 GSL BLAS Interfacegsl_blas_ztrsm: Level 3 GSL BLAS Interfacegsl_blas_ztrsv: Level 2 GSL BLAS Interfacegsl_block_alloc: Block allocationgsl_block_calloc: Block allocationgsl_block_fprintf: Reading and writing blocksgsl_block_fread: Reading and writing blocksgsl_block_free: Block allocationgsl_block_fscanf: Reading and writing blocksgsl_block_fwrite: Reading and writing blocksgsl_cdf_beta_P: The Beta Distributiongsl_cdf_beta_Pinv: The Beta Distributiongsl_cdf_beta_Q: The Beta Distributiongsl_cdf_beta_Qinv: The Beta Distributiongsl_cdf_binomial_P: The Binomial Distributiongsl_cdf_binomial_Q: The Binomial Distributiongsl_cdf_cauchy_P: The Cauchy Distributiongsl_cdf_cauchy_Pinv: The Cauchy Distributiongsl_cdf_cauchy_Q: The Cauchy Distributiongsl_cdf_cauchy_Qinv: The Cauchy Distributiongsl_cdf_chisq_P: The Chi-squared Distributiongsl_cdf_chisq_Pinv: The Chi-squared Distributiongsl_cdf_chisq_Q: The Chi-squared Distributiongsl_cdf_chisq_Qinv: The Chi-squared Distributiongsl_cdf_exponential_P: The Exponential Distributiongsl_cdf_exponential_Pinv: The Exponential Distributiongsl_cdf_exponential_Q: The Exponential Distributiongsl_cdf_exponential_Qinv: The Exponential Distributiongsl_cdf_exppow_P: The Exponential Power Distributiongsl_cdf_exppow_Q: The Exponential Power Distributiongsl_cdf_fdist_P: The F-distributiongsl_cdf_fdist_Pinv: The F-distributiongsl_cdf_fdist_Q: The F-distributiongsl_cdf_fdist_Qinv: The F-distributiongsl_cdf_flat_P: The Flat (Uniform) Distributiongsl_cdf_flat_Pinv: The Flat (Uniform) Distributiongsl_cdf_flat_Q: The Flat (Uniform) Distributiongsl_cdf_flat_Qinv: The Flat (Uniform) Distributiongsl_cdf_gamma_P: The Gamma Distributiongsl_cdf_gamma_Pinv: The Gamma Distributiongsl_cdf_gamma_Q: The Gamma Distributiongsl_cdf_gamma_Qinv: The Gamma Distributiongsl_cdf_gaussian_P: The Gaussian Distributiongsl_cdf_gaussian_Pinv: The Gaussian Distributiongsl_cdf_gaussian_Q: The Gaussian Distributiongsl_cdf_gaussian_Qinv: The Gaussian Distributiongsl_cdf_geometric_P: The Geometric Distributiongsl_cdf_geometric_Q: The Geometric Distributiongsl_cdf_gumbel1_P: The Type-1 Gumbel Distributiongsl_cdf_gumbel1_Pinv: The Type-1 Gumbel Distributiongsl_cdf_gumbel1_Q: The Type-1 Gumbel Distributiongsl_cdf_gumbel1_Qinv: The Type-1 Gumbel Distributiongsl_cdf_gumbel2_P: The Type-2 Gumbel Distributiongsl_cdf_gumbel2_Pinv: The Type-2 Gumbel Distributiongsl_cdf_gumbel2_Q: The Type-2 Gumbel Distributiongsl_cdf_gumbel2_Qinv: The Type-2 Gumbel Distributiongsl_cdf_hypergeometric_P: The Hypergeometric Distributiongsl_cdf_hypergeometric_Q: The Hypergeometric Distributiongsl_cdf_laplace_P: The Laplace Distributiongsl_cdf_laplace_Pinv: The Laplace Distributiongsl_cdf_laplace_Q: The Laplace Distributiongsl_cdf_laplace_Qinv: The Laplace Distributiongsl_cdf_logistic_P: The Logistic Distributiongsl_cdf_logistic_Pinv: The Logistic Distributiongsl_cdf_logistic_Q: The Logistic Distributiongsl_cdf_logistic_Qinv: The Logistic Distributiongsl_cdf_lognormal_P: The Lognormal Distributiongsl_cdf_lognormal_Pinv: The Lognormal Distributiongsl_cdf_lognormal_Q: The Lognormal Distributiongsl_cdf_lognormal_Qinv: The Lognormal Distributiongsl_cdf_negative_binomial_P: The Negative Binomial Distributiongsl_cdf_negative_binomial_Q: The Negative Binomial Distributiongsl_cdf_pareto_P: The Pareto Distributiongsl_cdf_pareto_Pinv: The Pareto Distributiongsl_cdf_pareto_Q: The Pareto Distributiongsl_cdf_pareto_Qinv: The Pareto Distributiongsl_cdf_pascal_P: The Pascal Distributiongsl_cdf_pascal_Q: The Pascal Distributiongsl_cdf_poisson_P: The Poisson Distributiongsl_cdf_poisson_Q: The Poisson Distributiongsl_cdf_rayleigh_P: The Rayleigh Distributiongsl_cdf_rayleigh_Pinv: The Rayleigh Distributiongsl_cdf_rayleigh_Q: The Rayleigh Distributiongsl_cdf_rayleigh_Qinv: The Rayleigh Distributiongsl_cdf_tdist_P: The t-distributiongsl_cdf_tdist_Pinv: The t-distributiongsl_cdf_tdist_Q: The t-distributiongsl_cdf_tdist_Qinv: The t-distributiongsl_cdf_ugaussian_P: The Gaussian Distributiongsl_cdf_ugaussian_Pinv: The Gaussian Distributiongsl_cdf_ugaussian_Q: The Gaussian Distributiongsl_cdf_ugaussian_Qinv: The Gaussian Distributiongsl_cdf_weibull_P: The Weibull Distributiongsl_cdf_weibull_Pinv: The Weibull Distributiongsl_cdf_weibull_Q: The Weibull Distributiongsl_cdf_weibull_Qinv: The Weibull Distributiongsl_cheb_alloc: Creation and Calculation of Chebyshev Seriesgsl_cheb_calc_deriv: Derivatives and Integralsgsl_cheb_calc_integ: Derivatives and Integralsgsl_cheb_eval: Chebyshev Series Evaluationgsl_cheb_eval_err: Chebyshev Series Evaluationgsl_cheb_eval_n: Chebyshev Series Evaluationgsl_cheb_eval_n_err: Chebyshev Series Evaluationgsl_cheb_free: Creation and Calculation of Chebyshev Seriesgsl_cheb_init: Creation and Calculation of Chebyshev Seriesgsl_combination_alloc: Combination allocationgsl_combination_calloc: Combination allocationgsl_combination_data: Combination propertiesgsl_combination_fprintf: Reading and writing combinationsgsl_combination_fread: Reading and writing combinationsgsl_combination_free: Combination allocationgsl_combination_fscanf: Reading and writing combinationsgsl_combination_fwrite: Reading and writing combinationsgsl_combination_get: Accessing combination elementsgsl_combination_init_first: Combination allocationgsl_combination_init_last: Combination allocationgsl_combination_k: Combination propertiesgsl_combination_memcpy: Combination allocationgsl_combination_n: Combination propertiesgsl_combination_next: Combination functionsgsl_combination_prev: Combination functionsgsl_combination_valid: Combination propertiesgsl_complex_abs: Properties of complex numbersgsl_complex_abs2: Properties of complex numbersgsl_complex_add: Complex arithmetic operatorsgsl_complex_add_imag: Complex arithmetic operatorsgsl_complex_add_real: Complex arithmetic operatorsgsl_complex_arccos: Inverse Complex Trigonometric Functionsgsl_complex_arccos_real: Inverse Complex Trigonometric Functionsgsl_complex_arccosh: Inverse Complex Hyperbolic Functionsgsl_complex_arccosh_real: Inverse Complex Hyperbolic Functionsgsl_complex_arccot: Inverse Complex Trigonometric Functionsgsl_complex_arccoth: Inverse Complex Hyperbolic Functionsgsl_complex_arccsc: Inverse Complex Trigonometric Functionsgsl_complex_arccsc_real: Inverse Complex Trigonometric Functionsgsl_complex_arccsch: Inverse Complex Hyperbolic Functionsgsl_complex_arcsec: Inverse Complex Trigonometric Functionsgsl_complex_arcsec_real: Inverse Complex Trigonometric Functionsgsl_complex_arcsech: Inverse Complex Hyperbolic Functionsgsl_complex_arcsin: Inverse Complex Trigonometric Functionsgsl_complex_arcsin_real: Inverse Complex Trigonometric Functionsgsl_complex_arcsinh: Inverse Complex Hyperbolic Functionsgsl_complex_arctan: Inverse Complex Trigonometric Functionsgsl_complex_arctanh: Inverse Complex Hyperbolic Functionsgsl_complex_arctanh_real: Inverse Complex Hyperbolic Functionsgsl_complex_arg: Properties of complex numbersgsl_complex_conjugate: Complex arithmetic operatorsgsl_complex_cos: Complex Trigonometric Functionsgsl_complex_cosh: Complex Hyperbolic Functionsgsl_complex_cot: Complex Trigonometric Functionsgsl_complex_coth: Complex Hyperbolic Functionsgsl_complex_csc: Complex Trigonometric Functionsgsl_complex_csch: Complex Hyperbolic Functionsgsl_complex_div: Complex arithmetic operatorsgsl_complex_div_imag: Complex arithmetic operatorsgsl_complex_div_real: Complex arithmetic operatorsgsl_complex_exp: Elementary Complex Functionsgsl_complex_inverse: Complex arithmetic operatorsgsl_complex_log: Elementary Complex Functionsgsl_complex_log10: Elementary Complex Functionsgsl_complex_log_b: Elementary Complex Functionsgsl_complex_logabs: Properties of complex numbersgsl_complex_mul: Complex arithmetic operatorsgsl_complex_mul_imag: Complex arithmetic operatorsgsl_complex_mul_real: Complex arithmetic operatorsgsl_complex_negative: Complex arithmetic operatorsgsl_complex_polar: Complex numbersgsl_complex_pow: Elementary Complex Functionsgsl_complex_pow_real: Elementary Complex Functionsgsl_complex_rect: Complex numbersgsl_complex_sec: Complex Trigonometric Functionsgsl_complex_sech: Complex Hyperbolic Functionsgsl_complex_sin: Complex Trigonometric Functionsgsl_complex_sinh: Complex Hyperbolic Functionsgsl_complex_sqrt: Elementary Complex Functionsgsl_complex_sqrt_real: Elementary Complex Functionsgsl_complex_sub: Complex arithmetic operatorsgsl_complex_sub_imag: Complex arithmetic operatorsgsl_complex_sub_real: Complex arithmetic operatorsgsl_complex_tan: Complex Trigonometric Functionsgsl_complex_tanh: Complex Hyperbolic Functionsgsl_deriv_backward: Numerical Differentiation functionsgsl_deriv_central: Numerical Differentiation functionsgsl_deriv_forward: Numerical Differentiation functionsgsl_dht_alloc: Discrete Hankel Transform Functionsgsl_dht_apply: Discrete Hankel Transform Functionsgsl_dht_free: Discrete Hankel Transform Functionsgsl_dht_init: Discrete Hankel Transform Functionsgsl_dht_k_sample: Discrete Hankel Transform Functionsgsl_dht_new: Discrete Hankel Transform Functionsgsl_dht_x_sample: Discrete Hankel Transform FunctionsGSL_EDOM: Error Codesgsl_eigen_herm: Complex Hermitian Matricesgsl_eigen_herm_alloc: Complex Hermitian Matricesgsl_eigen_herm_free: Complex Hermitian Matricesgsl_eigen_hermv: Complex Hermitian Matricesgsl_eigen_hermv_alloc: Complex Hermitian Matricesgsl_eigen_hermv_free: Complex Hermitian Matricesgsl_eigen_hermv_sort: Sorting Eigenvalues and Eigenvectorsgsl_eigen_symm: Real Symmetric Matricesgsl_eigen_symm_alloc: Real Symmetric Matricesgsl_eigen_symm_free: Real Symmetric Matricesgsl_eigen_symmv: Real Symmetric Matricesgsl_eigen_symmv_alloc: Real Symmetric Matricesgsl_eigen_symmv_free: Real Symmetric Matricesgsl_eigen_symmv_sort: Sorting Eigenvalues and EigenvectorsGSL_EINVAL: Error CodesGSL_ENOMEM: Error CodesGSL_ERANGE: Error CodesGSL_ERROR: Using GSL error reporting in your own functionsGSL_ERROR_VAL: Using GSL error reporting in your own functionsgsl_expm1: Elementary Functionsgsl_fcmp: Approximate Comparison of Floating Point Numbersgsl_fft_complex_backward: Mixed-radix FFT routines for complex datagsl_fft_complex_forward: Mixed-radix FFT routines for complex datagsl_fft_complex_inverse: Mixed-radix FFT routines for complex datagsl_fft_complex_radix2_backward: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_dif_backward: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_dif_forward: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_dif_inverse: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_dif_transform: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_forward: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_inverse: Radix-2 FFT routines for complex datagsl_fft_complex_radix2_transform: Radix-2 FFT routines for complex datagsl_fft_complex_transform: Mixed-radix FFT routines for complex datagsl_fft_complex_wavetable_alloc: Mixed-radix FFT routines for complex datagsl_fft_complex_wavetable_free: Mixed-radix FFT routines for complex datagsl_fft_complex_workspace_alloc: Mixed-radix FFT routines for complex datagsl_fft_complex_workspace_free: Mixed-radix FFT routines for complex datagsl_fft_halfcomplex_radix2_backward: Radix-2 FFT routines for real datagsl_fft_halfcomplex_radix2_inverse: Radix-2 FFT routines for real datagsl_fft_halfcomplex_transform: Mixed-radix FFT routines for real datagsl_fft_halfcomplex_unpack: Mixed-radix FFT routines for real datagsl_fft_halfcomplex_wavetable_alloc: Mixed-radix FFT routines for real datagsl_fft_halfcomplex_wavetable_free: Mixed-radix FFT routines for real datagsl_fft_real_radix2_transform: Radix-2 FFT routines for real datagsl_fft_real_transform: Mixed-radix FFT routines for real datagsl_fft_real_unpack: Mixed-radix FFT routines for real datagsl_fft_real_wavetable_alloc: Mixed-radix FFT routines for real datagsl_fft_real_wavetable_free: Mixed-radix FFT routines for real datagsl_fft_real_workspace_alloc: Mixed-radix FFT routines for real datagsl_fft_real_workspace_free: Mixed-radix FFT routines for real datagsl_finite: Infinities and Not-a-numbergsl_fit_linear: Linear regressiongsl_fit_linear_est: Linear regressiongsl_fit_mul: Linear fitting without a constant termgsl_fit_mul_est: Linear fitting without a constant termgsl_fit_wlinear: Linear regressiongsl_fit_wmul: Linear fitting without a constant termgsl_frexp: Elementary Functionsgsl_heapsort: Sorting objectsgsl_heapsort_index: Sorting objectsgsl_histogram2d_accumulate: Updating and accessing 2D histogram elementsgsl_histogram2d_add: 2D Histogram Operationsgsl_histogram2d_alloc: 2D Histogram allocationgsl_histogram2d_clone: Copying 2D Histogramsgsl_histogram2d_cov: 2D Histogram Statisticsgsl_histogram2d_div: 2D Histogram Operationsgsl_histogram2d_equal_bins_p: 2D Histogram Operationsgsl_histogram2d_find: Searching 2D histogram rangesgsl_histogram2d_fprintf: Reading and writing 2D histogramsgsl_histogram2d_fread: Reading and writing 2D histogramsgsl_histogram2d_free: 2D Histogram allocationgsl_histogram2d_fscanf: Reading and writing 2D histogramsgsl_histogram2d_fwrite: Reading and writing 2D histogramsgsl_histogram2d_get: Updating and accessing 2D histogram elementsgsl_histogram2d_get_xrange: Updating and accessing 2D histogram elementsgsl_histogram2d_get_yrange: Updating and accessing 2D histogram elementsgsl_histogram2d_increment: Updating and accessing 2D histogram elementsgsl_histogram2d_max_bin: 2D Histogram Statisticsgsl_histogram2d_max_val: 2D Histogram Statisticsgsl_histogram2d_memcpy: Copying 2D Histogramsgsl_histogram2d_min_bin: 2D Histogram Statisticsgsl_histogram2d_min_val: 2D Histogram Statisticsgsl_histogram2d_mul: 2D Histogram Operationsgsl_histogram2d_nx: Updating and accessing 2D histogram elementsgsl_histogram2d_ny: Updating and accessing 2D histogram elementsgsl_histogram2d_pdf_alloc: Resampling from 2D histogramsgsl_histogram2d_pdf_free: Resampling from 2D histogramsgsl_histogram2d_pdf_init: Resampling from 2D histogramsgsl_histogram2d_pdf_sample: Resampling from 2D histogramsgsl_histogram2d_reset: Updating and accessing 2D histogram elementsgsl_histogram2d_scale: 2D Histogram Operationsgsl_histogram2d_set_ranges: 2D Histogram allocationgsl_histogram2d_set_ranges_uniform: 2D Histogram allocationgsl_histogram2d_shift: 2D Histogram Operationsgsl_histogram2d_sub: 2D Histogram Operationsgsl_histogram2d_sum: 2D Histogram Statisticsgsl_histogram2d_xmax: Updating and accessing 2D histogram elementsgsl_histogram2d_xmean: 2D Histogram Statisticsgsl_histogram2d_xmin: Updating and accessing 2D histogram elementsgsl_histogram2d_xsigma: 2D Histogram Statisticsgsl_histogram2d_ymax: Updating and accessing 2D histogram elementsgsl_histogram2d_ymean: 2D Histogram Statisticsgsl_histogram2d_ymin: Updating and accessing 2D histogram elementsgsl_histogram2d_ysigma: 2D Histogram Statisticsgsl_histogram_accumulate: Updating and accessing histogram elementsgsl_histogram_add: Histogram Operationsgsl_histogram_alloc: Histogram allocationgsl_histogram_bins: Updating and accessing histogram elementsgsl_histogram_clone: Copying Histogramsgsl_histogram_div: Histogram Operationsgsl_histogram_equal_bins_p: Histogram Operationsgsl_histogram_find: Searching histogram rangesgsl_histogram_fprintf: Reading and writing histogramsgsl_histogram_fread: Reading and writing histogramsgsl_histogram_free: Histogram allocationgsl_histogram_fscanf: Reading and writing histogramsgsl_histogram_fwrite: Reading and writing histogramsgsl_histogram_get: Updating and accessing histogram elementsgsl_histogram_get_range: Updating and accessing histogram elementsgsl_histogram_increment: Updating and accessing histogram elementsgsl_histogram_max: Updating and accessing histogram elementsgsl_histogram_max_bin: Histogram Statisticsgsl_histogram_max_val: Histogram Statisticsgsl_histogram_mean: Histogram Statisticsgsl_histogram_memcpy: Copying Histogramsgsl_histogram_min: Updating and accessing histogram elementsgsl_histogram_min_bin: Histogram Statisticsgsl_histogram_min_val: Histogram Statisticsgsl_histogram_mul: Histogram Operationsgsl_histogram_pdf_alloc: The histogram probability distribution structgsl_histogram_pdf_free: The histogram probability distribution structgsl_histogram_pdf_init: The histogram probability distribution structgsl_histogram_pdf_sample: The histogram probability distribution structgsl_histogram_reset: Updating and accessing histogram elementsgsl_histogram_scale: Histogram Operationsgsl_histogram_set_ranges: Histogram allocationgsl_histogram_set_ranges_uniform: Histogram allocationgsl_histogram_shift: Histogram Operationsgsl_histogram_sigma: Histogram Statisticsgsl_histogram_sub: Histogram Operationsgsl_histogram_sum: Histogram Statisticsgsl_hypot: Elementary Functionsgsl_ieee_env_setup: Setting up your IEEE environmentgsl_ieee_fprintf_double: Representation of floating point numbersgsl_ieee_fprintf_float: Representation of floating point numbersgsl_ieee_printf_double: Representation of floating point numbersgsl_ieee_printf_float: Representation of floating point numbersGSL_IMAG: Complex numbersgsl_integration_qag: QAG adaptive integrationgsl_integration_qagi: QAGI adaptive integration on infinite intervalsgsl_integration_qagil: QAGI adaptive integration on infinite intervalsgsl_integration_qagiu: QAGI adaptive integration on infinite intervalsgsl_integration_qagp: QAGP adaptive integration with known singular pointsgsl_integration_qags: QAGS adaptive integration with singularitiesgsl_integration_qawc: QAWC adaptive integration for Cauchy principal valuesgsl_integration_qawf: QAWF adaptive integration for Fourier integralsgsl_integration_qawo: QAWO adaptive integration for oscillatory functionsgsl_integration_qawo_table_alloc: QAWO adaptive integration for oscillatory functionsgsl_integration_qawo_table_free: QAWO adaptive integration for oscillatory functionsgsl_integration_qawo_table_set: QAWO adaptive integration for oscillatory functionsgsl_integration_qawo_table_set_length: QAWO adaptive integration for oscillatory functionsgsl_integration_qaws: QAWS adaptive integration for singular functionsgsl_integration_qaws_table_alloc: QAWS adaptive integration for singular functionsgsl_integration_qaws_table_free: QAWS adaptive integration for singular functionsgsl_integration_qaws_table_set: QAWS adaptive integration for singular functionsgsl_integration_qng: QNG non-adaptive Gauss-Kronrod integrationgsl_integration_workspace_alloc: QAG adaptive integrationgsl_integration_workspace_free: QAG adaptive integrationgsl_interp_accel_alloc: Index Look-up and Accelerationgsl_interp_accel_find: Index Look-up and Accelerationgsl_interp_accel_free: Index Look-up and Accelerationgsl_interp_akima: Interpolation Typesgsl_interp_akima_periodic: Interpolation Typesgsl_interp_alloc: Interpolation Functionsgsl_interp_bsearch: Index Look-up and Accelerationgsl_interp_cspline: Interpolation Typesgsl_interp_cspline_periodic: Interpolation Typesgsl_interp_eval: Evaluation of Interpolating Functionsgsl_interp_eval_deriv: Evaluation of Interpolating Functionsgsl_interp_eval_deriv2: Evaluation of Interpolating Functionsgsl_interp_eval_deriv2_e: Evaluation of Interpolating Functionsgsl_interp_eval_deriv_e: Evaluation of Interpolating Functionsgsl_interp_eval_e: Evaluation of Interpolating Functionsgsl_interp_eval_integ: Evaluation of Interpolating Functionsgsl_interp_eval_integ_e: Evaluation of Interpolating Functionsgsl_interp_free: Interpolation Functionsgsl_interp_init: Interpolation Functionsgsl_interp_linear: Interpolation Typesgsl_interp_min_size: Interpolation Typesgsl_interp_name: Interpolation Typesgsl_interp_polynomial: Interpolation TypesGSL_IS_EVEN: Testing for Odd and Even NumbersGSL_IS_ODD: Testing for Odd and Even Numbersgsl_isinf: Infinities and Not-a-numbergsl_isnan: Infinities and Not-a-numbergsl_ldexp: Elementary Functionsgsl_linalg_bidiag_decomp: Bidiagonalizationgsl_linalg_bidiag_unpack: Bidiagonalizationgsl_linalg_bidiag_unpack2: Bidiagonalizationgsl_linalg_bidiag_unpack_B: Bidiagonalizationgsl_linalg_cholesky_decomp: Cholesky Decompositiongsl_linalg_cholesky_solve: Cholesky Decompositiongsl_linalg_cholesky_svx: Cholesky Decompositiongsl_linalg_complex_LU_decomp: LU Decompositiongsl_linalg_complex_LU_det: LU Decompositiongsl_linalg_complex_LU_invert: LU Decompositiongsl_linalg_complex_LU_lndet: LU Decompositiongsl_linalg_complex_LU_refine: LU Decompositiongsl_linalg_complex_LU_sgndet: LU Decompositiongsl_linalg_complex_LU_solve: LU Decompositiongsl_linalg_complex_LU_svx: LU Decompositiongsl_linalg_hermtd_decomp: Tridiagonal Decomposition of Hermitian Matricesgsl_linalg_hermtd_unpack: Tridiagonal Decomposition of Hermitian Matricesgsl_linalg_hermtd_unpack_T: Tridiagonal Decomposition of Hermitian Matricesgsl_linalg_HH_solve: Householder solver for linear systemsgsl_linalg_HH_svx: Householder solver for linear systemsgsl_linalg_householder_hm: Householder Transformationsgsl_linalg_householder_hv: Householder Transformationsgsl_linalg_householder_mh: Householder Transformationsgsl_linalg_householder_transform: Householder Transformationsgsl_linalg_LU_decomp: LU Decompositiongsl_linalg_LU_det: LU Decompositiongsl_linalg_LU_invert: LU Decompositiongsl_linalg_LU_lndet: LU Decompositiongsl_linalg_LU_refine: LU Decompositiongsl_linalg_LU_sgndet: LU Decompositiongsl_linalg_LU_solve: LU Decompositiongsl_linalg_LU_svx: LU Decompositiongsl_linalg_QR_decomp: QR Decompositiongsl_linalg_QR_lssolve: QR Decompositiongsl_linalg_QR_QRsolve: QR Decompositiongsl_linalg_QR_QTvec: QR Decompositiongsl_linalg_QR_Qvec: QR Decompositiongsl_linalg_QR_Rsolve: QR Decompositiongsl_linalg_QR_Rsvx: QR Decompositiongsl_linalg_QR_solve: QR Decompositiongsl_linalg_QR_svx: QR Decompositiongsl_linalg_QR_unpack: QR Decompositiongsl_linalg_QR_update: QR Decompositiongsl_linalg_QRPT_decomp: QR Decomposition with Column Pivotinggsl_linalg_QRPT_decomp2: QR Decomposition with Column Pivotinggsl_linalg_QRPT_QRsolve: QR Decomposition with Column Pivotinggsl_linalg_QRPT_Rsolve: QR Decomposition with Column Pivotinggsl_linalg_QRPT_Rsvx: QR Decomposition with Column Pivotinggsl_linalg_QRPT_solve: QR Decomposition with Column Pivotinggsl_linalg_QRPT_svx: QR Decomposition with Column Pivotinggsl_linalg_QRPT_update: QR Decomposition with Column Pivotinggsl_linalg_R_solve: QR Decompositiongsl_linalg_R_svx: QR Decompositiongsl_linalg_solve_cyc_tridiag: Tridiagonal Systemsgsl_linalg_solve_symm_cyc_tridiag: Tridiagonal Systemsgsl_linalg_solve_symm_tridiag: Tridiagonal Systemsgsl_linalg_solve_tridiag: Tridiagonal Systemsgsl_linalg_SV_decomp: Singular Value Decompositiongsl_linalg_SV_decomp_jacobi: Singular Value Decompositiongsl_linalg_SV_decomp_mod: Singular Value Decompositiongsl_linalg_SV_solve: Singular Value Decompositiongsl_linalg_symmtd_decomp: Tridiagonal Decomposition of Real Symmetric Matricesgsl_linalg_symmtd_unpack: Tridiagonal Decomposition of Real Symmetric Matricesgsl_linalg_symmtd_unpack_T: Tridiagonal Decomposition of Real Symmetric Matricesgsl_log1p: Elementary Functionsgsl_matrix_add: Matrix operationsgsl_matrix_add_constant: Matrix operationsgsl_matrix_alloc: Matrix allocationgsl_matrix_calloc: Matrix allocationgsl_matrix_column: Creating row and column viewsgsl_matrix_const_column: Creating row and column viewsgsl_matrix_const_diagonal: Creating row and column viewsgsl_matrix_const_ptr: Accessing matrix elementsgsl_matrix_const_row: Creating row and column viewsgsl_matrix_const_subdiagonal: Creating row and column viewsgsl_matrix_const_submatrix: Matrix viewsgsl_matrix_const_superdiagonal: Creating row and column viewsgsl_matrix_const_view_array: Matrix viewsgsl_matrix_const_view_array_with_tda: Matrix viewsgsl_matrix_const_view_vector: Matrix viewsgsl_matrix_const_view_vector_with_tda: Matrix viewsgsl_matrix_diagonal: Creating row and column viewsgsl_matrix_div_elements: Matrix operationsgsl_matrix_fprintf: Reading and writing matricesgsl_matrix_fread: Reading and writing matricesgsl_matrix_free: Matrix allocationgsl_matrix_fscanf: Reading and writing matricesgsl_matrix_fwrite: Reading and writing matricesgsl_matrix_get: Accessing matrix elementsgsl_matrix_get_col: Copying rows and columnsgsl_matrix_get_row: Copying rows and columnsgsl_matrix_isnull: Matrix propertiesgsl_matrix_max: Finding maximum and minimum elements of matricesgsl_matrix_max_index: Finding maximum and minimum elements of matricesgsl_matrix_memcpy: Copying matricesgsl_matrix_min: Finding maximum and minimum elements of matricesgsl_matrix_min_index: Finding maximum and minimum elements of matricesgsl_matrix_minmax: Finding maximum and minimum elements of matricesgsl_matrix_minmax_index: Finding maximum and minimum elements of matricesgsl_matrix_mul_elements: Matrix operationsgsl_matrix_ptr: Accessing matrix elementsgsl_matrix_row: Creating row and column viewsgsl_matrix_scale: Matrix operationsgsl_matrix_set: Accessing matrix elementsgsl_matrix_set_all: Initializing matrix elementsgsl_matrix_set_col: Copying rows and columnsgsl_matrix_set_identity: Initializing matrix elementsgsl_matrix_set_row: Copying rows and columnsgsl_matrix_set_zero: Initializing matrix elementsgsl_matrix_sub: Matrix operationsgsl_matrix_subdiagonal: Creating row and column viewsgsl_matrix_submatrix: Matrix viewsgsl_matrix_superdiagonal: Creating row and column viewsgsl_matrix_swap: Copying matricesgsl_matrix_swap_columns: Exchanging rows and columnsgsl_matrix_swap_rowcol: Exchanging rows and columnsgsl_matrix_swap_rows: Exchanging rows and columnsgsl_matrix_transpose: Exchanging rows and columnsgsl_matrix_transpose_memcpy: Exchanging rows and columnsgsl_matrix_view_array: Matrix viewsgsl_matrix_view_array_with_tda: Matrix viewsgsl_matrix_view_vector: Matrix viewsgsl_matrix_view_vector_with_tda: Matrix viewsGSL_MAX: Maximum and Minimum functionsGSL_MAX_DBL: Maximum and Minimum functionsGSL_MAX_INT: Maximum and Minimum functionsGSL_MAX_LDBL: Maximum and Minimum functionsGSL_MIN: Maximum and Minimum functionsGSL_MIN_DBL: Maximum and Minimum functionsgsl_min_fminimizer_alloc: Initializing the Minimizergsl_min_fminimizer_brent: Minimization Algorithmsgsl_min_fminimizer_f_lower: Minimization Iterationgsl_min_fminimizer_f_minimum: Minimization Iterationgsl_min_fminimizer_f_upper: Minimization Iterationgsl_min_fminimizer_free: Initializing the Minimizergsl_min_fminimizer_goldensection: Minimization Algorithmsgsl_min_fminimizer_iterate: Minimization Iterationgsl_min_fminimizer_name: Initializing the Minimizergsl_min_fminimizer_set: Initializing the Minimizergsl_min_fminimizer_set_with_values: Initializing the Minimizergsl_min_fminimizer_x_lower: Minimization Iterationgsl_min_fminimizer_x_minimum: Minimization Iterationgsl_min_fminimizer_x_upper: Minimization IterationGSL_MIN_INT: Maximum and Minimum functionsGSL_MIN_LDBL: Maximum and Minimum functionsgsl_min_test_interval: Minimization Stopping Parametersgsl_monte_miser_alloc: MISERgsl_monte_miser_free: MISERgsl_monte_miser_init: MISERgsl_monte_miser_integrate: MISERgsl_monte_plain_alloc: PLAIN Monte Carlogsl_monte_plain_free: PLAIN Monte Carlogsl_monte_plain_init: PLAIN Monte Carlogsl_monte_plain_integrate: PLAIN Monte Carlogsl_monte_vegas_alloc: VEGASgsl_monte_vegas_free: VEGASgsl_monte_vegas_init: VEGASgsl_monte_vegas_integrate: VEGASgsl_multifit_covar: Computing the covariance matrix of best fit parametersgsl_multifit_fdfsolver_alloc: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fdfsolver_free: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fdfsolver_iterate: Iteration of the Minimization Algorithmgsl_multifit_fdfsolver_lmder: Minimization Algorithms using Derivativesgsl_multifit_fdfsolver_lmsder: Minimization Algorithms using Derivativesgsl_multifit_fdfsolver_name: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fdfsolver_position: Iteration of the Minimization Algorithmgsl_multifit_fdfsolver_set: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fsolver_alloc: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fsolver_free: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fsolver_iterate: Iteration of the Minimization Algorithmgsl_multifit_fsolver_name: Initializing the Nonlinear Least-Squares Solvergsl_multifit_fsolver_position: Iteration of the Minimization Algorithmgsl_multifit_fsolver_set: Initializing the Nonlinear Least-Squares Solvergsl_multifit_gradient: Search Stopping Parameters for Minimization Algorithmsgsl_multifit_linear: Multi-parameter fittinggsl_multifit_linear_alloc: Multi-parameter fittinggsl_multifit_linear_est: Multi-parameter fittinggsl_multifit_linear_free: Multi-parameter fittinggsl_multifit_linear_svd: Multi-parameter fittinggsl_multifit_test_delta: Search Stopping Parameters for Minimization Algorithmsgsl_multifit_test_gradient: Search Stopping Parameters for Minimization Algorithmsgsl_multifit_wlinear: Multi-parameter fittinggsl_multifit_wlinear_svd: Multi-parameter fittinggsl_multimin_fdfminimizer_alloc: Initializing the Multidimensional Minimizergsl_multimin_fdfminimizer_conjugate_fr: Multimin Algorithmsgsl_multimin_fdfminimizer_conjugate_pr: Multimin Algorithmsgsl_multimin_fdfminimizer_free: Initializing the Multidimensional Minimizergsl_multimin_fdfminimizer_gradient: Multimin Iterationgsl_multimin_fdfminimizer_iterate: Multimin Iterationgsl_multimin_fdfminimizer_minimum: Multimin Iterationgsl_multimin_fdfminimizer_name: Initializing the Multidimensional Minimizergsl_multimin_fdfminimizer_restart: Multimin Iterationgsl_multimin_fdfminimizer_set: Initializing the Multidimensional Minimizergsl_multimin_fdfminimizer_steepest_descent: Multimin Algorithmsgsl_multimin_fdfminimizer_vector_bfgs: Multimin Algorithmsgsl_multimin_fdfminimizer_x: Multimin Iterationgsl_multimin_fminimizer_alloc: Initializing the Multidimensional Minimizergsl_multimin_fminimizer_free: Initializing the Multidimensional Minimizergsl_multimin_fminimizer_iterate: Multimin Iterationgsl_multimin_fminimizer_minimum: Multimin Iterationgsl_multimin_fminimizer_name: Initializing the Multidimensional Minimizergsl_multimin_fminimizer_nmsimplex: Multimin Algorithmsgsl_multimin_fminimizer_set: Initializing the Multidimensional Minimizergsl_multimin_fminimizer_size: Multimin Iterationgsl_multimin_fminimizer_x: Multimin Iterationgsl_multimin_test_gradient: Multimin Stopping Criteriagsl_multimin_test_size: Multimin Stopping Criteriagsl_multiroot_fdfsolver_alloc: Initializing the Multidimensional Solvergsl_multiroot_fdfsolver_dx: Iteration of the multidimensional solvergsl_multiroot_fdfsolver_f: Iteration of the multidimensional solvergsl_multiroot_fdfsolver_free: Initializing the Multidimensional Solvergsl_multiroot_fdfsolver_gnewton: Algorithms using Derivativesgsl_multiroot_fdfsolver_hybridj: Algorithms using Derivativesgsl_multiroot_fdfsolver_hybridsj: Algorithms using Derivativesgsl_multiroot_fdfsolver_iterate: Iteration of the multidimensional solvergsl_multiroot_fdfsolver_name: Initializing the Multidimensional Solvergsl_multiroot_fdfsolver_newton: Algorithms using Derivativesgsl_multiroot_fdfsolver_root: Iteration of the multidimensional solvergsl_multiroot_fdfsolver_set: Initializing the Multidimensional Solvergsl_multiroot_fsolver_alloc: Initializing the Multidimensional Solvergsl_multiroot_fsolver_broyden: Algorithms without Derivativesgsl_multiroot_fsolver_dnewton: Algorithms without Derivativesgsl_multiroot_fsolver_dx: Iteration of the multidimensional solvergsl_multiroot_fsolver_f: Iteration of the multidimensional solvergsl_multiroot_fsolver_free: Initializing the Multidimensional Solvergsl_multiroot_fsolver_hybrid: Algorithms without Derivativesgsl_multiroot_fsolver_hybrids: Algorithms without Derivativesgsl_multiroot_fsolver_iterate: Iteration of the multidimensional solvergsl_multiroot_fsolver_name: Initializing the Multidimensional Solvergsl_multiroot_fsolver_root: Iteration of the multidimensional solvergsl_multiroot_fsolver_set: Initializing the Multidimensional Solvergsl_multiroot_test_delta: Search Stopping Parameters for the multidimensional solvergsl_multiroot_test_residual: Search Stopping Parameters for the multidimensional solverGSL_NAN: Infinities and Not-a-numberGSL_NEGINF: Infinities and Not-a-numbergsl_ntuple_bookdata: Writing ntuplesgsl_ntuple_close: Closing an ntuple filegsl_ntuple_create: Creating ntuplesgsl_ntuple_open: Opening an existing ntuple filegsl_ntuple_project: Histogramming ntuple valuesgsl_ntuple_read: Reading ntuplesgsl_ntuple_write: Writing ntuplesgsl_odeiv_control_alloc: Adaptive Step-size Controlgsl_odeiv_control_free: Adaptive Step-size Controlgsl_odeiv_control_hadjust: Adaptive Step-size Controlgsl_odeiv_control_init: Adaptive Step-size Controlgsl_odeiv_control_name: Adaptive Step-size Controlgsl_odeiv_control_scaled_new: Adaptive Step-size Controlgsl_odeiv_control_standard_new: Adaptive Step-size Controlgsl_odeiv_control_y_new: Adaptive Step-size Controlgsl_odeiv_control_yp_new: Adaptive Step-size Controlgsl_odeiv_evolve_alloc: Evolutiongsl_odeiv_evolve_apply: Evolutiongsl_odeiv_evolve_free: Evolutiongsl_odeiv_evolve_reset: Evolutiongsl_odeiv_step_alloc: Stepping Functionsgsl_odeiv_step_apply: Stepping Functionsgsl_odeiv_step_bsimp: Stepping Functionsgsl_odeiv_step_free: Stepping Functionsgsl_odeiv_step_gear1: Stepping Functionsgsl_odeiv_step_gear2: Stepping Functionsgsl_odeiv_step_name: Stepping Functionsgsl_odeiv_step_order: Stepping Functionsgsl_odeiv_step_reset: Stepping Functionsgsl_odeiv_step_rk2: Stepping Functionsgsl_odeiv_step_rk2imp: Stepping Functionsgsl_odeiv_step_rk4: Stepping Functionsgsl_odeiv_step_rk4imp: Stepping Functionsgsl_odeiv_step_rk8pd: Stepping Functionsgsl_odeiv_step_rkck: Stepping Functionsgsl_odeiv_step_rkf45: Stepping Functionsgsl_permutation_alloc: Permutation allocationgsl_permutation_calloc: Permutation allocationgsl_permutation_canonical_cycles: Permutations in cyclic formgsl_permutation_canonical_to_linear: Permutations in cyclic formgsl_permutation_data: Permutation propertiesgsl_permutation_fprintf: Reading and writing permutationsgsl_permutation_fread: Reading and writing permutationsgsl_permutation_free: Permutation allocationgsl_permutation_fscanf: Reading and writing permutationsgsl_permutation_fwrite: Reading and writing permutationsgsl_permutation_get: Accessing permutation elementsgsl_permutation_init: Permutation allocationgsl_permutation_inverse: Permutation functionsgsl_permutation_inversions: Permutations in cyclic formgsl_permutation_linear_cycles: Permutations in cyclic formgsl_permutation_linear_to_canonical: Permutations in cyclic formgsl_permutation_memcpy: Permutation allocationgsl_permutation_mul: Applying Permutationsgsl_permutation_next: Permutation functionsgsl_permutation_prev: Permutation functionsgsl_permutation_reverse: Permutation functionsgsl_permutation_size: Permutation propertiesgsl_permutation_swap: Accessing permutation elementsgsl_permutation_valid: Permutation propertiesgsl_permute: Applying Permutationsgsl_permute_inverse: Applying Permutationsgsl_permute_vector: Applying Permutationsgsl_permute_vector_inverse: Applying Permutationsgsl_poly_complex_solve: General Polynomial Equationsgsl_poly_complex_solve_cubic: Cubic Equationsgsl_poly_complex_solve_quadratic: Quadratic Equationsgsl_poly_complex_workspace_alloc: General Polynomial Equationsgsl_poly_complex_workspace_free: General Polynomial Equationsgsl_poly_dd_eval: Divided Difference Representation of Polynomialsgsl_poly_dd_init: Divided Difference Representation of Polynomialsgsl_poly_dd_taylor: Divided Difference Representation of Polynomialsgsl_poly_eval: Polynomial Evaluationgsl_poly_solve_cubic: Cubic Equationsgsl_poly_solve_quadratic: Quadratic EquationsGSL_POSINF: Infinities and Not-a-numbergsl_pow_2: Small integer powersgsl_pow_3: Small integer powersgsl_pow_4: Small integer powersgsl_pow_5: Small integer powersgsl_pow_6: Small integer powersgsl_pow_7: Small integer powersgsl_pow_8: Small integer powersgsl_pow_9: Small integer powersgsl_pow_int: Small integer powersgsl_qrng_alloc: Quasi-random number generator initializationgsl_qrng_clone: Saving and resorting quasi-random number generator stategsl_qrng_free: Quasi-random number generator initializationgsl_qrng_get: Sampling from a quasi-random number generatorgsl_qrng_init: Quasi-random number generator initializationgsl_qrng_memcpy: Saving and resorting quasi-random number generator stategsl_qrng_name: Auxiliary quasi-random number generator functionsgsl_qrng_niederreiter_2: Quasi-random number generator algorithmsgsl_qrng_size: Auxiliary quasi-random number generator functionsgsl_qrng_sobol: Quasi-random number generator algorithmsgsl_qrng_state: Auxiliary quasi-random number generator functionsgsl_ran_bernoulli: The Bernoulli Distributiongsl_ran_bernoulli_pdf: The Bernoulli Distributiongsl_ran_beta: The Beta Distributiongsl_ran_beta_pdf: The Beta Distributiongsl_ran_binomial: The Binomial Distributiongsl_ran_binomial_pdf: The Binomial Distributiongsl_ran_bivariate_gaussian: The Bivariate Gaussian Distributiongsl_ran_bivariate_gaussian_pdf: The Bivariate Gaussian Distributiongsl_ran_cauchy: The Cauchy Distributiongsl_ran_cauchy_pdf: The Cauchy Distributiongsl_ran_chisq: The Chi-squared Distributiongsl_ran_chisq_pdf: The Chi-squared Distributiongsl_ran_choose: Shuffling and Samplinggsl_ran_dir_2d: Spherical Vector Distributionsgsl_ran_dir_2d_trig_method: Spherical Vector Distributionsgsl_ran_dir_3d: Spherical Vector Distributionsgsl_ran_dir_nd: Spherical Vector Distributionsgsl_ran_dirichlet: The Dirichlet Distributiongsl_ran_dirichlet_lnpdf: The Dirichlet Distributiongsl_ran_dirichlet_pdf: The Dirichlet Distributiongsl_ran_discrete: General Discrete Distributionsgsl_ran_discrete_free: General Discrete Distributionsgsl_ran_discrete_pdf: General Discrete Distributionsgsl_ran_discrete_preproc: General Discrete Distributionsgsl_ran_exponential: The Exponential Distributiongsl_ran_exponential_pdf: The Exponential Distributiongsl_ran_exppow: The Exponential Power Distributiongsl_ran_exppow_pdf: The Exponential Power Distributiongsl_ran_fdist: The F-distributiongsl_ran_fdist_pdf: The F-distributiongsl_ran_flat: The Flat (Uniform) Distributiongsl_ran_flat_pdf: The Flat (Uniform) Distributiongsl_ran_gamma: The Gamma Distributiongsl_ran_gamma_mt: The Gamma Distributiongsl_ran_gamma_pdf: The Gamma Distributiongsl_ran_gaussian: The Gaussian Distributiongsl_ran_gaussian_pdf: The Gaussian Distributiongsl_ran_gaussian_ratio_method: The Gaussian Distributiongsl_ran_gaussian_tail: The Gaussian Tail Distributiongsl_ran_gaussian_tail_pdf: The Gaussian Tail Distributiongsl_ran_gaussian_ziggurat: The Gaussian Distributiongsl_ran_geometric: The Geometric Distributiongsl_ran_geometric_pdf: The Geometric Distributiongsl_ran_gumbel1: The Type-1 Gumbel Distributiongsl_ran_gumbel1_pdf: The Type-1 Gumbel Distributiongsl_ran_gumbel2: The Type-2 Gumbel Distributiongsl_ran_gumbel2_pdf: The Type-2 Gumbel Distributiongsl_ran_hypergeometric: The Hypergeometric Distributiongsl_ran_hypergeometric_pdf: The Hypergeometric Distributiongsl_ran_landau: The Landau Distributiongsl_ran_landau_pdf: The Landau Distributiongsl_ran_laplace: The Laplace Distributiongsl_ran_laplace_pdf: The Laplace Distributiongsl_ran_levy: The Levy alpha-Stable Distributionsgsl_ran_levy_skew: The Levy skew alpha-Stable Distributiongsl_ran_logarithmic: The Logarithmic Distributiongsl_ran_logarithmic_pdf: The Logarithmic Distributiongsl_ran_logistic: The Logistic Distributiongsl_ran_logistic_pdf: The Logistic Distributiongsl_ran_lognormal: The Lognormal Distributiongsl_ran_lognormal_pdf: The Lognormal Distributiongsl_ran_multinomial: The Multinomial Distributiongsl_ran_multinomial_lnpdf: The Multinomial Distributiongsl_ran_multinomial_pdf: The Multinomial Distributiongsl_ran_negative_binomial: The Negative Binomial Distributiongsl_ran_negative_binomial_pdf: The Negative Binomial Distributiongsl_ran_pareto: The Pareto Distributiongsl_ran_pareto_pdf: The Pareto Distributiongsl_ran_pascal: The Pascal Distributiongsl_ran_pascal_pdf: The Pascal Distributiongsl_ran_poisson: The Poisson Distributiongsl_ran_poisson_pdf: The Poisson Distributiongsl_ran_rayleigh: The Rayleigh Distributiongsl_ran_rayleigh_pdf: The Rayleigh Distributiongsl_ran_rayleigh_tail: The Rayleigh Tail Distributiongsl_ran_rayleigh_tail_pdf: The Rayleigh Tail Distributiongsl_ran_sample: Shuffling and Samplinggsl_ran_shuffle: Shuffling and Samplinggsl_ran_tdist: The t-distributiongsl_ran_tdist_pdf: The t-distributiongsl_ran_ugaussian: The Gaussian Distributiongsl_ran_ugaussian_pdf: The Gaussian Distributiongsl_ran_ugaussian_ratio_method: The Gaussian Distributiongsl_ran_ugaussian_tail: The Gaussian Tail Distributiongsl_ran_ugaussian_tail_pdf: The Gaussian Tail Distributiongsl_ran_weibull: The Weibull Distributiongsl_ran_weibull_pdf: The Weibull DistributionGSL_REAL: Complex numbersgsl_rng_alloc: Random number generator initializationgsl_rng_borosh13: Other random number generatorsgsl_rng_clone: Copying random number generator stategsl_rng_cmrg: Random number generator algorithmsgsl_rng_coveyou: Other random number generatorsgsl_rng_env_setup: Random number environment variablesgsl_rng_fishman18: Other random number generatorsgsl_rng_fishman20: Other random number generatorsgsl_rng_fishman2x: Other random number generatorsgsl_rng_fread: Reading and writing random number generator stategsl_rng_free: Random number generator initializationgsl_rng_fwrite: Reading and writing random number generator stategsl_rng_get: Sampling from a random number generatorgsl_rng_gfsr4: Random number generator algorithmsgsl_rng_knuthran: Other random number generatorsgsl_rng_knuthran2: Other random number generatorsgsl_rng_lecuyer21: Other random number generatorsgsl_rng_max: Auxiliary random number generator functionsgsl_rng_memcpy: Copying random number generator stategsl_rng_min: Auxiliary random number generator functionsgsl_rng_minstd: Other random number generatorsgsl_rng_mrg: Random number generator algorithmsgsl_rng_mt19937: Random number generator algorithmsgsl_rng_name: Auxiliary random number generator functionsgsl_rng_r250: Other random number generatorsgsl_rng_rand: Unix random number generatorsgsl_rng_rand48: Unix random number generatorsgsl_rng_random_bsd: Unix random number generatorsgsl_rng_random_glibc2: Unix random number generatorsgsl_rng_random_libc5: Unix random number generatorsgsl_rng_randu: Other random number generatorsgsl_rng_ranf: Other random number generatorsgsl_rng_ranlux: Random number generator algorithmsgsl_rng_ranlux389: Random number generator algorithmsgsl_rng_ranlxd1: Random number generator algorithmsgsl_rng_ranlxd2: Random number generator algorithmsgsl_rng_ranlxs0: Random number generator algorithmsgsl_rng_ranlxs1: Random number generator algorithmsgsl_rng_ranlxs2: Random number generator algorithmsgsl_rng_ranmar: Other random number generatorsgsl_rng_set: Random number generator initializationgsl_rng_size: Auxiliary random number generator functionsgsl_rng_slatec: Other random number generatorsgsl_rng_state: Auxiliary random number generator functionsgsl_rng_taus: Random number generator algorithmsgsl_rng_taus2: Random number generator algorithmsgsl_rng_transputer: Other random number generatorsgsl_rng_tt800: Other random number generatorsgsl_rng_types_setup: Auxiliary random number generator functionsgsl_rng_uni: Other random number generatorsgsl_rng_uni32: Other random number generatorsgsl_rng_uniform: Sampling from a random number generatorgsl_rng_uniform_int: Sampling from a random number generatorgsl_rng_uniform_pos: Sampling from a random number generatorgsl_rng_vax: Other random number generatorsgsl_rng_waterman14: Other random number generatorsgsl_rng_zuf: Other random number generatorsgsl_root_fdfsolver_alloc: Initializing the Solvergsl_root_fdfsolver_free: Initializing the Solvergsl_root_fdfsolver_iterate: Root Finding Iterationgsl_root_fdfsolver_name: Initializing the Solvergsl_root_fdfsolver_newton: Root Finding Algorithms using Derivativesgsl_root_fdfsolver_root: Root Finding Iterationgsl_root_fdfsolver_secant: Root Finding Algorithms using Derivativesgsl_root_fdfsolver_set: Initializing the Solvergsl_root_fdfsolver_steffenson: Root Finding Algorithms using Derivativesgsl_root_fsolver_alloc: Initializing the Solvergsl_root_fsolver_bisection: Root Bracketing Algorithmsgsl_root_fsolver_brent: Root Bracketing Algorithmsgsl_root_fsolver_falsepos: Root Bracketing Algorithmsgsl_root_fsolver_free: Initializing the Solvergsl_root_fsolver_iterate: Root Finding Iterationgsl_root_fsolver_name: Initializing the Solvergsl_root_fsolver_root: Root Finding Iterationgsl_root_fsolver_set: Initializing the Solvergsl_root_fsolver_x_lower: Root Finding Iterationgsl_root_fsolver_x_upper: Root Finding Iterationgsl_root_test_delta: Search Stopping Parametersgsl_root_test_interval: Search Stopping Parametersgsl_root_test_residual: Search Stopping ParametersGSL_SET_COMPLEX: Complex numbersgsl_set_error_handler: Error Handlersgsl_set_error_handler_off: Error HandlersGSL_SET_IMAG: Complex numbersGSL_SET_REAL: Complex numbersgsl_sf_airy_Ai: Airy Functionsgsl_sf_airy_Ai_deriv: Derivatives of Airy Functionsgsl_sf_airy_Ai_deriv_e: Derivatives of Airy Functionsgsl_sf_airy_Ai_deriv_scaled: Derivatives of Airy Functionsgsl_sf_airy_Ai_deriv_scaled_e: Derivatives of Airy Functionsgsl_sf_airy_Ai_e: Airy Functionsgsl_sf_airy_Ai_scaled: Airy Functionsgsl_sf_airy_Ai_scaled_e: Airy Functionsgsl_sf_airy_Bi: Airy Functionsgsl_sf_airy_Bi_deriv: Derivatives of Airy Functionsgsl_sf_airy_Bi_deriv_e: Derivatives of Airy Functionsgsl_sf_airy_Bi_deriv_scaled: Derivatives of Airy Functionsgsl_sf_airy_Bi_deriv_scaled_e: Derivatives of Airy Functionsgsl_sf_airy_Bi_e: Airy Functionsgsl_sf_airy_Bi_scaled: Airy Functionsgsl_sf_airy_Bi_scaled_e: Airy Functionsgsl_sf_airy_zero_Ai: Zeros of Airy Functionsgsl_sf_airy_zero_Ai_deriv: Zeros of Derivatives of Airy Functionsgsl_sf_airy_zero_Ai_deriv_e: Zeros of Derivatives of Airy Functionsgsl_sf_airy_zero_Ai_e: Zeros of Airy Functionsgsl_sf_airy_zero_Bi: Zeros of Airy Functionsgsl_sf_airy_zero_Bi_deriv: Zeros of Derivatives of Airy Functionsgsl_sf_airy_zero_Bi_deriv_e: Zeros of Derivatives of Airy Functionsgsl_sf_airy_zero_Bi_e: Zeros of Airy Functionsgsl_sf_angle_restrict_pos: Restriction Functionsgsl_sf_angle_restrict_pos_e: Restriction Functionsgsl_sf_angle_restrict_symm: Restriction Functionsgsl_sf_angle_restrict_symm_e: Restriction Functionsgsl_sf_atanint: Arctangent Integralgsl_sf_atanint_e: Arctangent Integralgsl_sf_bessel_I0: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_I0_e: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_i0_scaled: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_I0_scaled: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_i0_scaled_e: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_I0_scaled_e: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_I1: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_I1_e: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_i1_scaled: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_I1_scaled: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_i1_scaled_e: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_I1_scaled_e: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_i2_scaled: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_i2_scaled_e: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_il_scaled: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_il_scaled_array: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_il_scaled_e: Regular Modified Spherical Bessel Functionsgsl_sf_bessel_In: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_In_array: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_In_e: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_In_scaled: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_In_scaled_array: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_In_scaled_e: Regular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Inu: Regular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_Inu_e: Regular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_Inu_scaled: Regular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_Inu_scaled_e: Regular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_j0: Regular Spherical Bessel Functionsgsl_sf_bessel_J0: Regular Cylindrical Bessel Functionsgsl_sf_bessel_j0_e: Regular Spherical Bessel Functionsgsl_sf_bessel_J0_e: Regular Cylindrical Bessel Functionsgsl_sf_bessel_j1: Regular Spherical Bessel Functionsgsl_sf_bessel_J1: Regular Cylindrical Bessel Functionsgsl_sf_bessel_j1_e: Regular Spherical Bessel Functionsgsl_sf_bessel_J1_e: Regular Cylindrical Bessel Functionsgsl_sf_bessel_j2: Regular Spherical Bessel Functionsgsl_sf_bessel_j2_e: Regular Spherical Bessel Functionsgsl_sf_bessel_jl: Regular Spherical Bessel Functionsgsl_sf_bessel_jl_array: Regular Spherical Bessel Functionsgsl_sf_bessel_jl_e: Regular Spherical Bessel Functionsgsl_sf_bessel_jl_steed_array: Regular Spherical Bessel Functionsgsl_sf_bessel_Jn: Regular Cylindrical Bessel Functionsgsl_sf_bessel_Jn_array: Regular Cylindrical Bessel Functionsgsl_sf_bessel_Jn_e: Regular Cylindrical Bessel Functionsgsl_sf_bessel_Jnu: Regular Bessel Function - Fractional Ordergsl_sf_bessel_Jnu_e: Regular Bessel Function - Fractional Ordergsl_sf_bessel_K0: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_K0_e: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_k0_scaled: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_K0_scaled: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_k0_scaled_e: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_K0_scaled_e: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_K1: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_K1_e: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_k1_scaled: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_K1_scaled: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_k1_scaled_e: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_K1_scaled_e: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_k2_scaled: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_k2_scaled_e: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_kl_scaled: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_kl_scaled_array: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_kl_scaled_e: Irregular Modified Spherical Bessel Functionsgsl_sf_bessel_Kn: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Kn_array: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Kn_e: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Kn_scaled: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Kn_scaled_array: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Kn_scaled_e: Irregular Modified Cylindrical Bessel Functionsgsl_sf_bessel_Knu: Irregular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_Knu_e: Irregular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_Knu_scaled: Irregular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_Knu_scaled_e: Irregular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_lnKnu: Irregular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_lnKnu_e: Irregular Modified Bessel Functions - Fractional Ordergsl_sf_bessel_sequence_Jnu_e: Regular Bessel Function - Fractional Ordergsl_sf_bessel_y0: Irregular Spherical Bessel Functionsgsl_sf_bessel_Y0: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_y0_e: Irregular Spherical Bessel Functionsgsl_sf_bessel_Y0_e: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_y1: Irregular Spherical Bessel Functionsgsl_sf_bessel_Y1: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_y1_e: Irregular Spherical Bessel Functionsgsl_sf_bessel_Y1_e: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_y2: Irregular Spherical Bessel Functionsgsl_sf_bessel_y2_e: Irregular Spherical Bessel Functionsgsl_sf_bessel_yl: Irregular Spherical Bessel Functionsgsl_sf_bessel_yl_array: Irregular Spherical Bessel Functionsgsl_sf_bessel_yl_e: Irregular Spherical Bessel Functionsgsl_sf_bessel_Yn: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_Yn_array: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_Yn_e: Irregular Cylindrical Bessel Functionsgsl_sf_bessel_Ynu: Irregular Bessel Functions - Fractional Ordergsl_sf_bessel_Ynu_e: Irregular Bessel Functions - Fractional Ordergsl_sf_bessel_zero_J0: Zeros of Regular Bessel Functionsgsl_sf_bessel_zero_J0_e: Zeros of Regular Bessel Functionsgsl_sf_bessel_zero_J1: Zeros of Regular Bessel Functionsgsl_sf_bessel_zero_J1_e: Zeros of Regular Bessel Functionsgsl_sf_bessel_zero_Jnu: Zeros of Regular Bessel Functionsgsl_sf_bessel_zero_Jnu_e: Zeros of Regular Bessel Functionsgsl_sf_beta: Beta Functionsgsl_sf_beta_e: Beta Functionsgsl_sf_beta_inc: Incomplete Beta Functiongsl_sf_beta_inc_e: Incomplete Beta Functiongsl_sf_Chi: Hyperbolic Integralsgsl_sf_Chi_e: Hyperbolic Integralsgsl_sf_choose: Factorialsgsl_sf_choose_e: Factorialsgsl_sf_Ci: Trigonometric Integralsgsl_sf_Ci_e: Trigonometric Integralsgsl_sf_clausen: Clausen Functionsgsl_sf_clausen_e: Clausen Functionsgsl_sf_complex_cos_e: Trigonometric Functions for Complex Argumentsgsl_sf_complex_dilog_e: Complex Argumentgsl_sf_complex_log_e: Logarithm and Related Functionsgsl_sf_complex_logsin_e: Trigonometric Functions for Complex Argumentsgsl_sf_complex_sin_e: Trigonometric Functions for Complex Argumentsgsl_sf_conicalP_0: Conical Functionsgsl_sf_conicalP_0_e: Conical Functionsgsl_sf_conicalP_1: Conical Functionsgsl_sf_conicalP_1_e: Conical Functionsgsl_sf_conicalP_cyl_reg: Conical Functionsgsl_sf_conicalP_cyl_reg_e: Conical Functionsgsl_sf_conicalP_half: Conical Functionsgsl_sf_conicalP_half_e: Conical Functionsgsl_sf_conicalP_mhalf: Conical Functionsgsl_sf_conicalP_mhalf_e: Conical Functionsgsl_sf_conicalP_sph_reg: Conical Functionsgsl_sf_conicalP_sph_reg_e: Conical Functionsgsl_sf_cos: Circular Trigonometric Functionsgsl_sf_cos_e: Circular Trigonometric Functionsgsl_sf_cos_err_e: Trigonometric Functions With Error Estimatesgsl_sf_coulomb_CL_array: Coulomb Wave Function Normalization Constantgsl_sf_coulomb_CL_e: Coulomb Wave Function Normalization Constantgsl_sf_coulomb_wave_F_array: Coulomb Wave Functionsgsl_sf_coulomb_wave_FG_array: Coulomb Wave Functionsgsl_sf_coulomb_wave_FG_e: Coulomb Wave Functionsgsl_sf_coulomb_wave_FGp_array: Coulomb Wave Functionsgsl_sf_coulomb_wave_sphF_array: Coulomb Wave Functionsgsl_sf_coupling_3j: 3-j Symbolsgsl_sf_coupling_3j_e: 3-j Symbolsgsl_sf_coupling_6j: 6-j Symbolsgsl_sf_coupling_6j_e: 6-j Symbolsgsl_sf_coupling_9j: 9-j Symbolsgsl_sf_coupling_9j_e: 9-j Symbolsgsl_sf_dawson: Dawson Functiongsl_sf_dawson_e: Dawson Functiongsl_sf_debye_1: Debye Functionsgsl_sf_debye_1_e: Debye Functionsgsl_sf_debye_2: Debye Functionsgsl_sf_debye_2_e: Debye Functionsgsl_sf_debye_3: Debye Functionsgsl_sf_debye_3_e: Debye Functionsgsl_sf_debye_4: Debye Functionsgsl_sf_debye_4_e: Debye Functionsgsl_sf_debye_5: Debye Functionsgsl_sf_debye_5_e: Debye Functionsgsl_sf_debye_6: Debye Functionsgsl_sf_debye_6_e: Debye Functionsgsl_sf_dilog: Real Argumentgsl_sf_dilog_e: Real Argumentgsl_sf_doublefact: Factorialsgsl_sf_doublefact_e: Factorialsgsl_sf_ellint_D: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_D_e: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_E: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_E_e: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_Ecomp: Legendre Form of Complete Elliptic Integralsgsl_sf_ellint_Ecomp_e: Legendre Form of Complete Elliptic Integralsgsl_sf_ellint_F: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_F_e: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_Kcomp: Legendre Form of Complete Elliptic Integralsgsl_sf_ellint_Kcomp_e: Legendre Form of Complete Elliptic Integralsgsl_sf_ellint_P: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_P_e: Legendre Form of Incomplete Elliptic Integralsgsl_sf_ellint_RC: Carlson Formsgsl_sf_ellint_RC_e: Carlson Formsgsl_sf_ellint_RD: Carlson Formsgsl_sf_ellint_RD_e: Carlson Formsgsl_sf_ellint_RF: Carlson Formsgsl_sf_ellint_RF_e: Carlson Formsgsl_sf_ellint_RJ: Carlson Formsgsl_sf_ellint_RJ_e: Carlson Formsgsl_sf_elljac_e: Elliptic Functions (Jacobi)gsl_sf_erf: Error Functiongsl_sf_erf_e: Error Functiongsl_sf_erf_Q: Probability functionsgsl_sf_erf_Q_e: Probability functionsgsl_sf_erf_Z: Probability functionsgsl_sf_erf_Z_e: Probability functionsgsl_sf_erfc: Complementary Error Functiongsl_sf_erfc_e: Complementary Error Functiongsl_sf_eta: Eta Functiongsl_sf_eta_e: Eta Functiongsl_sf_eta_int: Eta Functiongsl_sf_eta_int_e: Eta Functiongsl_sf_exp: Exponential Functiongsl_sf_exp_e: Exponential Functiongsl_sf_exp_e10_e: Exponential Functiongsl_sf_exp_err_e: Exponentiation With Error Estimategsl_sf_exp_err_e10_e: Exponentiation With Error Estimategsl_sf_exp_mult: Exponential Functiongsl_sf_exp_mult_e: Exponential Functiongsl_sf_exp_mult_e10_e: Exponential Functiongsl_sf_exp_mult_err_e: Exponentiation With Error Estimategsl_sf_exp_mult_err_e10_e: Exponentiation With Error Estimategsl_sf_expint_3: Ei_3(x)gsl_sf_expint_3_e: Ei_3(x)gsl_sf_expint_E1: Exponential Integralgsl_sf_expint_E1_e: Exponential Integralgsl_sf_expint_E2: Exponential Integralgsl_sf_expint_E2_e: Exponential Integralgsl_sf_expint_Ei: Ei(x)gsl_sf_expint_Ei_e: Ei(x)gsl_sf_expm1: Relative Exponential Functionsgsl_sf_expm1_e: Relative Exponential Functionsgsl_sf_exprel: Relative Exponential Functionsgsl_sf_exprel_2: Relative Exponential Functionsgsl_sf_exprel_2_e: Relative Exponential Functionsgsl_sf_exprel_e: Relative Exponential Functionsgsl_sf_exprel_n: Relative Exponential Functionsgsl_sf_exprel_n_e: Relative Exponential Functionsgsl_sf_fact: Factorialsgsl_sf_fact_e: Factorialsgsl_sf_fermi_dirac_0: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_0_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_1: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_1_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_2: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_2_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_3half: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_3half_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_half: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_half_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_inc_0: Incomplete Fermi-Dirac Integralsgsl_sf_fermi_dirac_inc_0_e: Incomplete Fermi-Dirac Integralsgsl_sf_fermi_dirac_int: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_int_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_m1: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_m1_e: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_mhalf: Complete Fermi-Dirac Integralsgsl_sf_fermi_dirac_mhalf_e: Complete Fermi-Dirac Integralsgsl_sf_gamma: Gamma Functionsgsl_sf_gamma_e: Gamma Functionsgsl_sf_gamma_inc: Incomplete Gamma Functionsgsl_sf_gamma_inc_e: Incomplete Gamma Functionsgsl_sf_gamma_inc_P: Incomplete Gamma Functionsgsl_sf_gamma_inc_P_e: Incomplete Gamma Functionsgsl_sf_gamma_inc_Q: Incomplete Gamma Functionsgsl_sf_gamma_inc_Q_e: Incomplete Gamma Functionsgsl_sf_gammainv: Gamma Functionsgsl_sf_gammainv_e: Gamma Functionsgsl_sf_gammastar: Gamma Functionsgsl_sf_gammastar_e: Gamma Functionsgsl_sf_gegenpoly_1: Gegenbauer Functionsgsl_sf_gegenpoly_1_e: Gegenbauer Functionsgsl_sf_gegenpoly_2: Gegenbauer Functionsgsl_sf_gegenpoly_2_e: Gegenbauer Functionsgsl_sf_gegenpoly_3: Gegenbauer Functionsgsl_sf_gegenpoly_3_e: Gegenbauer Functionsgsl_sf_gegenpoly_array: Gegenbauer Functionsgsl_sf_gegenpoly_n: Gegenbauer Functionsgsl_sf_gegenpoly_n_e: Gegenbauer Functionsgsl_sf_hazard: Probability functionsgsl_sf_hazard_e: Probability functionsgsl_sf_hydrogenicR: Normalized Hydrogenic Bound Statesgsl_sf_hydrogenicR_1: Normalized Hydrogenic Bound Statesgsl_sf_hydrogenicR_1_e: Normalized Hydrogenic Bound Statesgsl_sf_hydrogenicR_e: Normalized Hydrogenic Bound Statesgsl_sf_hyperg_0F1: Hypergeometric Functionsgsl_sf_hyperg_0F1_e: Hypergeometric Functionsgsl_sf_hyperg_1F1: Hypergeometric Functionsgsl_sf_hyperg_1F1_e: Hypergeometric Functionsgsl_sf_hyperg_1F1_int: Hypergeometric Functionsgsl_sf_hyperg_1F1_int_e: Hypergeometric Functionsgsl_sf_hyperg_2F0: Hypergeometric Functionsgsl_sf_hyperg_2F0_e: Hypergeometric Functionsgsl_sf_hyperg_2F1: Hypergeometric Functionsgsl_sf_hyperg_2F1_conj: Hypergeometric Functionsgsl_sf_hyperg_2F1_conj_e: Hypergeometric Functionsgsl_sf_hyperg_2F1_conj_renorm: Hypergeometric Functionsgsl_sf_hyperg_2F1_conj_renorm_e: Hypergeometric Functionsgsl_sf_hyperg_2F1_e: Hypergeometric Functionsgsl_sf_hyperg_2F1_renorm: Hypergeometric Functionsgsl_sf_hyperg_2F1_renorm_e: Hypergeometric Functionsgsl_sf_hyperg_U: Hypergeometric Functionsgsl_sf_hyperg_U_e: Hypergeometric Functionsgsl_sf_hyperg_U_e10_e: Hypergeometric Functionsgsl_sf_hyperg_U_int: Hypergeometric Functionsgsl_sf_hyperg_U_int_e: Hypergeometric Functionsgsl_sf_hyperg_U_int_e10_e: Hypergeometric Functionsgsl_sf_hypot: Circular Trigonometric Functionsgsl_sf_hypot_e: Circular Trigonometric Functionsgsl_sf_hzeta: Hurwitz Zeta Functiongsl_sf_hzeta_e: Hurwitz Zeta Functiongsl_sf_laguerre_1: Laguerre Functionsgsl_sf_laguerre_1_e: Laguerre Functionsgsl_sf_laguerre_2: Laguerre Functionsgsl_sf_laguerre_2_e: Laguerre Functionsgsl_sf_laguerre_3: Laguerre Functionsgsl_sf_laguerre_3_e: Laguerre Functionsgsl_sf_laguerre_n: Laguerre Functionsgsl_sf_laguerre_n_e: Laguerre Functionsgsl_sf_lambert_W0: Lambert W Functionsgsl_sf_lambert_W0_e: Lambert W Functionsgsl_sf_lambert_Wm1: Lambert W Functionsgsl_sf_lambert_Wm1_e: Lambert W Functionsgsl_sf_legendre_array_size: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_H3d: Radial Functions for Hyperbolic Spacegsl_sf_legendre_H3d_0: Radial Functions for Hyperbolic Spacegsl_sf_legendre_H3d_0_e: Radial Functions for Hyperbolic Spacegsl_sf_legendre_H3d_1: Radial Functions for Hyperbolic Spacegsl_sf_legendre_H3d_1_e: Radial Functions for Hyperbolic Spacegsl_sf_legendre_H3d_array: Radial Functions for Hyperbolic Spacegsl_sf_legendre_H3d_e: Radial Functions for Hyperbolic Spacegsl_sf_legendre_P1: Legendre Polynomialsgsl_sf_legendre_P1_e: Legendre Polynomialsgsl_sf_legendre_P2: Legendre Polynomialsgsl_sf_legendre_P2_e: Legendre Polynomialsgsl_sf_legendre_P3: Legendre Polynomialsgsl_sf_legendre_P3_e: Legendre Polynomialsgsl_sf_legendre_Pl: Legendre Polynomialsgsl_sf_legendre_Pl_array: Legendre Polynomialsgsl_sf_legendre_Pl_deriv_array: Legendre Polynomialsgsl_sf_legendre_Pl_e: Legendre Polynomialsgsl_sf_legendre_Plm: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_Plm_array: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_Plm_deriv_array: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_Plm_e: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_Q0: Legendre Polynomialsgsl_sf_legendre_Q0_e: Legendre Polynomialsgsl_sf_legendre_Q1: Legendre Polynomialsgsl_sf_legendre_Q1_e: Legendre Polynomialsgsl_sf_legendre_Ql: Legendre Polynomialsgsl_sf_legendre_Ql_e: Legendre Polynomialsgsl_sf_legendre_sphPlm: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_sphPlm_array: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_sphPlm_deriv_array: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_legendre_sphPlm_e: Associated Legendre Polynomials and Spherical Harmonicsgsl_sf_lnbeta: Beta Functionsgsl_sf_lnbeta_e: Beta Functionsgsl_sf_lnchoose: Factorialsgsl_sf_lnchoose_e: Factorialsgsl_sf_lncosh: Hyperbolic Trigonometric Functionsgsl_sf_lncosh_e: Hyperbolic Trigonometric Functionsgsl_sf_lndoublefact: Factorialsgsl_sf_lndoublefact_e: Factorialsgsl_sf_lnfact: Factorialsgsl_sf_lnfact_e: Factorialsgsl_sf_lngamma: Gamma Functionsgsl_sf_lngamma_complex_e: Gamma Functionsgsl_sf_lngamma_e: Gamma Functionsgsl_sf_lngamma_sgn_e: Gamma Functionsgsl_sf_lnpoch: Pochhammer Symbolgsl_sf_lnpoch_e: Pochhammer Symbolgsl_sf_lnpoch_sgn_e: Pochhammer Symbolgsl_sf_lnsinh: Hyperbolic Trigonometric Functionsgsl_sf_lnsinh_e: Hyperbolic Trigonometric Functionsgsl_sf_log: Logarithm and Related Functionsgsl_sf_log_1plusx: Logarithm and Related Functionsgsl_sf_log_1plusx_e: Logarithm and Related Functionsgsl_sf_log_1plusx_mx: Logarithm and Related Functionsgsl_sf_log_1plusx_mx_e: Logarithm and Related Functionsgsl_sf_log_abs: Logarithm and Related Functionsgsl_sf_log_abs_e: Logarithm and Related Functionsgsl_sf_log_e: Logarithm and Related Functionsgsl_sf_log_erfc: Log Complementary Error Functiongsl_sf_log_erfc_e: Log Complementary Error Functiongsl_sf_multiply_e: Elementary Operationsgsl_sf_multiply_err_e: Elementary Operationsgsl_sf_poch: Pochhammer Symbolgsl_sf_poch_e: Pochhammer Symbolgsl_sf_pochrel: Pochhammer Symbolgsl_sf_pochrel_e: Pochhammer Symbolgsl_sf_polar_to_rect: Conversion Functionsgsl_sf_pow_int: Power Functiongsl_sf_pow_int_e: Power Functiongsl_sf_psi: Digamma Functiongsl_sf_psi_1: Trigamma Functiongsl_sf_psi_1_e: Trigamma Functiongsl_sf_psi_1_int: Trigamma Functiongsl_sf_psi_1_int_e: Trigamma Functiongsl_sf_psi_1piy: Digamma Functiongsl_sf_psi_1piy_e: Digamma Functiongsl_sf_psi_e: Digamma Functiongsl_sf_psi_int: Digamma Functiongsl_sf_psi_int_e: Digamma Functiongsl_sf_psi_n: Polygamma Functiongsl_sf_psi_n_e: Polygamma Functiongsl_sf_rect_to_polar: Conversion Functionsgsl_sf_Shi: Hyperbolic Integralsgsl_sf_Shi_e: Hyperbolic Integralsgsl_sf_Si: Trigonometric Integralsgsl_sf_Si_e: Trigonometric Integralsgsl_sf_sin: Circular Trigonometric Functionsgsl_sf_sin_e: Circular Trigonometric Functionsgsl_sf_sin_err_e: Trigonometric Functions With Error Estimatesgsl_sf_sinc: Circular Trigonometric Functionsgsl_sf_sinc_e: Circular Trigonometric Functionsgsl_sf_synchrotron_1: Synchrotron Functionsgsl_sf_synchrotron_1_e: Synchrotron Functionsgsl_sf_synchrotron_2: Synchrotron Functionsgsl_sf_synchrotron_2_e: Synchrotron Functionsgsl_sf_taylorcoeff: Factorialsgsl_sf_taylorcoeff_e: Factorialsgsl_sf_transport_2: Transport Functionsgsl_sf_transport_2_e: Transport Functionsgsl_sf_transport_3: Transport Functionsgsl_sf_transport_3_e: Transport Functionsgsl_sf_transport_4: Transport Functionsgsl_sf_transport_4_e: Transport Functionsgsl_sf_transport_5: Transport Functionsgsl_sf_transport_5_e: Transport Functionsgsl_sf_zeta: Riemann Zeta Functiongsl_sf_zeta_e: Riemann Zeta Functiongsl_sf_zeta_int: Riemann Zeta Functiongsl_sf_zeta_int_e: Riemann Zeta Functiongsl_sf_zetam1: Riemann Zeta Function Minus Onegsl_sf_zetam1_e: Riemann Zeta Function Minus Onegsl_sf_zetam1_int: Riemann Zeta Function Minus Onegsl_sf_zetam1_int_e: Riemann Zeta Function Minus OneGSL_SIGN: Testing the Sign of Numbersgsl_siman_solve: Simulated Annealing functionsgsl_sort: Sorting vectorsgsl_sort_index: Sorting vectorsgsl_sort_largest: Selecting the k smallest or largest elementsgsl_sort_largest_index: Selecting the k smallest or largest elementsgsl_sort_smallest: Selecting the k smallest or largest elementsgsl_sort_smallest_index: Selecting the k smallest or largest elementsgsl_sort_vector: Sorting vectorsgsl_sort_vector_index: Sorting vectorsgsl_sort_vector_largest: Selecting the k smallest or largest elementsgsl_sort_vector_largest_index: Selecting the k smallest or largest elementsgsl_sort_vector_smallest: Selecting the k smallest or largest elementsgsl_sort_vector_smallest_index: Selecting the k smallest or largest elementsgsl_spline_alloc: Higher-level Interfacegsl_spline_eval: Higher-level Interfacegsl_spline_eval_deriv: Higher-level Interfacegsl_spline_eval_deriv2: Higher-level Interfacegsl_spline_eval_deriv2_e: Higher-level Interfacegsl_spline_eval_deriv_e: Higher-level Interfacegsl_spline_eval_e: Higher-level Interfacegsl_spline_eval_integ: Higher-level Interfacegsl_spline_eval_integ_e: Higher-level Interfacegsl_spline_free: Higher-level Interfacegsl_spline_init: Higher-level Interfacegsl_spline_min_size: Higher-level Interfacegsl_spline_name: Higher-level Interfacegsl_stats_absdev: Absolute deviationgsl_stats_absdev_m: Absolute deviationgsl_stats_covariance: Covariancegsl_stats_covariance_m: Covariancegsl_stats_kurtosis: Higher moments (skewness and kurtosis)gsl_stats_kurtosis_m_sd: Higher moments (skewness and kurtosis)gsl_stats_lag1_autocorrelation: Autocorrelationgsl_stats_lag1_autocorrelation_m: Autocorrelationgsl_stats_max: Maximum and Minimum valuesgsl_stats_max_index: Maximum and Minimum valuesgsl_stats_mean: Mean and standard deviation and variancegsl_stats_median_from_sorted_data: Median and Percentilesgsl_stats_min: Maximum and Minimum valuesgsl_stats_min_index: Maximum and Minimum valuesgsl_stats_minmax: Maximum and Minimum valuesgsl_stats_minmax_index: Maximum and Minimum valuesgsl_stats_quantile_from_sorted_data: Median and Percentilesgsl_stats_sd: Mean and standard deviation and variancegsl_stats_sd_m: Mean and standard deviation and variancegsl_stats_sd_with_fixed_mean: Mean and standard deviation and variancegsl_stats_skew: Higher moments (skewness and kurtosis)gsl_stats_skew_m_sd: Higher moments (skewness and kurtosis)gsl_stats_variance: Mean and standard deviation and variancegsl_stats_variance_m: Mean and standard deviation and variancegsl_stats_variance_with_fixed_mean: Mean and standard deviation and variancegsl_stats_wabsdev: Weighted Samplesgsl_stats_wabsdev_m: Weighted Samplesgsl_stats_wkurtosis: Weighted Samplesgsl_stats_wkurtosis_m_sd: Weighted Samplesgsl_stats_wmean: Weighted Samplesgsl_stats_wsd: Weighted Samplesgsl_stats_wsd_m: Weighted Samplesgsl_stats_wsd_with_fixed_mean: Weighted Samplesgsl_stats_wskew: Weighted Samplesgsl_stats_wskew_m_sd: Weighted Samplesgsl_stats_wvariance: Weighted Samplesgsl_stats_wvariance_m: Weighted Samplesgsl_stats_wvariance_with_fixed_mean: Weighted Samplesgsl_strerror: Error Codesgsl_sum_levin_u_accel: Acceleration functionsgsl_sum_levin_u_alloc: Acceleration functionsgsl_sum_levin_u_free: Acceleration functionsgsl_sum_levin_utrunc_accel: Acceleration functions without error estimationgsl_sum_levin_utrunc_alloc: Acceleration functions without error estimationgsl_sum_levin_utrunc_free: Acceleration functions without error estimationgsl_vector_add: Vector operationsgsl_vector_add_constant: Vector operationsgsl_vector_alloc: Vector allocationgsl_vector_calloc: Vector allocationgsl_vector_complex_const_imag: Vector viewsgsl_vector_complex_const_real: Vector viewsgsl_vector_complex_imag: Vector viewsgsl_vector_complex_real: Vector viewsgsl_vector_const_ptr: Accessing vector elementsgsl_vector_const_subvector: Vector viewsgsl_vector_const_subvector_with_stride: Vector viewsgsl_vector_const_view_array: Vector viewsgsl_vector_const_view_array_with_stride: Vector viewsgsl_vector_div: Vector operationsgsl_vector_fprintf: Reading and writing vectorsgsl_vector_fread: Reading and writing vectorsgsl_vector_free: Vector allocationgsl_vector_fscanf: Reading and writing vectorsgsl_vector_fwrite: Reading and writing vectorsgsl_vector_get: Accessing vector elementsgsl_vector_isnull: Vector propertiesgsl_vector_max: Finding maximum and minimum elements of vectorsgsl_vector_max_index: Finding maximum and minimum elements of vectorsgsl_vector_memcpy: Copying vectorsgsl_vector_min: Finding maximum and minimum elements of vectorsgsl_vector_min_index: Finding maximum and minimum elements of vectorsgsl_vector_minmax: Finding maximum and minimum elements of vectorsgsl_vector_minmax_index: Finding maximum and minimum elements of vectorsgsl_vector_mul: Vector operationsgsl_vector_ptr: Accessing vector elementsgsl_vector_reverse: Exchanging elementsgsl_vector_scale: Vector operationsgsl_vector_set: Accessing vector elementsgsl_vector_set_all: Initializing vector elementsgsl_vector_set_basis: Initializing vector elementsgsl_vector_set_zero: Initializing vector elementsgsl_vector_sub: Vector operationsgsl_vector_subvector: Vector viewsgsl_vector_subvector_with_stride: Vector viewsgsl_vector_swap: Copying vectorsgsl_vector_swap_elements: Exchanging elementsgsl_vector_view_array: Vector viewsgsl_vector_view_array_with_stride: Vector viewsgsl_wavelet2d_nstransform: DWT in two dimensiongsl_wavelet2d_nstransform_forward: DWT in two dimensiongsl_wavelet2d_nstransform_inverse: DWT in two dimensiongsl_wavelet2d_nstransform_matrix: DWT in two dimensiongsl_wavelet2d_nstransform_matrix_forward: DWT in two dimensiongsl_wavelet2d_nstransform_matrix_inverse: DWT in two dimensiongsl_wavelet2d_transform: DWT in two dimensiongsl_wavelet2d_transform_forward: DWT in two dimensiongsl_wavelet2d_transform_inverse: DWT in two dimensiongsl_wavelet2d_transform_matrix: DWT in two dimensiongsl_wavelet2d_transform_matrix_forward: DWT in two dimensiongsl_wavelet2d_transform_matrix_inverse: DWT in two dimensiongsl_wavelet_alloc: DWT Initializationgsl_wavelet_bspline: DWT Initializationgsl_wavelet_bspline_centered: DWT Initializationgsl_wavelet_daubechies: DWT Initializationgsl_wavelet_daubechies_centered: DWT Initializationgsl_wavelet_free: DWT Initializationgsl_wavelet_haar: DWT Initializationgsl_wavelet_haar_centered: DWT Initializationgsl_wavelet_name: DWT Initializationgsl_wavelet_transform: DWT in one dimensiongsl_wavelet_transform_forward: DWT in one dimensiongsl_wavelet_transform_inverse: DWT in one dimensiongsl_wavelet_workspace_alloc: DWT Initializationgsl_wavelet_workspace_free: DWT Initializationalpha: VEGASalpha: MISERchisq: VEGASdither: MISERestimate_frac: MISERiterations: VEGASmin_calls: MISERmin_calls_per_bisection: MISERmode: VEGASostream: VEGASresult: VEGASsigma: VEGASstage: VEGASverbose: VEGASgsl_error_handler_t: Error Handlersgsl_fft_complex_wavetable: Mixed-radix FFT routines for complex datagsl_function: Providing the function to solvegsl_function_fdf: Providing the function to solvegsl_histogram: The histogram structgsl_histogram2d: The 2D histogram structgsl_histogram2d_pdf: Resampling from 2D histogramsgsl_histogram_pdf: The histogram probability distribution structgsl_monte_function: Monte Carlo Interfacegsl_multifit_function: Providing the Function to be Minimizedgsl_multifit_function_fdf: Providing the Function to be Minimizedgsl_multimin_function: Providing a function to minimizegsl_multimin_function_fdf: Providing a function to minimizegsl_multiroot_function: Providing the multidimensional system of equations to solvegsl_multiroot_function_fdf: Providing the multidimensional system of equations to solvegsl_odeiv_system: Defining the ODE Systemgsl_siman_copy_construct_t: Simulated Annealing functionsgsl_siman_copy_t: Simulated Annealing functionsgsl_siman_destroy_t: Simulated Annealing functionsgsl_siman_Efunc_t: Simulated Annealing functionsgsl_siman_metric_t: Simulated Annealing functionsgsl_siman_params_t: Simulated Annealing functionsgsl_siman_print_t: Simulated Annealing functionsgsl_siman_step_t: Simulated Annealing functions$, shell prompt: Conventions used in this manual$, shell prompt: Conventions used in this manual