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This file documents the GNU Scientific Library (GSL), a collection of numerical routines for scientific computing. It corresponds to release 2.2.1 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.
A Japanese translation of this manual is available from the GSL project homepage thanks to Daisuke Tominaga.
Copyright © 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 The GSL Team.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with the Invariant Sections being “GNU General Public License” and “Free Software Needs Free Documentation”, the Front-Cover text being “A GNU Manual”, and with the Back-Cover Text being (a) (see below). A copy of the license is included in the section entitled “GNU Free Documentation License”.
(a) The Back-Cover Text is: “You have the freedom to copy and modify this GNU Manual.”
Next: Using the library, Previous: Top, Up: Top [Index]
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.
• Routines available in GSL: | ||
• GSL is Free Software: | ||
• Obtaining GSL: | ||
• No Warranty: | ||
• Reporting Bugs: | ||
• Further Information: | ||
• Conventions used in this manual: |
Next: GSL is Free Software, Up: Introduction [Index]
The library covers a wide range of topics in numerical computing. Routines are available for the following areas,
Complex Numbers | Roots of Polynomials | |
Special Functions | Vectors and Matrices | |
Permutations | Combinations | |
Sorting | BLAS Support | |
Linear Algebra | CBLAS Library | |
Fast Fourier Transforms | Eigensystems | |
Random Numbers | Quadrature | |
Random Distributions | Quasi-Random Sequences | |
Histograms | Statistics | |
Monte Carlo Integration | N-Tuples | |
Differential Equations | Simulated Annealing | |
Numerical Differentiation | Interpolation | |
Series Acceleration | Chebyshev Approximations | |
Root-Finding | Discrete Hankel Transforms | |
Least-Squares Fitting | Minimization | |
IEEE Floating-Point | Physical Constants | |
Basis Splines | Wavelets |
The use of these routines is described in this manual. Each chapter provides detailed definitions of the functions, followed by example programs and references to the articles on which the algorithms are based.
Where possible the routines have been based on reliable public-domain packages such as FFTPACK and QUADPACK, which the developers of GSL have reimplemented in C with modern coding conventions.
Next: Obtaining GSL, Previous: Routines available in GSL, Up: Introduction [Index]
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/).
Next: No Warranty, Previous: GSL is Free Software, Up: Introduction [Index]
The source code for the library can be obtained in different ways, by copying it from a friend, purchasing it on CDROM or downloading it from the internet. A list of public ftp servers which carry the source code can be found on the GNU website,
The preferred platform for the library is a GNU system, which allows it to take advantage of additional features in the GNU C compiler and GNU C library. However, the library is fully portable and should compile on most systems with a C compiler.
Announcements of new releases, updates and other relevant events are
made on the info-gsl@gnu.org
mailing list. To subscribe to this
low-volume list, send an email of the following form:
To: info-gsl-request@gnu.org Subject: subscribe
You will receive a response asking you to reply in order to confirm your subscription.
Next: Reporting Bugs, Previous: Obtaining GSL, Up: Introduction [Index]
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).
Next: Further Information, Previous: No Warranty, Up: Introduction [Index]
A list of known bugs can be found in the BUGS file included in the GSL distribution or online in the GSL bug tracker.^{1} Details of compilation problems can be found in the INSTALL file.
If you find a bug which is not listed in these files, please report it to 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.
Next: Conventions used in this manual, Previous: Reporting Bugs, Up: Introduction [Index]
Additional information, including online copies of this manual, links to related projects, and mailing list archives are available from the website mentioned above.
Any questions about the use and installation of the library can be asked
on the mailing list help-gsl@gnu.org
. To subscribe to this
list, send an email of the following form:
To: help-gsl-request@gnu.org Subject: subscribe
This mailing list can be used to ask questions not covered by this manual, and to contact the developers of the library.
If you would like to refer to the GNU Scientific Library in a journal article, the recommended way is to cite this reference manual, e.g. M. Galassi et al, GNU Scientific Library Reference Manual (3rd Ed.), ISBN 0954612078.
If you want to give a url, use “http://www.gnu.org/software/gsl/”.
Previous: Further Information, Up: Introduction [Index]
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
).
Next: Error Handling, Previous: Introduction, Up: Top [Index]
This chapter describes how to compile programs that use GSL, and introduces its conventions.
Next: Compiling and Linking, Up: Using the library [Index]
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,^{2}
J0(5) = -1.775967713143382642e-01
The steps needed to compile this program are described in the following sections.
Next: Shared Libraries, Previous: An Example Program, Up: Using the library [Index]
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.
• Linking programs with the library: | ||
• Linking with an alternative BLAS library: |
The library is installed as a single file, libgsl.a. A shared version of the library libgsl.so is also installed on systems that support shared libraries. The default location of these files is /usr/local/lib. If this directory is not on the standard search path of your linker you will also need to provide its location as a command line flag.
To link against the library you need to specify both the main library and a supporting CBLAS library, which provides standard basic linear algebra subroutines. A suitable CBLAS implementation is provided in the library libgslcblas.a if your system does not provide one. The following example shows how to link an application with the library,
$ gcc -L/usr/local/lib example.o -lgsl -lgslcblas -lm
The default library path for gcc
searches /usr/local/lib
automatically so the -L
option can be omitted when GSL is
installed in its default location.
The option -lm
links with the system math library. On some
systems it is not needed.^{3}
For a tutorial introduction to the GNU C Compiler and related programs, see An Introduction to GCC (ISBN 0954161793).^{4}
Previous: Linking programs with the library, Up: Compiling and Linking [Index]
The following command line shows how you would link the same application with an alternative CBLAS library libcblas.a,
$ gcc example.o -lgsl -lcblas -lm
For the best performance an optimized platform-specific CBLAS
library should be used for -lcblas
. The library must conform to
the CBLAS standard. The ATLAS package provides a portable
high-performance BLAS library with a CBLAS interface. It is
free software and should be installed for any work requiring fast vector
and matrix operations. The following command line will link with the
ATLAS library and its CBLAS interface,
$ gcc example.o -lgsl -lcblas -latlas -lm
If the ATLAS library is installed in a non-standard directory use
the -L
option to add it to the search path, as described above.
For more information about BLAS functions see BLAS Support.
Next: ANSI C Compliance, Previous: Compiling and Linking, Up: Using the library [Index]
To run a program linked with the shared version of the library the operating system must be able to locate the corresponding .so file at runtime. If the library cannot be found, the following error will occur:
$ ./a.out ./a.out: error while loading shared libraries: libgsl.so.0: cannot open shared object file: No such file or directory
To avoid this error, either modify the system dynamic linker
configuration^{5} or
define the shell variable LD_LIBRARY_PATH
to include the
directory where the library is installed.
For example, in the Bourne shell (/bin/sh
or /bin/bash
),
the library search path can be set with the following commands:
$ LD_LIBRARY_PATH=/usr/local/lib $ export LD_LIBRARY_PATH $ ./example
In the C-shell (/bin/csh
or /bin/tcsh
) the equivalent
command is,
% setenv LD_LIBRARY_PATH /usr/local/lib
The standard prompt for the C-shell in the example above is the percent character ‘%’, and should not be typed as part of the command.
To save retyping these commands each session they can be placed in an individual or system-wide login file.
To compile a statically linked version of the program, use the
-static
flag in gcc
,
$ gcc -static example.o -lgsl -lgslcblas -lm
Next: Inline functions, Previous: Shared Libraries, Up: Using the library [Index]
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_
.
Next: Long double, Previous: ANSI C Compliance, Up: Using the library [Index]
The inline
keyword is not part of the original ANSI C standard
(C89) so the library does not export any inline function definitions
by default. Inline functions were introduced officially in the newer
C99 standard but most C89 compilers have also included inline
as
an extension for a long time.
To allow the use of inline functions, the library provides optional
inline versions of performance-critical routines by conditional
compilation in the exported header files. The inline versions of these
functions can be included by defining the macro HAVE_INLINE
when compiling an application,
$ gcc -Wall -c -DHAVE_INLINE example.c
If you use autoconf
this macro can be defined automatically. If
you do not define the macro HAVE_INLINE
then the slower
non-inlined versions of the functions will be used instead.
By default, the actual form of the inline keyword is extern
inline
, which is a gcc
extension that eliminates unnecessary
function definitions. If the form extern inline
causes
problems with other compilers a stricter autoconf test can be used,
see Autoconf Macros.
When compiling with gcc
in C99 mode (gcc -std=c99
) the
header files automatically switch to C99-compatible inline function
declarations instead of extern inline
. With other C99
compilers, define the macro GSL_C99_INLINE
to use these
declarations.
Next: Portability functions, Previous: Inline functions, Up: Using the library [Index]
In general, the algorithms in the library are written for double
precision only. The long double
type is not supported for
actual computation.
One reason for this choice is that the precision of long double
is platform dependent. The IEEE standard only specifies the minimum
precision of extended precision numbers, while the precision of
double
is the same on all platforms.
However, it is sometimes necessary to interact with external data in long-double format, so the vector and matrix datatypes include long-double versions.
It should be noted that in some system libraries the stdio.h
formatted input/output functions printf
and scanf
are
not implemented correctly for long double
. Undefined or
incorrect results are avoided by testing these functions during the
configure
stage of library compilation and eliminating certain
GSL functions which depend on them if necessary. The corresponding
line in the configure
output looks like this,
checking whether printf works with long double... no
Consequently when long double
formatted input/output does not
work on a given system it should be impossible to link a program which
uses GSL functions dependent on this.
If it is necessary to work on a system which does not support formatted
long double
input/output then the options are to use binary
formats or to convert long double
results into double
for
reading and writing.
Next: Alternative optimized functions, Previous: Long double, Up: Using the library [Index]
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.
Next: Support for different numeric types, Previous: Portability functions, Up: Using the library [Index]
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.
Next: Compatibility with C++, Previous: Alternative optimized functions, Up: Using the library [Index]
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
Next: Aliasing of arrays, Previous: Support for different numeric types, Up: Using the library [Index]
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.
Next: Thread-safety, Previous: Compatibility with C++, Up: Using the library [Index]
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.
Next: Deprecated Functions, Previous: Aliasing of arrays, Up: Using the library [Index]
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.
Next: Code Reuse, Previous: Thread-safety, Up: Using the library [Index]
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.
Previous: Deprecated Functions, Up: Using the library [Index]
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.
Next: Mathematical Functions, Previous: Using the library, Up: Top [Index]
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.
• Error Reporting: | ||
• Error Codes: | ||
• Error Handlers: | ||
• Using GSL error reporting in your own functions: | ||
• Error Reporting Examples: |
Next: Error Codes, Up: Error Handling [Index]
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.
Next: Error Handlers, Previous: Error Reporting, Up: Error Handling [Index]
The error code numbers returned by library functions are defined in
the file gsl_errno.h. They all have the prefix GSL_
and
expand to non-zero constant integer values. Error codes above 1024 are
reserved for applications, and are not used by the library. Many of
the error codes use the same base name as the corresponding error code
in the C library. Here are some of the most common error codes,
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 error
for a
status value of GSL_ERANGE
.
Next: Using GSL error reporting in your own functions, Previous: Error Codes, Up: Error Handling [Index]
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 type void
.
Error handler functions should be defined like this,
void handler (const char * reason, const char * file, int line, int gsl_errno)
To request the use of your own error handler you need to call the
function gsl_set_error_handler
which is also declared in
gsl_errno.h,
This function sets a new error handler, new_handler, for the GSL library routines. The previous handler is returned (so that you can restore it later). Note that the pointer to a user defined error handler function is stored in a static variable, so there can be only one error handler per program. This function should be not be used in multi-threaded programs except to set up a program-wide error handler from a master thread. The following example shows how to set and restore a new error handler,
/* save original handler, install new handler */ old_handler = gsl_set_error_handler (&my_handler); /* code uses new handler */ ..... /* restore original handler */ gsl_set_error_handler (old_handler);
To use the default behavior (abort
on error) set the error
handler to NULL
,
old_handler = gsl_set_error_handler (NULL);
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.
Next: Error Reporting Examples, Previous: Error Handlers, Up: Error Handling [Index]
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_ERROR
but returns a user-defined
value of value instead of an error code. It can be used for
mathematical functions that return a floating point value.
The following example shows how to return a NaN
at a mathematical
singularity using the GSL_ERROR_VAL
macro,
if (x == 0) { GSL_ERROR_VAL("argument lies on singularity", GSL_ERANGE, GSL_NAN); }
Previous: Using GSL error reporting in your own functions, Up: Error Handling [Index]
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, stride, n); if (status) { if (status == GSL_EINVAL) { fprintf (stderr, "invalid argument, n=%d\n", n); } else { fprintf (stderr, "failed, gsl_errno=%d\n", status); } exit (-1); } ...
The function gsl_fft_complex_radix2
only accepts integer lengths
which are a power of two. If the variable n
is not a power of
two then the call to the library function will return GSL_EINVAL
,
indicating that the length argument is invalid. The function call to
gsl_set_error_handler_off
stops the default error handler from
aborting the program. The else
clause catches any other possible
errors.
Next: Complex Numbers, Previous: Error Handling, Up: Top [Index]
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.
Next: Infinities and Not-a-number, Up: Mathematical Functions [Index]
The library ensures that the standard BSD mathematical constants are defined. For reference, here is a list of the constants:
M_E
The base of exponentials, e
M_LOG2E
The base-2 logarithm of e, \log_2 (e)
M_LOG10E
The base-10 logarithm of e, \log_10 (e)
M_SQRT2
The square root of two, \sqrt 2
M_SQRT1_2
The square root of one-half, \sqrt{1/2}
M_SQRT3
The square root of three, \sqrt 3
M_PI
The constant pi, \pi
M_PI_2
Pi divided by two, \pi/2
M_PI_4
Pi divided by four, \pi/4
M_SQRTPI
The square root of pi, \sqrt\pi
M_2_SQRTPI
Two divided by the square root of pi, 2/\sqrt\pi
M_1_PI
The reciprocal of pi, 1/\pi
M_2_PI
Twice the reciprocal of pi, 2/\pi
M_LN10
The natural logarithm of ten, \ln(10)
M_LN2
The natural logarithm of two, \ln(2)
M_LNPI
The natural logarithm of pi, \ln(\pi)
M_EULER
Euler’s constant, \gamma
Next: Elementary Functions, Previous: Mathematical Constants, Up: Mathematical Functions [Index]
This 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 ratio 0.0/0.0
.
This function returns 1 if x is not-a-number.
This function returns +1 if x is positive infinity, -1 if x is negative infinity and 0 otherwise.^{6}
This function returns 1 if x is a real number, and 0 if it is infinite or not-a-number.
Next: Small integer powers, Previous: Infinities and Not-a-number, Up: Mathematical Functions [Index]
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 \sqrt{x^2 + y^2 + z^2} in a way that avoids overflow.
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)
.
Next: Testing the Sign of Numbers, Previous: Elementary Functions, Up: Mathematical Functions [Index]
A common complaint about the standard C library is its lack of a function for calculating (small) integer powers. GSL provides some simple functions to fill this gap. For reasons of efficiency, these functions do not check for overflow or underflow conditions.
These routines computes the power x^n for integer n. The
power is computed efficiently—for example, x^8 is computed as
((x^2)^2)^2, requiring only 3 multiplications. A version of this
function which also computes the numerical error in the result is
available as gsl_sf_pow_int_e
.
These functions can be used to compute small integer powers x^2,
x^3, etc. efficiently. The functions will be inlined when
HAVE_INLINE
is defined, so that use of these functions
should be as efficient as explicitly writing the corresponding
product expression.
#include <gsl/gsl_math.h> double y = gsl_pow_4 (3.141) /* compute 3.141**4 */
Next: Testing for Odd and Even Numbers, Previous: Small integer powers, Up: Mathematical Functions [Index]
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).
Next: Maximum and Minimum functions, Previous: Testing the Sign of Numbers, Up: Mathematical Functions [Index]
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.
Next: Approximate Comparison of Floating Point Numbers, Previous: Testing for Odd and Even Numbers, Up: Mathematical Functions [Index]
Note that the following macros perform multiple evaluations of their arguments, so they should not be used with arguments that have side effects (such as a call to a random number generator).
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_MAX
will 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_MIN
will 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_MAX
or GSL_MIN
will 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_MAX
or GSL_MIN
will be automatically substituted.
Previous: Maximum and Minimum functions, Up: Mathematical Functions [Index]
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.
Note that x and y are compared to relative accuracy, so this function is not suitable for testing whether a value is approximately zero.
The implementation is based on the package fcmp
by T.C. Belding.
Next: Polynomials, Previous: Mathematical Functions, Up: Top [Index]
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)^{7} 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.
Next: Properties of complex numbers, Up: Complex Numbers [Index]
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. An inline version of this function is used when HAVE_INLINE
is defined.
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.
Next: Complex arithmetic operators, Previous: Representation of complex numbers, Up: Complex Numbers [Index]
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.
Next: Elementary Complex Functions, Previous: Properties of complex numbers, Up: Complex Numbers [Index]
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).
Next: Complex Trigonometric Functions, Previous: Complex arithmetic operators, Up: Complex Numbers [Index]
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).
Next: Inverse Complex Trigonometric Functions, Previous: Elementary Complex Functions, Up: Complex Numbers [Index]
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).
Next: Complex Hyperbolic Functions, Previous: Complex Trigonometric Functions, Up: Complex Numbers [Index]
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).
Next: Inverse Complex Hyperbolic Functions, Previous: Inverse Complex Trigonometric Functions, Up: Complex Numbers [Index]
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).
Next: Complex Number References and Further Reading, Previous: Complex Hyperbolic Functions, Up: Complex Numbers [Index]
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. Note that in this case we use the negative square root in formula 4.6.21 of Abramowitz & Stegun giving \arccosh(z)=\log(z-\sqrt{z^2-1}).
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).
Previous: Inverse Complex Hyperbolic Functions, Up: Complex Numbers [Index]
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,
Next: Special Functions, Previous: Complex Numbers, Up: Top [Index]
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.
The functions described here evaluate the polynomial
P(x) = c[0] + c[1] x + c[2] x^2 + \dots + c[len-1] x^{len-1} using
Horner’s method for stability. Inline versions of these functions are used when HAVE_INLINE
is defined.
This function evaluates a polynomial with real coefficients for the real variable x.
This function evaluates a polynomial with real coefficients for the complex variable z.
This function evaluates a polynomial with complex coefficients for the complex variable z.
This function evaluates a polynomial and its derivatives storing the results in the array res of size lenres. The output array contains the values of d^k P/d x^k for the specified value of x starting with k = 0.
Next: Quadratic Equations, Previous: Polynomial Evaluation, Up: Polynomials [Index]
The functions described here manipulate polynomials stored in Newton’s divided-difference representation. The use of divided-differences is described in Abramowitz & Stegun sections 25.1.4 and 25.2.26, and Burden and Faires, chapter 3, and discussed briefly below.
Given a function f(x), an nth degree interpolating polynomial P_{n}(x) can be constructed which agrees with f at n+1 distinct points x_0,x_1,...,x_{n}. This polynomial can be written in a form known as Newton’s divided-difference representation:
P_n(x) = f(x_0) + \sum_(k=1)^n [x_0,x_1,...,x_k] (x-x_0)(x-x_1)...(x-x_(k-1))
where the divided differences [x_0,x_1,...,x_k] are defined in section 25.1.4 of Abramowitz and Stegun. Additionally, it is possible to construct an interpolating polynomial of degree 2n+1 which also matches the first derivatives of f at the points x_0,x_1,...,x_n. This is called the Hermite interpolating polynomial and is defined as
H_(2n+1)(x) = f(z_0) + \sum_(k=1)^(2n+1) [z_0,z_1,...,z_k] (x-z_0)(x-z_1)...(x-z_(k-1))
where the elements of z = \{x_0,x_0,x_1,x_1,...,x_n,x_n\} are defined by z_{2k} = z_{2k+1} = x_k. The divided-differences [z_0,z_1,...,z_k] are discussed in Burden and Faires, section 3.4.
This function computes a divided-difference representation of the interpolating polynomial for the points (x, y) stored in the arrays xa and ya of length size. On output the divided-differences of (xa,ya) are stored in the array dd, also of length size. Using the notation above, dd[k] = [x_0,x_1,...,x_k].
This function evaluates the polynomial stored in divided-difference form
in the arrays dd and xa of length size at the point
x. An inline version of this function is used when HAVE_INLINE
is defined.
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 computes a divided-difference representation of the
interpolating Hermite polynomial for the points (x, y) stored in
the arrays xa and ya of length size. Hermite interpolation
constructs polynomials which also match first derivatives dy/dx which are
provided in the array dya also of length size. The first derivatives can be
incorported into the usual divided-difference algorithm by forming a new
dataset z = \{x_0,x_0,x_1,x_1,...\}, which is stored in the array
za of length 2*size on output. On output the
divided-differences of the Hermite representation are stored in the array
dd, also of length 2*size. Using the notation above,
dd[k] = [z_0,z_1,...,z_k]. The resulting Hermite polynomial
can be evaluated by calling gsl_poly_dd_eval
and using
za for the input argument xa.
Next: Cubic Equations, Previous: Divided Difference Representation of Polynomials, Up: Polynomials [Index]
This function finds the real roots of the quadratic equation,
a x^2 + b x + c = 0
The number of real roots (either zero, one or two) is returned, and their locations are stored in x0 and x1. If no real roots are found then x0 and x1 are not modified. If one real root is found (i.e. if a=0) then it is stored in x0. When two real roots are found they are stored in x0 and x1 in ascending order. The case of coincident roots is not considered special. For example (x-1)^2=0 will have two roots, which happen to have exactly equal values.
The number of roots found depends on the sign of the discriminant b^2 - 4 a c. This will be subject to rounding and cancellation errors when computed in double precision, and will also be subject to errors if the coefficients of the polynomial are inexact. These errors may cause a discrete change in the number of roots. However, for polynomials with small integer coefficients the discriminant can always be computed exactly.
This function finds the complex roots of the quadratic equation,
a z^2 + b z + c = 0
The number of complex roots is returned (either one or two) and the locations of the roots are stored in z0 and z1. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components. If only one real root is found (i.e. if a=0) then it is stored in z0.
Next: General Polynomial Equations, Previous: Quadratic Equations, Up: Polynomials [Index]
This function finds the real roots of the cubic equation,
x^3 + a x^2 + b x + c = 0
with a leading coefficient of unity. The number of real roots (either one or three) is returned, and their locations are stored in x0, x1 and x2. If one real root is found then only x0 is modified. When three real roots are found they are stored in x0, x1 and x2 in ascending order. The case of coincident roots is not considered special. For example, the equation (x-1)^3=0 will have three roots with exactly equal values. As in the quadratic case, finite precision may cause equal or closely-spaced real roots to move off the real axis into the complex plane, leading to a discrete change in the number of real roots.
This function finds the complex roots of the cubic equation,
z^3 + a z^2 + b z + c = 0
The number of complex roots is returned (always three) and the locations of the roots are stored in z0, z1 and z2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.
Next: Roots of Polynomials Examples, Previous: Cubic Equations, Up: Polynomials [Index]
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_workspace
struct and a workspace suitable for solving a polynomial with n
coefficients using the routine gsl_poly_complex_solve
.
The function returns a pointer to the newly allocated
gsl_poly_complex_workspace
if no errors were detected, and a null
pointer in the case of error.
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_SUCCESS
if all the roots are found. If
the QR reduction does not converge, the error handler is invoked with
an error code of GSL_EFAILED
. Note that due to finite precision,
roots of higher multiplicity are returned as a cluster of simple roots
with reduced accuracy. The solution of polynomials with higher-order
roots requires specialized algorithms that take the multiplicity
structure into account (see e.g. Z. Zeng, Algorithm 835, ACM
Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp
218–236).
Next: Roots of Polynomials References and Further Reading, Previous: General Polynomial Equations, Up: Polynomials [Index]
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 = -0.809016994374947673 +0.587785252292473359 z1 = -0.809016994374947673 -0.587785252292473359 z2 = +0.309016994374947507 +0.951056516295152976 z3 = +0.309016994374947507 -0.951056516295152976 z4 = +0.999999999999999889 +0.000000000000000000
which agrees with the analytic result, z_n = \exp(2 \pi n i/5).
Previous: Roots of Polynomials Examples, Up: Polynomials [Index]
The balanced-QR method and its error analysis are described in the following papers,
The formulas for divided differences are given in the following texts,
Next: Vectors and Matrices, Previous: Polynomials, Up: Top [Index]
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.
Next: The gsl_sf_result struct, Up: Special Functions [Index]
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
.
Next: Special Function Modes, Previous: Special Function Usage, Up: Special Functions [Index]
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;
Next: Airy Functions and Derivatives, Previous: The gsl_sf_result struct, Up: Special Functions [Index]
The goal of the library is to achieve double precision accuracy wherever
possible. However the cost of evaluating some special functions to
double precision can be significant, particularly where very high order
terms are required. In these cases a mode
argument allows the
accuracy of the function to be reduced in order to improve performance.
The following precision levels are available for the mode argument,
GSL_PREC_DOUBLE
Double-precision, a relative accuracy of approximately 2 * 10^-16.
GSL_PREC_SINGLE
Single-precision, a relative accuracy of approximately 10^-7.
GSL_PREC_APPROX
Approximate values, a relative accuracy of approximately 5 * 10^-4.
The approximate mode provides the fastest evaluation at the lowest accuracy.
Next: Bessel Functions, Previous: Special Function Modes, Up: Special Functions [Index]
The Airy functions Ai(x) and Bi(x) are defined by the integral representations,
Ai(x) = (1/\pi) \int_0^\infty \cos((1/3) t^3 + xt) dt Bi(x) = (1/\pi) \int_0^\infty (e^(-(1/3) t^3 + xt) + \sin((1/3) t^3 + xt)) dt
For further information see Abramowitz & Stegun, Section 10.4. The Airy functions are defined in the header file gsl_sf_airy.h.
• Airy Functions: | ||
• Derivatives of Airy Functions: | ||
• Zeros of Airy Functions: | ||
• Zeros of Derivatives of Airy Functions: |
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.
Next: Zeros of Airy Functions, Previous: Airy Functions, Up: Airy Functions and Derivatives [Index]
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.
Next: Zeros of Derivatives of Airy Functions, Previous: Derivatives of Airy Functions, Up: Airy Functions and Derivatives [Index]
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).
Previous: Zeros of Airy Functions, Up: Airy Functions and Derivatives [Index]
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).
Next: Clausen Functions, Previous: Airy Functions and Derivatives, Up: Special Functions [Index]
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.
Next: Irregular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Regular Modified Cylindrical Bessel Functions, Previous: Regular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Irregular Modified Cylindrical Bessel Functions, Previous: Irregular Cylindrical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Regular Spherical Bessel Functions, Previous: Regular Modified Cylindrical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Irregular Spherical Bessel Functions, Previous: Irregular Modified Cylindrical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Regular Modified Spherical Bessel Functions, Previous: Regular Spherical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Irregular Modified Spherical Bessel Functions, Previous: Irregular Spherical Bessel Functions, Up: Bessel Functions [Index]
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 spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to lmax inclusive for lmax >= 0, storing the results in the array result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values.
Next: Regular Bessel Function - Fractional Order, Previous: Regular Modified Spherical Bessel Functions, Up: Bessel Functions [Index]
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.
Next: Irregular Bessel Functions - Fractional Order, Previous: Irregular Modified Spherical Bessel Functions, Up: Bessel Functions [Index]
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).
Next: Regular Modified Bessel Functions - Fractional Order, Previous: Regular Bessel Function - Fractional Order, Up: Bessel Functions [Index]
These routines compute the irregular cylindrical Bessel function of fractional order \nu, Y_\nu(x).
Next: Irregular Modified Bessel Functions - Fractional Order, Previous: Irregular Bessel Functions - Fractional Order, Up: Bessel Functions [Index]
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.
Next: Zeros of Regular Bessel Functions, Previous: Regular Modified Bessel Functions - Fractional Order, Up: Bessel Functions [Index]
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.
Previous: Irregular Modified Bessel Functions - Fractional Order, Up: Bessel Functions [Index]
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.
Next: Coulomb Functions, Previous: Bessel Functions, Up: Special Functions [Index]
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).
Next: Coupling Coefficients, Previous: Clausen Functions, Up: Special Functions [Index]
The prototypes of the Coulomb functions are declared in the header file gsl_sf_coulomb.h. Both bound state and scattering solutions are available.
• Normalized Hydrogenic Bound States: | ||
• Coulomb Wave Functions: | ||
• Coulomb Wave Function Normalization Constant: |
Next: Coulomb Wave Functions, Up: Coulomb Functions [Index]
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}.
Next: Coulomb Wave Function Normalization Constant, Previous: Normalized Hydrogenic Bound States, Up: Coulomb Functions [Index]
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_EOVRFLW
is 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.
Previous: Coulomb Wave Functions, Up: Coulomb Functions [Index]
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.
Next: Dawson Function, Previous: Coulomb Functions, Up: Special Functions [Index]
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.
• 3-j Symbols: | ||
• 6-j Symbols: | ||
• 9-j Symbols: |
Next: 6-j Symbols, Up: Coupling Coefficients [Index]
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.
Next: 9-j Symbols, Previous: 3-j Symbols, Up: Coupling Coefficients [Index]
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.
Previous: 6-j Symbols, Up: Coupling Coefficients [Index]
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.
Next: Debye Functions, Previous: Coupling Coefficients, Up: Special Functions [Index]
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.
Next: Dilogarithm, Previous: Dawson Function, Up: Special Functions [Index]
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 sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).
Next: Elementary Operations, Previous: Debye Functions, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_dilog.h.
• Real Argument: | ||
• Complex Argument: |
Next: Complex Argument, Up: Dilogarithm [Index]
These routines compute the dilogarithm for a real argument. In Lewin’s notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for x <= 1, and -\pi\log(x) for x > 1.
Note that Abramowitz & Stegun refer to the Spence integral S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x).
Previous: Real Argument, Up: Dilogarithm [Index]
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.
Next: Elliptic Integrals, Previous: Dilogarithm, Up: Special Functions [Index]
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.
Next: Elliptic Functions (Jacobi), Previous: Elementary Operations, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_ellint.h. Further information about the elliptic integrals can be found in Abramowitz & Stegun, Chapter 17.
• Definition of Legendre Forms: | ||
• Definition of Carlson Forms: | ||
• Legendre Form of Complete Elliptic Integrals: | ||
• Legendre Form of Incomplete Elliptic Integrals: | ||
• Carlson Forms: |
Next: Definition of Carlson Forms, Up: Elliptic Integrals [Index]
The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and \Pi(\phi,k,n) are defined by,
F(\phi,k) = \int_0^\phi dt 1/\sqrt((1 - k^2 \sin^2(t))) E(\phi,k) = \int_0^\phi dt \sqrt((1 - k^2 \sin^2(t))) Pi(\phi,k,n) = \int_0^\phi dt 1/((1 + n \sin^2(t))\sqrt(1 - k^2 \sin^2(t)))
The complete Legendre forms are denoted by K(k) = F(\pi/2, k) and E(k) = E(\pi/2, k).
The notation used here is based on Carlson, Numerische Mathematik 33 (1979) 1 and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter m = k^2 and n is replaced by -n.
Next: Legendre Form of Complete Elliptic Integrals, Previous: Definition of Legendre Forms, Up: Elliptic Integrals [Index]
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)
Next: Legendre Form of Incomplete Elliptic Integrals, Previous: Definition of Carlson Forms, Up: Elliptic Integrals [Index]
These routines compute the complete elliptic integral K(k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the complete elliptic integral \Pi(k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.
Next: Carlson Forms, Previous: Legendre Form of Complete Elliptic Integrals, Up: Elliptic Integrals [Index]
These routines compute the incomplete elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the incomplete elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2.
These routines compute the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n.
These functions compute the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation,
D(\phi,k) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1).
Previous: Legendre Form of Incomplete Elliptic Integrals, Up: Elliptic Integrals [Index]
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.
Next: Error Functions, Previous: Elliptic Integrals, Up: Special Functions [Index]
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.
Next: Exponential Functions, Previous: Elliptic Functions (Jacobi), Up: Special Functions [Index]
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.
• Error Function: | ||
• Complementary Error Function: | ||
• Log Complementary Error Function: | ||
• Probability functions: |
Next: Complementary Error Function, Up: Error Functions [Index]
These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).
Next: Log Complementary Error Function, Previous: Error Function, Up: Error Functions [Index]
These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).
Next: Probability functions, Previous: Complementary Error Function, Up: Error Functions [Index]
These routines compute the logarithm of the complementary error function \log(\erfc(x)).
Previous: Log Complementary Error Function, Up: Error Functions [Index]
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 Mills’ ratio, is defined as,
h(x) = Z(x)/Q(x) = \sqrt{2/\pi} \exp(-x^2 / 2) / \erfc(x/\sqrt 2)
It decreases rapidly as x approaches -\infty and asymptotes to h(x) \sim x as x approaches +\infty.
These routines compute the hazard function for the normal distribution.
Next: Exponential Integrals, Previous: Error Functions, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_exp.h.
• Exponential Function: | ||
• Relative Exponential Functions: | ||
• Exponentiation With Error Estimate: |
Next: Relative Exponential Functions, Up: Exponential Functions [Index]
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_e10
type to return a result with extended range.
This function may be useful if the value of \exp(x) would
overflow the numeric range of double
.
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_e10
type to return a result with extended numeric
range.
Next: Exponentiation With Error Estimate, Previous: Exponential Function, Up: Exponential Functions [Index]
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_exprel
and
gsl_sf_exprel_2
. The N-relative exponential is given by,
exprel_N(x) = N!/x^N (\exp(x) - \sum_{k=0}^{N-1} x^k/k!) = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ... = 1F1 (1,1+N,x)
Previous: Relative Exponential Functions, Up: Exponential Functions [Index]
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_e10
type 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_e10
type to return a result with extended range.
Next: Fermi-Dirac Function, Previous: Exponential Functions, Up: Special Functions [Index]
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.
• Exponential Integral: | ||
• Ei(x): | ||
• Hyperbolic Integrals: | ||
• Ei_3(x): | ||
• Trigonometric Integrals: | ||
• Arctangent Integral: |
Next: Ei(x), Up: Exponential Integrals [Index]
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 E_n(x) of order n,
E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.
Next: Hyperbolic Integrals, Previous: Exponential Integral, Up: Exponential Integrals [Index]
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.
Next: Ei_3(x), Previous: Ei(x), Up: Exponential Integrals [Index]
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
).
Next: Trigonometric Integrals, Previous: Hyperbolic Integrals, Up: Exponential Integrals [Index]
These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.
Next: Arctangent Integral, Previous: Ei_3(x), Up: Exponential Integrals [Index]
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.
Previous: Trigonometric Integrals, Up: Exponential Integrals [Index]
These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.
Next: Gamma and Beta Functions, Previous: Exponential Integrals, Up: Special Functions [Index]
The functions described in this section are declared in the header file gsl_sf_fermi_dirac.h.
• Complete Fermi-Dirac Integrals: | ||
• Incomplete Fermi-Dirac Integrals: |
Next: Incomplete Fermi-Dirac Integrals, Up: Fermi-Dirac Function [Index]
The complete Fermi-Dirac integral F_j(x) is given by,
F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))
Note that the Fermi-Dirac integral is sometimes defined without the normalisation factor in other texts.
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).
Previous: Complete Fermi-Dirac Integrals, Up: Fermi-Dirac Function [Index]
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).
Next: Gegenbauer Functions, Previous: Fermi-Dirac Function, Up: Special Functions [Index]
This following routines compute the gamma and beta functions in their full and incomplete forms, as well as various kinds of factorials. The functions described in this section are declared in the header file gsl_sf_gamma.h.
• Gamma Functions: | ||
• Factorials: | ||
• Pochhammer Symbol: | ||
• Incomplete Gamma Functions: | ||
• Beta Functions: | ||
• Incomplete Beta Function: |
Next: Factorials, Up: Gamma and Beta Functions [Index]
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.
These routines compute the Gamma function \Gamma(x), subject to x
not being a negative integer or zero. The function is computed using the real
Lanczos method. The maximum value of x such that \Gamma(x) is not
considered an overflow is given by the macro GSL_SF_GAMMA_XMAX
and is 171.0.
These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not being a negative integer or zero. For x<0 the real part of \log(\Gamma(x)) is returned, which is equivalent to \log(|\Gamma(x)|). The function is computed using the real Lanczos method.
This routine computes the sign of the gamma function and the logarithm of its magnitude, subject to x not being a negative integer or zero. The function is computed using the real Lanczos method. The value of the gamma function and its error can be reconstructed using the relation \Gamma(x) = sgn * \exp(result\_lg), taking into account the two components of result_lg.
These routines compute the regulated Gamma Function \Gamma^*(x) for x > 0. The regulated gamma function is given by,
\Gamma^*(x) = \Gamma(x)/(\sqrt{2\pi} x^{(x-1/2)} \exp(-x)) = (1 + (1/12x) + ...) for x \to \infty
and is a useful suggestion of Temme.
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 or zero, using the complex Lanczos
method. The returned parameters are lnr = \log|\Gamma(z)| and
arg = \arg(\Gamma(z)) in (-\pi,\pi]. Note that the phase
part (arg) is not well-determined when |z| is very large,
due to inevitable roundoff in restricting to (-\pi,\pi]. This
will result in a GSL_ELOSS
error when it occurs. The absolute
value part (lnr), however, never suffers from loss of precision.
Next: Pochhammer Symbol, Previous: Gamma Functions, Up: Gamma and Beta Functions [Index]
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_NMAX
and 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_NMAX
and 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_lngamma
for 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.
Next: Incomplete Gamma Functions, Previous: Factorials, Up: Gamma and Beta Functions [Index]
These routines compute the Pochhammer symbol (a)_x = \Gamma(a + x)/\Gamma(a). The Pochhammer symbol is also known as the Apell symbol and sometimes written as (a,x). When a and a+x are negative integers or zero, the limiting value of the ratio is returned.
These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)).
These routines compute the sign of the Pochhammer symbol and the logarithm of its magnitude. The computed parameters are result = \log(|(a)_x|) with a corresponding error term, and sgn = \sgn((a)_x) where (a)_x = \Gamma(a + x)/\Gamma(a).
These routines compute the relative Pochhammer symbol ((a)_x - 1)/x where (a)_x = \Gamma(a + x)/\Gamma(a).
Next: Beta Functions, Previous: Pochhammer Symbol, Up: Gamma and Beta Functions [Index]
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).
Next: Incomplete Beta Function, Previous: Incomplete Gamma Functions, Up: Gamma and Beta Functions [Index]
These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being negative integers.
These routines compute the logarithm of the Beta Function, \log(B(a,b)) subject to a and b not being negative integers.
Previous: Beta Functions, Up: Gamma and Beta Functions [Index]
These routines compute the normalized incomplete Beta function I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt for 0 <= x <= 1. For a > 0, b > 0 the value is computed using a continued fraction expansion. For all other values it is computed using the relation I_x(a,b,x) = (1/a) x^a 2F1(a,1-b,a+1,x)/B(a,b).
Next: Hypergeometric Functions, Previous: Gamma and Beta Functions, Up: Special Functions [Index]
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.
Next: Laguerre Functions, Previous: Gegenbauer Functions, Up: Special Functions [Index]
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_e10
type 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_e10
type to return a
result with extended range.
These routines compute the Gauss hypergeometric function 2F1(a,b,c,x) = F(a,b,c,x) for |x| < 1.
If the arguments (a,b,c,x) are too close to a singularity then
the function can return the error code GSL_EMAXITER
when the
series approximation converges too slowly. This occurs in the region of
x=1, c - a - b = m for integer m.
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.
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)
Next: Lambert W Functions, Previous: Hypergeometric Functions, Up: Special Functions [Index]
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.
Next: Legendre Functions and Spherical Harmonics, Previous: Laguerre Functions, Up: Special Functions [Index]
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).
Next: Logarithm and Related Functions, Previous: Lambert W Functions, Up: Special Functions [Index]
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.
• Legendre Polynomials: | ||
• Associated Legendre Polynomials and Spherical Harmonics: | ||
• Conical Functions: | ||
• Radial Functions for Hyperbolic Space: |
Next: Associated Legendre Polynomials and Spherical Harmonics, Up: Legendre Functions and Spherical Harmonics [Index]
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 arrays of Legendre polynomials P_l(x) and 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.
Next: Conical Functions, Previous: Legendre Polynomials, Up: Legendre Functions and Spherical Harmonics [Index]
The following functions compute the associated Legendre polynomials P_l^m(x) which are solutions of the differential equation
(1 - x^2) d^2 P_l^m(x) / dx^2 P_l^m(x) - 2x d/dx P_l^m(x) + ( l(l+1) - m^2 / (1 - x^2) ) P_l^m(x) = 0
where the degree l and order m satisfy 0 \le l and 0 \le m \le l. The functions P_l^m(x) grow combinatorially with l and can overflow for l larger than about 150. Alternatively, one may calculate normalized associated Legendre polynomials. There are a number of different normalization conventions, and these functions can be stably computed up to degree and order 2700. The following normalizations are provided:
Schmidt semi-normalization
Schmidt semi-normalized associated Legendre polynomials are often used in the magnetics community and are defined as
S_l^0(x) = P_l^0(x) S_l^m(x) = (-1)^m \sqrt((2(l-m)! / (l+m)!)) P_l^m(x), m > 0
The factor of (-1)^m is called the Condon-Shortley phase
factor and can be excluded if desired by setting the parameter
csphase = 1
in the functions below.
Spherical Harmonic Normalization
The associated Legendre polynomials suitable for calculating spherical harmonics are defined as
Y_l^m(x) = (-1)^m \sqrt((2l + 1) * (l-m)! / (4 \pi) / (l+m)!) P_l^m(x)
where again the phase factor (-1)^m can be included or excluded if desired.
Full Normalization
The fully normalized associated Legendre polynomials are defined as
N_l^m(x) = (-1)^m \sqrt((l + 1/2) * (l-m)! / (l+m)!) P_l^m(x)
and have the property
\int_(-1)^1 ( N_l^m(x) )^2 dx = 1
The normalized associated Legendre routines below use a recurrence relation which is stable up to a degree and order of about 2700. Beyond this, the computed functions could suffer from underflow leading to incorrect results. Routines are provided to compute first and second derivatives dP_l^m(x)/dx and d^2 P_l^m(x)/dx^2 as well as their alternate versions d P_l^m(\cos{\theta})/d\theta and d^2 P_l^m(\cos{\theta})/d\theta^2. While there is a simple scaling relationship between the two forms, the derivatives involving \theta are heavily used in spherical harmonic expansions and so these routines are also provided.
In the functions below, a parameter of type gsl_sf_legendre_t
specifies the type of normalization to use. The possible values are
GSL_SF_LEGENDRE_NONE
This specifies the computation of the unnormalized associated Legendre polynomials P_l^m(x).
GSL_SF_LEGENDRE_SCHMIDT
This specifies the computation of the Schmidt semi-normalized associated Legendre polynomials S_l^m(x).
GSL_SF_LEGENDRE_SPHARM
This specifies the computation of the spherical harmonic associated Legendre polynomials Y_l^m(x).
GSL_SF_LEGENDRE_FULL
This specifies the computation of the fully normalized associated Legendre polynomials N_l^m(x).
These functions calculate all normalized associated Legendre
polynomials for 0 \le l \le lmax and
0 \le m \le l for
|x| <= 1.
The norm parameter specifies which normalization is used.
The normalized P_l^m(x) values are stored in result_array, whose
minimum size can be obtained from calling gsl_sf_legendre_array_n
.
The array index of P_l^m(x) is obtained from calling
gsl_sf_legendre_array_index(l, m)
. To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e
function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their first derivatives up to degree lmax for
|x| < 1.
The parameter norm specifies the normalization used. The
normalized P_l^m(x) values and their derivatives
dP_l^m(x)/dx are stored in result_array and
result_deriv_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e
function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their (alternate) first derivatives up to degree lmax for
|x| < 1.
The normalized P_l^m(x) values and their derivatives
dP_l^m(\cos{\theta})/d\theta are stored in result_array and
result_deriv_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e
function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their first and second derivatives up to degree lmax for
|x| < 1.
The parameter norm specifies the normalization used. The
normalized P_l^m(x), their first derivatives
dP_l^m(x)/dx, and their second derivatives
d^2 P_l^m(x)/dx^2 are stored in result_array,
result_deriv_array, and result_deriv2_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e
function. This factor is included by default.
These functions calculate all normalized associated Legendre
functions and their (alternate) first and second derivatives up to degree
lmax for
|x| < 1.
The parameter norm specifies the normalization used. The
normalized P_l^m(x), their first derivatives
dP_l^m(\cos{\theta})/d\theta, and their second derivatives
d^2 P_l^m(\cos{\theta})/d\theta^2 are stored in result_array,
result_deriv_array, and result_deriv2_array respectively.
To include or exclude
the Condon-Shortley phase factor of (-1)^m, set the parameter
csphase to either -1 or 1 respectively in the
_e
function. This factor is included by default.
This function returns the minimum array size for maximum degree lmax needed for the array versions of the associated Legendre functions. Size is calculated as the total number of P_l^m(x) functions, plus extra space for precomputing multiplicative factors used in the recurrence relations.
This function returns the index into result_array, result_deriv_array, or result_deriv2_array corresponding to P_l^m(x), P_l^{'m}(x), or P_l^{''m}(x). The index is given by l(l+1)/2 + m.
These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |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 are now deprecated and will be removed in a future
release; see gsl_sf_legendre_array
and
gsl_sf_legendre_deriv_array
.
These functions are now deprecated and will be removed in a future
release; see gsl_sf_legendre_array
and
gsl_sf_legendre_deriv_array
.
This function is now deprecated and will be removed in a future release.
Next: Radial Functions for Hyperbolic Space, Previous: Associated Legendre Polynomials and Spherical Harmonics, Up: Legendre Functions and Spherical Harmonics [Index]
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.
Previous: Conical Functions, Up: Legendre Functions and Spherical Harmonics [Index]
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.
Next: Mathieu Functions, Previous: Legendre Functions and Spherical Harmonics, Up: Special Functions [Index]
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.
Next: Power Function, Previous: Logarithm and Related Functions, Up: Special Functions [Index]
The routines described in this section compute the angular and radial Mathieu functions, and their characteristic values. Mathieu functions are the solutions of the following two differential equations:
d^2y/dv^2 + (a - 2q\cos 2v)y = 0 d^2f/du^2 - (a - 2q\cosh 2u)f = 0
The angular Mathieu functions ce_r(x,q), se_r(x,q) are the even and odd periodic solutions of the first equation, which is known as Mathieu’s equation. These exist only for the discrete sequence of characteristic values a=a_r(q) (even-periodic) and a=b_r(q) (odd-periodic).
The radial Mathieu functions Mc^{(j)}_{r}(z,q), Ms^{(j)}_{r}(z,q) are the solutions of the second equation, which is referred to as Mathieu’s modified equation. The radial Mathieu functions of the first, second, third and fourth kind are denoted by the parameter j, which takes the value 1, 2, 3 or 4.
For more information on the Mathieu functions, see Abramowitz and Stegun, Chapter 20. These functions are defined in the header file gsl_sf_mathieu.h.
• Mathieu Function Workspace: | ||
• Mathieu Function Characteristic Values: | ||
• Angular Mathieu Functions: | ||
• Radial Mathieu Functions: |
Next: Mathieu Function Characteristic Values, Up: Mathieu Functions [Index]
The Mathieu functions can be computed for a single order or for multiple orders, using array-based routines. The array-based routines require a preallocated workspace.
This function returns a workspace for the array versions of the Mathieu routines. The arguments n and qmax specify the maximum order and q-value of Mathieu functions which can be computed with this workspace.
This function frees the workspace work.
Next: Angular Mathieu Functions, Previous: Mathieu Function Workspace, Up: Mathieu Functions [Index]
These routines compute the characteristic values a_n(q), b_n(q) of the Mathieu functions ce_n(q,x) and se_n(q,x), respectively.
These routines compute a series of Mathieu characteristic values a_n(q), b_n(q) for n from order_min to order_max inclusive, storing the results in the array result_array.
Next: Radial Mathieu Functions, Previous: Mathieu Function Characteristic Values, Up: Mathieu Functions [Index]
These routines compute the angular Mathieu functions ce_n(q,x) and se_n(q,x), respectively.
These routines compute a series of the angular Mathieu functions ce_n(q,x) and se_n(q,x) of order n from nmin to nmax inclusive, storing the results in the array result_array.
Previous: Angular Mathieu Functions, Up: Mathieu Functions [Index]
These routines compute the radial j-th kind Mathieu functions Mc_n^{(j)}(q,x) and Ms_n^{(j)}(q,x) of order n.
The allowed values of j are 1 and 2. The functions for j = 3,4 can be computed as M_n^{(3)} = M_n^{(1)} + iM_n^{(2)} and M_n^{(4)} = M_n^{(1)} - iM_n^{(2)}, where M_n^{(j)} = Mc_n^{(j)} or Ms_n^{(j)}.
These routines compute a series of the radial Mathieu functions of kind j, with order from nmin to nmax inclusive, storing the results in the array result_array.
Next: Psi (Digamma) Function, Previous: Mathieu Functions, Up: Special Functions [Index]
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);
Next: Synchrotron Functions, Previous: Power Function, Up: Special Functions [Index]
The polygamma functions of order n are defined by
\psi^{(n)}(x) = (d/dx)^n \psi(x) = (d/dx)^{n+1} \log(\Gamma(x))
where \psi(x) = \Gamma'(x)/\Gamma(x) is known as the digamma function. These functions are declared in the header file gsl_sf_psi.h.
• Digamma Function: | ||
• Trigamma Function: | ||
• Polygamma Function: |
Next: Trigamma Function, Up: Psi (Digamma) Function [Index]
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)].
Next: Polygamma Function, Previous: Digamma Function, Up: Psi (Digamma) Function [Index]
These routines compute the Trigamma function \psi'(n) for positive integer n.
These routines compute the Trigamma function \psi'(x) for general x.
Previous: Trigamma Function, Up: Psi (Digamma) Function [Index]
These routines compute the polygamma function \psi^{(n)}(x) for n >= 0, x > 0.
Next: Transport Functions, Previous: Psi (Digamma) Function, Up: Special Functions [Index]
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.
Next: Trigonometric Functions, Previous: Synchrotron Functions, Up: Special Functions [Index]
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).
Next: Zeta Functions, Previous: Transport Functions, Up: Special Functions [Index]
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 sine function \sin(x).
These routines compute the cosine function \cos(x).
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.
Next: Hyperbolic Trigonometric Functions, Previous: Circular Trigonometric Functions, Up: Trigonometric Functions [Index]
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 czr, czi.
This function computes the logarithm of the complex sine, \log(\sin(z_r + i z_i)) storing the real and imaginary parts in lszr, lszi.
Next: Conversion Functions, Previous: Trigonometric Functions for Complex Arguments, Up: Trigonometric Functions [Index]
These routines compute \log(\sinh(x)) for x > 0.
These routines compute \log(\cosh(x)) for any x.
Next: Restriction Functions, Previous: Hyperbolic Trigonometric Functions, Up: Trigonometric Functions [Index]
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].
Next: Trigonometric Functions With Error Estimates, Previous: Conversion Functions, Up: Trigonometric Functions [Index]
These routines force the angle theta to lie in the range (-\pi,\pi].
Note that the mathematical value of \pi is slightly greater
than M_PI
, so the machine numbers M_PI
and -M_PI
are included in the range.
These routines force the angle theta to lie in the range [0, 2\pi).
Note that the mathematical value of 2\pi is slightly greater
than 2*M_PI
, so the machine number 2*M_PI
is included in
the range.
Previous: Restriction Functions, Up: Trigonometric Functions [Index]
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.
Next: Special Functions Examples, Previous: Trigonometric Functions, Up: Special Functions [Index]
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.
• Riemann Zeta Function: | ||
• Riemann Zeta Function Minus One: | ||
• Hurwitz Zeta Function: | ||
• Eta Function: |
Next: Riemann Zeta Function Minus One, Up: Zeta Functions [Index]
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.
Next: Hurwitz Zeta Function, Previous: Riemann Zeta Function, Up: Zeta Functions [Index]
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.
Next: Eta Function, Previous: Riemann Zeta Function Minus One, Up: Zeta Functions [Index]
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.
Previous: Hurwitz Zeta Function, Up: Zeta Functions [Index]
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.
Next: Special Functions References and Further Reading, Previous: Zeta Functions, Up: Special Functions [Index]
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) = -0.177596771314338264 +/- 0.000000000000000193 exact = -0.177596771314338292
The next program computes the same quantity using the natural form of the function. In this case the error term result.err and return status are not accessible.
#include <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) = -0.177596771314338264 exact = -0.177596771314338292
Previous: Special Functions Examples, Up: Special Functions [Index]
The library follows the conventions of Abramowitz & Stegun where possible,
The following papers contain information on the algorithms used to compute the special functions,
Next: Permutations, Previous: Special Functions, Up: Top [Index]
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.
• Data types: | ||
• Blocks: | ||
• Vectors: | ||
• Matrices: | ||
• Vector and Matrix References and Further Reading: |
Next: Blocks, Up: Vectors and Matrices [Index]
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.
Next: Vectors, Previous: Data types, Up: Vectors and Matrices [Index]
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
• Block allocation: | ||
• Reading and writing blocks: | ||
• Example programs for blocks: |
Next: Reading and writing blocks, Up: Blocks [Index]
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_calloc
if you want to ensure that all the
elements are initialized to zero.
A null pointer is returned if insufficient memory is available to create the block.
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_alloc
or gsl_block_calloc
.
Next: Example programs for blocks, Previous: Block allocation, Up: Blocks [Index]
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_EFAILED
if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the block b from the open stream
stream in binary format. The block b must be preallocated
with the correct length since the function uses the size of b to
determine how many bytes to read. The return value is 0 for success and
GSL_EFAILED
if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the block b line-by-line to
the stream stream using the format specifier format, which
should be one of the %g
, %e
or %f
formats for
floating point numbers and %d
for integers. The function returns
0 for success and GSL_EFAILED
if there was a problem writing to
the file.
This function reads formatted data from the stream stream into the
block b. The block b must be preallocated with the correct
length since the function uses the size of b to determine how many
numbers to read. The function returns 0 for success and
GSL_EFAILED
if there was a problem reading from the file.
Previous: Reading and writing blocks, Up: Blocks [Index]
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 = %zu\n", b->size); printf ("block data address = %p\n", b->data); gsl_block_free (b); return 0; }
Here is the output from the program,
length of block = 100 block data address = 0x804b0d8
Next: Matrices, Previous: Blocks, Up: Vectors and Matrices [Index]
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
Next: Accessing vector elements, Up: Vectors [Index]
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_alloc
then the block
underlying the vector will also be deallocated. If the vector has
been created from another object then the memory is still owned by
that object and will not be deallocated.
Next: Initializing vector elements, Previous: Vector allocation, Up: Vectors [Index]
Unlike FORTRAN compilers, C compilers do not usually provide
support for range checking of vectors and matrices.^{8} The functions gsl_vector_get
and
gsl_vector_set
can perform portable range checking for you and
report an error if you attempt to access elements outside the allowed
range.
The functions for accessing the elements of a vector or matrix are
defined in gsl_vector.h and declared extern inline
to
eliminate function-call overhead. You must compile your program with
the preprocessor macro HAVE_INLINE
defined to use these
functions.
If necessary you can turn off range checking completely without
modifying any source files by recompiling your program with the
preprocessor definition GSL_RANGE_CHECK_OFF
. Provided your
compiler supports inline functions the effect of turning off range
checking is to replace calls to gsl_vector_get(v,i)
by
v->data[i*v->stride]
and calls to gsl_vector_set(v,i,x)
by
v->data[i*v->stride]=x
. Thus there should be no performance
penalty for using the range checking functions when range checking is
turned off.
If you use a C99 compiler which requires inline functions in header
files to be declared inline
instead of extern inline
,
define the macro GSL_C99_INLINE
(see Inline functions).
With GCC this is selected automatically when compiling in C99 mode
(-std=c99
).
If inline functions are not used, calls to the functions
gsl_vector_get
and gsl_vector_set
will link to the
compiled versions of these functions in the library itself. The range
checking in these functions is controlled by the global integer
variable gsl_check_range
. It is enabled by default—to
disable it, set gsl_check_range
to zero. Due to function-call
overhead, there is less benefit in disabling range checking here than
for inline functions.
This function returns the i-th element of a vector v. If
i lies outside the allowed range of 0 to n-1 then the error
handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE
is defined.
This function sets the value of the i-th element of a vector
v to x. If i lies outside the allowed range of 0 to
n-1 then the error handler is invoked. An inline version of this function is used when HAVE_INLINE
is defined.
These functions return a pointer to the i-th element of a vector
v. If i lies outside the allowed range of 0 to n-1
then the error handler is invoked and a null pointer is returned. Inline versions of these functions are used when HAVE_INLINE
is defined.
Next: Reading and writing vectors, Previous: Accessing vector elements, Up: Vectors [Index]
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.
Next: Vector views, Previous: Initializing vector elements, Up: Vectors [Index]
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_EFAILED
if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the vector v from the open stream
stream in binary format. The vector v must be preallocated
with the correct length since the function uses the size of v to
determine how many bytes to read. The return value is 0 for success and
GSL_EFAILED
if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the vector v line-by-line to
the stream stream using the format specifier format, which
should be one of the %g
, %e
or %f
formats for
floating point numbers and %d
for integers. The function returns
0 for success and GSL_EFAILED
if there was a problem writing to
the file.
This function reads formatted data from the stream stream into the
vector v. The vector v must be preallocated with the correct
length since the function uses the size of v to determine how many
numbers to read. The function returns 0 for success and
GSL_EFAILED
if there was a problem reading from the file.
Next: Copying vectors, Previous: Reading and writing vectors, Up: Vectors [Index]
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 data
pointer of the returned vector struct is set to null if
the combined parameters (offset,n) overrun the end of the
original vector.
The new vector is only a view of the block underlying the original vector, v. The block containing the elements of v is not owned by the new vector. When the view goes out of scope the original vector v and its block will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.
The function gsl_vector_const_subvector
is equivalent to
gsl_vector_subvector
but can be used for vectors which are
declared const
.
These functions return a vector view of a subvector of another vector
v with an additional stride argument. The subvector is formed in
the same way as for gsl_vector_subvector
but the new vector has
n elements with a step-size of stride from one element to
the next in the original vector. Mathematically, the i-th element
of the new vector v’ is given by,
v'(i) = v->data[(offset + i*stride)*v->stride]
where the index i runs from 0 to n-1
.
Note that subvector views give direct access to the underlying elements
of the original vector. For example, the following code will zero the
even elements of the vector v
of length n
, while leaving the
odd elements untouched,
gsl_vector_view v_even = gsl_vector_subvector_with_stride (v, 0, 2, n/2); gsl_vector_set_zero (&v_even.vector);
A vector view can be passed to any subroutine which takes a vector
argument just as a directly allocated vector would be, using
&
view.vector
. For example, the following code
computes the norm of the odd elements of v
using the BLAS
routine DNRM2,
gsl_vector_view v_odd = gsl_vector_subvector_with_stride (v, 1, 2, n/2); double r = gsl_blas_dnrm2 (&v_odd.vector);
The function gsl_vector_const_subvector_with_stride
is equivalent
to gsl_vector_subvector_with_stride
but can be used for vectors
which are declared const
.
These functions return a vector view of the real parts of the complex vector v.
The function gsl_vector_complex_const_real
is equivalent to
gsl_vector_complex_real
but can be used for vectors which are
declared const
.
These functions return a vector view of the imaginary parts of the complex vector v.
The function gsl_vector_complex_const_imag
is equivalent to
gsl_vector_complex_imag
but can be used for vectors which are
declared const
.
These functions return a vector view of an array. The start of the new vector is given by base and has n elements. Mathematically, the i-th element of the new vector v’ is given by,
v'(i) = base[i]
where the index i runs from 0 to n-1
.
The array containing the elements of v is not owned by the new vector view. When the view goes out of scope the original array will continue to exist. The original memory can only be deallocated by freeing the original pointer base. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_vector_const_view_array
is equivalent to
gsl_vector_view_array
but can be used for arrays which are
declared const
.
These functions return a vector view of an array base with an
additional stride argument. The subvector is formed in the same way as
for gsl_vector_view_array
but the new vector has n elements
with a step-size of stride from one element to the next in the
original array. Mathematically, the i-th element of the new
vector v’ is given by,
v'(i) = base[i*stride]
where the index i runs from 0 to n-1
.
Note that the view gives direct access to the underlying elements of the
original array. A vector view can be passed to any subroutine which
takes a vector argument just as a directly allocated vector would be,
using &
view.vector
.
The function gsl_vector_const_view_array_with_stride
is
equivalent to gsl_vector_view_array_with_stride
but can be used
for arrays which are declared const
.
Next: Exchanging elements, Previous: Vector views, Up: Vectors [Index]
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.
Next: Vector operations, Previous: Copying vectors, Up: Vectors [Index]
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.
Next: Finding maximum and minimum elements of vectors, Previous: Exchanging elements, Up: Vectors [Index]
This function adds the elements of vector b to the elements of vector a. The result a_i \leftarrow a_i + b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function subtracts the elements of vector b from the elements of vector a. The result a_i \leftarrow a_i - b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function multiplies the elements of vector a by the elements of vector b. The result a_i \leftarrow a_i * b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function divides the elements of vector a by the elements of vector b. The result a_i \leftarrow a_i / b_i is stored in a and b remains unchanged. The two vectors must have the same length.
This function multiplies the elements of vector a by the constant factor x. The result a_i \leftarrow x a_i is stored in a.
This function adds the constant value x to the elements of the vector a. The result a_i \leftarrow a_i + x is stored in a.
Next: Vector properties, Previous: Vector operations, Up: Vectors [Index]
The following operations are only defined for real vectors.
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.
Next: Example programs for vectors, Previous: Finding maximum and minimum elements of vectors, Up: Vectors [Index]
The following functions are defined for real and complex vectors. For complex vectors both the real and imaginary parts must satisfy the conditions.
These functions return 1 if all the elements of the vector v are zero, strictly positive, strictly negative, or non-negative respectively, and 0 otherwise.
This function returns 1 if the vectors u and v are equal (by comparison of element values) and 0 otherwise.
Previous: Vector properties, Up: Vectors [Index]
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++) /* OUT OF RANGE ERROR */ { printf ("v_%d = %g\n", i, gsl_vector_get (v, i)); } gsl_vector_free (v); return 0; }
Here is the output from the program. The final loop attempts to read
outside the range of the vector v
, and the error is trapped by
the range-checking code in gsl_vector_get
.
$ ./a.out v_0 = 1.23 v_1 = 2.23 v_2 = 3.23 gsl: vector_source.c:12: ERROR: index out of range Default GSL error handler invoked. Aborted (core dumped)
The next program shows how to write a vector to a file.
#include <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); } gsl_vector_free (v); return 0; }
After running this program the file test.dat should contain the
elements of v
, written using the format specifier
%.5g
. The vector could then be read back in using the function
gsl_vector_fscanf (f, v)
as follows:
#include <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)); } gsl_vector_free (v); return 0; }
Next: Vector and Matrix References and Further Reading, Previous: Vectors, Up: Vectors and Matrices [Index]
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
Next: Accessing matrix elements, Up: Matrices [Index]
The functions for allocating memory to a matrix follow the style of
malloc
and free
. They also perform their own error
checking. If there is insufficient memory available to allocate a matrix
then the functions call the GSL error handler (with an error number of
GSL_ENOMEM
) in addition to returning a null pointer. Thus if you
use the library error handler to abort your program then it isn’t
necessary to check every alloc
.
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_alloc
then the block
underlying the matrix will also be deallocated. If the matrix has
been created from another object then the memory is still owned by
that object and will not be deallocated.
Next: Initializing matrix elements, Previous: Matrix allocation, Up: Matrices [Index]
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. An inline version of this function is used when HAVE_INLINE
is defined.
This function sets the value of the (i,j)-th element of a
matrix m to x. If i or j lies outside the
allowed range of 0 to n1-1 and 0 to n2-1 then the error
handler is invoked. An inline version of this function is used when HAVE_INLINE
is defined.
These functions return a pointer to the (i,j)-th element of a
matrix m. If i or j lie outside the allowed range of
0 to n1-1 and 0 to n2-1 then the error handler is invoked
and a null pointer is returned. Inline versions of these functions are used when HAVE_INLINE
is defined.
Next: Reading and writing matrices, Previous: Accessing matrix elements, Up: Matrices [Index]
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.
Next: Matrix views, Previous: Initializing matrix elements, Up: Matrices [Index]
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_EFAILED
if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the matrix m from the open stream
stream in binary format. The matrix m must be preallocated
with the correct dimensions since the function uses the size of m to
determine how many bytes to read. The return value is 0 for success and
GSL_EFAILED
if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the matrix m line-by-line to
the stream stream using the format specifier format, which
should be one of the %g
, %e
or %f
formats for
floating point numbers and %d
for integers. The function returns
0 for success and GSL_EFAILED
if there was a problem writing to
the file.
This function reads formatted data from the stream stream into the
matrix m. The matrix m must be preallocated with the correct
dimensions since the function uses the size of m to determine how many
numbers to read. The function returns 0 for success and
GSL_EFAILED
if there was a problem reading from the file.
Next: Creating row and column views, Previous: Reading and writing matrices, Up: Matrices [Index]
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 n1-1
and the index j
runs from 0 to n2-1
.
The data
pointer of the returned matrix struct is set to null if
the combined parameters (i,j,n1,n2,tda)
overrun the ends of the original matrix.
The new matrix view is only a view of the block underlying the existing matrix, m. The block containing the elements of m is not owned by the new matrix view. When the view goes out of scope the original matrix m and its block will continue to exist. The original memory can only be deallocated by freeing the original matrix. Of course, the original matrix should not be deallocated while the view is still in use.
The function gsl_matrix_const_submatrix
is equivalent to
gsl_matrix_submatrix
but can be used for matrices which are
declared const
.
These functions return a matrix view of the array base. The matrix has n1 rows and n2 columns. The physical number of columns in memory is also given by n2. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = base[i*n2 + j]
where the index i runs from 0 to n1-1
and the index j
runs from 0 to n2-1
.
The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_array
is equivalent to
gsl_matrix_view_array
but can be used for matrices which are
declared const
.
These functions return a matrix view of the array base with a physical number of columns tda which may differ from the corresponding dimension of the matrix. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = base[i*tda + j]
where the index i runs from 0 to n1-1
and the index j
runs from 0 to n2-1
.
The new matrix is only a view of the array base. When the view goes out of scope the original array base will continue to exist. The original memory can only be deallocated by freeing the original array. Of course, the original array should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_array_with_tda
is equivalent
to gsl_matrix_view_array_with_tda
but can be used for matrices
which are declared const
.
These functions return a matrix view of the vector v. The matrix has n1 rows and n2 columns. The vector must have unit stride. The physical number of columns in memory is also given by n2. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*n2 + j]
where the index i runs from 0 to n1-1
and the index j
runs from 0 to n2-1
.
The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_vector
is equivalent to
gsl_matrix_view_vector
but can be used for matrices which are
declared const
.
These functions return a matrix view of the vector v with a physical number of columns tda which may differ from the corresponding matrix dimension. The vector must have unit stride. The matrix has n1 rows and n2 columns, and the physical number of columns in memory is given by tda. Mathematically, the (i,j)-th element of the new matrix is given by,
m'(i,j) = v->data[i*tda + j]
where the index i runs from 0 to n1-1
and the index j
runs from 0 to n2-1
.
The new matrix is only a view of the vector v. When the view goes out of scope the original vector v will continue to exist. The original memory can only be deallocated by freeing the original vector. Of course, the original vector should not be deallocated while the view is still in use.
The function gsl_matrix_const_view_vector_with_tda
is equivalent
to gsl_matrix_view_vector_with_tda
but can be used for matrices
which are declared const
.
Next: Copying matrices, Previous: Matrix views, Up: Matrices [Index]
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 data
pointer of the new vector is set to null if
i is out of range.
The function gsl_vector_const_row
is equivalent to
gsl_matrix_row
but can be used for matrices which are declared
const
.
These functions return a vector view of the j-th column of the
matrix m. The data
pointer of the new vector is set to
null if j is out of range.
The function gsl_vector_const_column
is equivalent to
gsl_matrix_column
but can be used for matrices which are declared
const
.
These functions return a vector view of the i-th row of the matrix
m beginning at offset elements past the first column and
containing n elements. The data
pointer of the new vector
is set to null if i, offset, or n are out of range.
The function gsl_vector_const_subrow
is equivalent to
gsl_matrix_subrow
but can be used for matrices which are declared
const
.
These functions return a vector view of the j-th column of the matrix
m beginning at offset elements past the first row and
containing n elements. The data
pointer of the new vector
is set to null if j, offset, or n are out of range.
The function gsl_vector_const_subcolumn
is equivalent to
gsl_matrix_subcolumn
but can be used for matrices which are declared
const
.
These functions return a vector view of the diagonal of the matrix m. The matrix m is not required to be square. For a rectangular matrix the length of the diagonal is the same as the smaller dimension of the matrix.
The function gsl_matrix_const_diagonal
is equivalent to
gsl_matrix_diagonal
but can be used for matrices which are
declared const
.
These functions return a vector view of the k-th subdiagonal of the matrix m. The matrix m is not required to be square. The diagonal of the matrix corresponds to k = 0.
The function gsl_matrix_const_subdiagonal
is equivalent to
gsl_matrix_subdiagonal
but can be used for matrices which are
declared const
.
These functions return a vector view of the k-th superdiagonal of the matrix m. The matrix m is not required to be square. The diagonal of the matrix corresponds to k = 0.
The function gsl_matrix_const_superdiagonal
is equivalent to
gsl_matrix_superdiagonal
but can be used for matrices which are
declared const
.
Next: Copying rows and columns, Previous: Creating row and column views, Up: Matrices [Index]
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.
Next: Exchanging rows and columns, Previous: Copying matrices, Up: Matrices [Index]
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.
Next: Matrix operations, Previous: Copying rows and columns, Up: Matrices [Index]
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.
Next: Finding maximum and minimum elements of matrices, Previous: Exchanging rows and columns, Up: Matrices [Index]
The following operations are defined for real and complex matrices.
This function adds the elements of matrix b to the elements of matrix a. The result a(i,j) \leftarrow a(i,j) + b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function subtracts the elements of matrix b from the elements of matrix a. The result a(i,j) \leftarrow a(i,j) - b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function multiplies the elements of matrix a by the elements of matrix b. The result a(i,j) \leftarrow a(i,j) * b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function divides the elements of matrix a by the elements of matrix b. The result a(i,j) \leftarrow a(i,j) / b(i,j) is stored in a and b remains unchanged. The two matrices must have the same dimensions.
This function multiplies the elements of matrix a by the constant factor x. The result a(i,j) \leftarrow x a(i,j) is stored in a.
This function adds the constant value x to the elements of the matrix a. The result a(i,j) \leftarrow a(i,j) + x is stored in a.
Next: Matrix properties, Previous: Matrix operations, Up: Matrices [Index]
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.
Next: Example programs for matrices, Previous: Finding maximum and minimum elements of matrices, Up: Matrices [Index]
The following functions are defined for real and complex matrices. For complex matrices both the real and imaginary parts must satisfy the conditions.
These functions return 1 if all the elements of the matrix m are zero, strictly positive, strictly negative, or non-negative respectively, and 0 otherwise. To test whether a matrix is positive-definite, use the Cholesky decomposition (see Cholesky Decomposition).
This function returns 1 if the matrices a and b are equal (by comparison of element values) and 0 otherwise.
Previous: Matrix properties, Up: Matrices [Index]
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++) /* OUT OF RANGE ERROR */ for (j = 0; j < 3; j++) printf ("m(%d,%d) = %g\n", i, j, gsl_matrix_get (m, i, j)); gsl_matrix_free (m); return 0; }
Here is the output from the program. The final loop attempts to read
outside the range of the matrix m
, and the error is trapped by
the range-checking code in gsl_matrix_get
.
$ ./a.out m(0,0) = 0.23 m(0,1) = 1.23 m(0,2) = 2.23 m(1,0) = 100.23 m(1,1) = 101.23 m(1,2) = 102.23 ... m(9,2) = 902.23 gsl: matrix_source.c:13: ERROR: first index out of range Default GSL error handler invoked. Aborted (core dumped)
The next program shows how to write a matrix to a file.
#include <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++; } gsl_matrix_free (m); gsl_matrix_free (a); printf ("differences = %d (should be zero)\n", k); return (k > 0); }
After running this program the file test.dat should contain the
elements of m
, written in binary format. The matrix which is read
back in using the function gsl_matrix_fread
should be exactly
equal to the original matrix.
The following program demonstrates the use of vector views. The program computes the column norms of a matrix.
#include <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 %zu, norm = %g\n", j, d); } gsl_matrix_free (m); return 0; }
Here is the output of the program,
$ ./a.out
matrix column 0, norm = 4.31461 matrix column 1, norm = 3.1205 matrix column 2, norm = 2.19316 matrix column 3, norm = 3.26114 matrix column 4, norm = 2.53416 matrix column 5, norm = 2.57281 matrix column 6, norm = 4.20469 matrix column 7, norm = 3.65202 matrix column 8, norm = 2.08524 matrix column 9, norm = 3.07313
The results can be confirmed using GNU OCTAVE,
$ octave GNU Octave, version 2.0.16.92 octave> m = sin(0:9)' * ones(1,10) + ones(10,1) * cos(0:9); octave> sqrt(sum(m.^2)) ans = 4.3146 3.1205 2.1932 3.2611 2.5342 2.5728 4.2047 3.6520 2.0852 3.0731
Previous: Matrices, Up: Vectors and Matrices [Index]
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,
Next: Combinations, Previous: Vectors and Matrices, Up: Top [Index]
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.
Next: Permutation allocation, Up: Permutations [Index]
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;
Next: Accessing permutation elements, Previous: The Permutation struct, Up: Permutations [Index]
This function allocates memory for a new permutation of size n.
The permutation is not initialized and its elements are undefined. Use
the function gsl_permutation_calloc
if you want to create a
permutation which is initialized to the identity. A null pointer is
returned if insufficient memory is available to create the permutation.
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.
Next: Permutation properties, Previous: Permutation allocation, Up: Permutations [Index]
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. An inline version of this function is used when HAVE_INLINE
is defined.
This function exchanges the i-th and j-th elements of the permutation p.
Next: Permutation functions, Previous: Accessing permutation elements, Up: Permutations [Index]
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.
Next: Applying Permutations, Previous: Permutation properties, Up: Permutations [Index]
This function reverses the elements of the permutation p.
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 returns GSL_FAILURE
and leaves
p unmodified. Starting with the identity permutation and
repeatedly applying this function will iterate through all possible
permutations of a given order.
This function steps backwards from the permutation p to the
previous permutation in lexicographic order, returning
GSL_SUCCESS
. If no previous permutation is available it returns
GSL_FAILURE
and leaves p unmodified.
Next: Reading and writing permutations, Previous: Permutation functions, Up: Permutations [Index]
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 applies the permutation p to the matrix A from the right, A' = A P. The j-th column of the permutation matrix P is given by the p_j-th column of the identity matrix. This effectively permutes the columns of A according to the permutation p, and so the number of columns of A must equal the size of the permutation p.
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.
Next: Permutations in cyclic form, Previous: Applying Permutations, Up: Permutations [Index]
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_EFAILED
if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads into the permutation p from the open stream
stream in binary format. The permutation p must be
preallocated with the correct length since the function uses the size of
p to determine how many bytes to read. The function returns
GSL_EFAILED
if there was a problem reading from the file. The
data is assumed to have been written in the native binary format on the
same architecture.
This function writes the elements of the permutation p
line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t.
In ISO C99 the type modifier z
represents size_t
, so
"%zu\n"
is a suitable format.^{9}
The function returns GSL_EFAILED
if 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_EFAILED
if there was a problem reading from the file.
Next: Permutation Examples, Previous: Reading and writing permutations, Up: Permutations [Index]
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.
Next: Permutation References and Further Reading, Previous: Permutations in cyclic form, Up: Permutations [Index]
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"); gsl_permutation_free (p); gsl_permutation_free (q); gsl_rng_free (r); return 0; }
Here is the output from the program,
$ ./a.out initial permutation: 0 1 2 3 4 5 6 7 8 9 random permutation: 1 3 5 2 7 6 0 4 9 8 inverse permutation: 6 0 3 1 7 2 5 4 9 8
The random permutation p[i]
and its inverse q[i]
are
related through the identity p[q[i]] = i
, which can be verified
from the output.
The next example program steps forwards through all possible third order permutations, starting from the identity,
#include <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); gsl_permutation_free (p); return 0; }
Here is the output from the program,
$ ./a.out 0 1 2 0 2 1 1 0 2 1 2 0 2 0 1 2 1 0
The permutations are generated in lexicographic order. To reverse the
sequence, begin with the final permutation (which is the reverse of the
identity) and replace gsl_permutation_next
with
gsl_permutation_prev
.
Previous: Permutation Examples, Up: Permutations [Index]
The subject of permutations is covered extensively in Knuth’s Sorting and Searching,
For the definition of the canonical form see,
Next: Multisets, Previous: Permutations, Up: Top [Index]
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.
Next: Combination allocation, Up: Combinations [Index]
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;
Next: Accessing combination elements, Previous: The Combination struct, Up: Combinations [Index]
This function allocates memory for a new combination with parameters
n, k. The combination is not initialized and its elements
are undefined. Use the function gsl_combination_calloc
if you
want to create a combination which is initialized to the
lexicographically first combination. A null pointer is returned if
insufficient memory is available to create the combination.
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.
Next: Combination properties, Previous: Combination allocation, Up: Combinations [Index]
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. An inline version of this function is used when HAVE_INLINE
is defined.
Next: Combination functions, Previous: Accessing combination elements, Up: Combinations [Index]
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.
Next: Reading and writing combinations, Previous: Combination properties, Up: Combinations [Index]
This function advances the combination c to the next combination
in lexicographic order and returns GSL_SUCCESS
. If no further
combinations are available it returns GSL_FAILURE
and leaves
c unmodified. Starting with the first combination and
repeatedly applying this function will iterate through all possible
combinations of a given order.
This function steps backwards from the combination c to the
previous combination in lexicographic order, returning
GSL_SUCCESS
. If no previous combination is available it returns
GSL_FAILURE
and leaves c unmodified.
Next: Combination Examples, Previous: Combination functions, Up: Combinations [Index]
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_EFAILED
if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads elements from the open stream stream into the
combination c in binary format. The combination c must be
preallocated with correct values of n and k since the
function uses the size of c to determine how many bytes to read.
The function returns GSL_EFAILED
if there was a problem reading
from the file. The data is assumed to have been written in the native
binary format on the same architecture.
This function writes the elements of the combination c
line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t.
In ISO C99 the type modifier z
represents size_t
, so
"%zu\n"
is a suitable format.^{10} The function returns
GSL_EFAILED
if 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_EFAILED
if there was a problem reading from the file.
Next: Combination References and Further Reading, Previous: Reading and writing combinations, Up: Combinations [Index]
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.
Previous: Combination Examples, Up: Combinations [Index]
Further information on combinations can be found in,
Next: Sorting, Previous: Combinations, Up: Top [Index]
This chapter describes functions for creating and manipulating multisets. A multiset c is represented by an array of k integers in the range 0 to n-1, where each value c_i may occur more than once. The multiset c corresponds to indices of k elements chosen from an n element vector with replacement. In mathematical terms, n is the cardinality of the multiset while k is the maximum multiplicity of any value. Multisets are useful, for example, when iterating over the indices of a k-th order symmetric tensor in n-space.
The functions described in this chapter are defined in the header file gsl_multiset.h.
• The Multiset struct: | ||
• Multiset allocation: | ||
• Accessing multiset elements: | ||
• Multiset properties: | ||
• Multiset functions: | ||
• Reading and writing multisets: | ||
• Multiset Examples: |
Next: Multiset allocation, Up: Multisets [Index]
A multiset is defined by a structure containing three components, the
values of n and k, and a pointer to the multiset array.
The elements of the multiset array are all of type size_t
, and
are stored in increasing order. The gsl_multiset
structure
looks like this,
typedef struct { size_t n; size_t k; size_t *data; } gsl_multiset;
Next: Accessing multiset elements, Previous: The Multiset struct, Up: Multisets [Index]
This function allocates memory for a new multiset with parameters n,
k. The multiset is not initialized and its elements are undefined. Use
the function gsl_multiset_calloc
if you want to create a multiset which
is initialized to the lexicographically first multiset element. A null pointer
is returned if insufficient memory is available to create the multiset.
This function allocates memory for a new multiset with parameters n, k and initializes it to the lexicographically first multiset element. A null pointer is returned if insufficient memory is available to create the multiset.
This function initializes the multiset c to the lexicographically first multiset element, i.e. 0 repeated k times.
This function initializes the multiset c to the lexicographically last multiset element, i.e. n-1 repeated k times.
This function frees all the memory used by the multiset c.
This function copies the elements of the multiset src into the multiset dest. The two multisets must have the same size.
Next: Multiset properties, Previous: Multiset allocation, Up: Multisets [Index]
The following function can be used to access the elements of a multiset.
This function returns the value of the i-th element of the
multiset c. If i lies outside the allowed range of 0 to
k-1 then the error handler is invoked and 0 is returned. An inline version of this function is used when HAVE_INLINE
is defined.
Next: Multiset functions, Previous: Accessing multiset elements, Up: Multisets [Index]
This function returns the range (n) of the multiset c.
This function returns the number of elements (k) in the multiset c.
This function returns a pointer to the array of elements in the multiset c.
This function checks that the multiset c is valid. The k elements should lie in the range 0 to n-1, with each value occurring in nondecreasing order.
Next: Reading and writing multisets, Previous: Multiset properties, Up: Multisets [Index]
This function advances the multiset c to the next multiset element in
lexicographic order and returns GSL_SUCCESS
. If no further multisets
elements are available it returns GSL_FAILURE
and leaves c
unmodified. Starting with the first multiset and repeatedly applying this
function will iterate through all possible multisets of a given order.
This function steps backwards from the multiset c to the previous
multiset element in lexicographic order, returning GSL_SUCCESS
. If no
previous multiset is available it returns GSL_FAILURE
and leaves c
unmodified.
Next: Multiset Examples, Previous: Multiset functions, Up: Multisets [Index]
The library provides functions for reading and writing multisets to a file as binary data or formatted text.
This function writes the elements of the multiset c to the
stream stream in binary format. The function returns
GSL_EFAILED
if there was a problem writing to the file. Since the
data is written in the native binary format it may not be portable
between different architectures.
This function reads elements from the open stream stream into the
multiset c in binary format. The multiset c must be
preallocated with correct values of n and k since the
function uses the size of c to determine how many bytes to read.
The function returns GSL_EFAILED
if there was a problem reading
from the file. The data is assumed to have been written in the native
binary format on the same architecture.
This function writes the elements of the multiset c
line-by-line to the stream stream using the format specifier
format, which should be suitable for a type of size_t.
In ISO C99 the type modifier z
represents size_t
, so
"%zu\n"
is a suitable format.^{11} The function returns
GSL_EFAILED
if there was a problem writing to the file.
This function reads formatted data from the stream stream into the
multiset c. The multiset c must be preallocated with
correct values of n and k since the function uses the size of c to
determine how many numbers to read. The function returns
GSL_EFAILED
if there was a problem reading from the file.
Previous: Reading and writing multisets, Up: Multisets [Index]
The example program below prints all multisets elements containing the values {0,1,2,3} ordered by size. Multiset elements of the same size are ordered lexicographically.
#include <stdio.h> #include <gsl/gsl_multiset.h> int main (void) { gsl_multiset * c; size_t i; printf ("All multisets of {0,1,2,3} by size:\n") ; for (i = 0; i <= 4; i++) { c = gsl_multiset_calloc (4, i); do { printf ("{"); gsl_multiset_fprintf (stdout, c, " %u"); printf (" }\n"); } while (gsl_multiset_next (c) == GSL_SUCCESS); gsl_multiset_free (c); } return 0; }
Here is the output from the program,
$ ./a.out
All multisets of {0,1,2,3} by size: { } { 0 } { 1 } { 2 } { 3 } { 0 0 } { 0 1 } { 0 2 } { 0 3 } { 1 1 } { 1 2 } { 1 3 } { 2 2 } { 2 3 } { 3 3 } { 0 0 0 } { 0 0 1 } { 0 0 2 } { 0 0 3 } { 0 1 1 } { 0 1 2 } { 0 1 3 } { 0 2 2 } { 0 2 3 } { 0 3 3 } { 1 1 1 } { 1 1 2 } { 1 1 3 } { 1 2 2 } { 1 2 3 } { 1 3 3 } { 2 2 2 } { 2 2 3 } { 2 3 3 } { 3 3 3 } { 0 0 0 0 } { 0 0 0 1 } { 0 0 0 2 } { 0 0 0 3 } { 0 0 1 1 } { 0 0 1 2 } { 0 0 1 3 } { 0 0 2 2 } { 0 0 2 3 } { 0 0 3 3 } { 0 1 1 1 } { 0 1 1 2 } { 0 1 1 3 } { 0 1 2 2 } { 0 1 2 3 } { 0 1 3 3 } { 0 2 2 2 } { 0 2 2 3 } { 0 2 3 3 } { 0 3 3 3 } { 1 1 1 1 } { 1 1 1 2 } { 1 1 1 3 } { 1 1 2 2 } { 1 1 2 3 } { 1 1 3 3 } { 1 2 2 2 } { 1 2 2 3 } { 1 2 3 3 } { 1 3 3 3 } { 2 2 2 2 } { 2 2 2 3 } { 2 2 3 3 } { 2 3 3 3 } { 3 3 3 3 }
All 70 multisets are generated and sorted lexicographically.
Next: BLAS Support, Previous: Multisets, Up: Top [Index]
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.
• Sorting objects: | ||
• Sorting vectors: | ||
• Selecting the k smallest or largest elements: | ||
• Computing the rank: | ||
• Sorting Examples: | ||
• Sorting References and Further Reading: |
Next: Sorting vectors, Up: Sorting [Index]
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, 0
if the two arguments
are equal and a positive integer if the first argument is greater than
the second argument.
For example, the following function can be used to sort doubles into ascending numerical order.
int compare_doubles (const double * a, const double * b) { if (*a > *b) return 1; else if (*a < *b) return -1; else return 0; }
The appropriate function call to perform the sort is,
gsl_heapsort (array, count, sizeof(double), compare_doubles);
Note that unlike qsort
the heapsort algorithm cannot be made into
a stable sort by pointer arithmetic. The trick of comparing pointers for
equal elements in the comparison function does not work for the heapsort
algorithm. The heapsort algorithm performs an internal rearrangement of
the data which destroys its initial ordering.
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.
Next: Selecting the k smallest or largest elements, Previous: Sorting objects, Up: Sorting [Index]
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 n elements of the array data1 with stride stride1 into ascending numerical order, while making the same rearrangement of the array data2 with stride stride2, also of size n.
This function sorts the elements of the vector v into ascending numerical order.
This function sorts the elements of the vector v1 into ascending numerical order, while making the same rearrangement of the vector v2.
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.
Next: Computing the rank, Previous: Sorting vectors, Up: Sorting [Index]
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.
Next: Sorting Examples, Previous: Selecting the k smallest or largest elements, Up: Sorting [Index]
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); }
Next: Sorting References and Further Reading, Previous: Computing the rank, Up: Sorting [Index]
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 ("%zu smallest values from %zu\n", k, N); for (i = 0; i < k; i++) { printf ("%zu: %.18f\n", i, small[i]); } free (x); free (small); gsl_rng_free (r); return 0; }
The output lists the 5 smallest values, in ascending order,
$ ./a.out
5 smallest values from 100000 0: 0.000003489200025797 1: 0.000008199829608202 2: 0.000008953968062997 3: 0.000010712770745158 4: 0.000033531803637743
Previous: Sorting Examples, Up: Sorting [Index]
The subject of sorting is covered extensively in Knuth’s Sorting and Searching,
The Heapsort algorithm is described in the following book,
Next: Linear Algebra, Previous: Sorting, Up: Top [Index]
The Basic Linear Algebra Subprograms (BLAS) define a set of fundamental operations on vectors and matrices which can be used to create optimized higher-level linear algebra functionality.
The library provides a low-level layer which corresponds directly to the C-language BLAS standard, referred to here as “CBLAS”, and a higher-level interface for operations on GSL vectors and matrices. Users who are interested in simple operations on GSL vector and matrix objects should use the high-level layer described in this chapter. The functions are declared in the file gsl_blas.h and should satisfy the needs of most users.
Note that GSL matrices are implemented using dense-storage so the interface only includes the corresponding dense-storage BLAS functions. The full BLAS functionality for band-format and packed-format matrices is available through the low-level CBLAS interface. Similarly, GSL vectors are restricted to positive strides, whereas the low-level CBLAS interface supports negative strides as specified in the BLAS standard.^{12}
The interface for the gsl_cblas
layer is specified in the file
gsl_cblas.h. This interface corresponds to the BLAS Technical
Forum’s standard for the C interface to legacy BLAS
implementations. Users who have access to other conforming CBLAS
implementations can use these in place of the version provided by the
library. Note that users who have only a Fortran BLAS library can
use a CBLAS conformant wrapper to convert it into a CBLAS
library. A reference CBLAS wrapper for legacy Fortran
implementations exists as part of the CBLAS standard and can
be obtained from Netlib. The complete set of CBLAS functions is
listed in an appendix (see GSL CBLAS Library).
There are three levels of BLAS operations,
Vector operations, e.g. y = \alpha x + y
Matrix-vector operations, e.g. y = \alpha A x + \beta y
Matrix-matrix operations, e.g. C = \alpha A B + C
Each routine has a name which specifies the operation, the type of matrices involved and their precisions. Some of the most common operations and their names are given below,
scalar product, x^T y
vector sum, \alpha x + y
matrix-vector product, A x
matrix-vector solve, inv(A) x
matrix-matrix product, A B
matrix-matrix solve, inv(A) B
The types of matrices are,
general
general band
symmetric
symmetric band
symmetric packed
hermitian
hermitian band
hermitian packed
triangular
triangular band
triangular packed
Each operation is defined for four precisions,
single real
double real
single complex
double complex
Thus, for example, the name SGEMM stands for “single-precision general matrix-matrix multiply” and ZGEMM stands for “double-precision complex matrix-matrix multiply”.
Note that the vector and matrix arguments to BLAS functions must not be aliased, as the results are undefined when the underlying arrays overlap (see Aliasing of arrays).
• GSL BLAS Interface: | ||
• BLAS Examples: | ||
• BLAS References and Further Reading: |
Next: BLAS Examples, Up: BLAS Support [Index]
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.
• Level 1 GSL BLAS Interface: | ||
• Level 2 GSL BLAS Interface: | ||
• Level 3 GSL BLAS Interface: |
Next: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
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 dotu
These functions compute the complex conjugate scalar product x^H y for the vectors x and y, returning the result in dotc
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.
Next: Level 3 GSL BLAS Interface, Previous: Level 1 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
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 = op(A) x for the triangular matrix A, where
op(A) = A, A^T, A^H for TransA =
CblasNoTrans
, CblasTrans
, CblasConjTrans
. When
Uplo is CblasUpper
then the upper triangle of A is
used, and when Uplo is CblasLower
then the lower triangle
of A is used. If Diag is CblasNonUnit
then the
diagonal of the matrix is used, but if Diag is CblasUnit
then the diagonal elements of the matrix A are taken as unity and
are not referenced.
These functions compute inv(op(A)) x for x, where
op(A) = A, A^T, A^H for TransA =
CblasNoTrans
, CblasTrans
, CblasConjTrans
. When
Uplo is CblasUpper
then the upper triangle of A is
used, and when Uplo is CblasLower
then the lower triangle
of A is used. If Diag is CblasNonUnit
then the
diagonal of the matrix is used, but if Diag is CblasUnit
then the diagonal elements of the matrix A are taken as unity and
are not referenced.
These functions compute the matrix-vector product and sum y =
\alpha A x + \beta y for the symmetric matrix A. Since the
matrix A is symmetric only its upper half or lower half need to be
stored. When Uplo is CblasUpper
then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower
then the lower triangle and diagonal of A are
used.
These functions compute the matrix-vector product and sum y =
\alpha A x + \beta y for the hermitian matrix A. Since the
matrix A is hermitian only its upper half or lower half need to be
stored. When Uplo is CblasUpper
then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower
then the lower triangle and diagonal of A are
used. The imaginary elements of the diagonal are automatically assumed
to be zero and are not referenced.
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 CblasUpper
then the upper triangle and diagonal of
A are used, and when Uplo is CblasLower
then the
lower triangle and diagonal of A are used.
These functions compute the hermitian rank-1 update A = \alpha x
x^H + A of the hermitian matrix A. Since the matrix A is
hermitian only its upper half or lower half need to be stored. When
Uplo is CblasUpper
then the upper triangle and diagonal of
A are used, and when Uplo is CblasLower
then the
lower triangle and diagonal of A are used. The imaginary elements
of the diagonal are automatically set to zero.
These functions compute the symmetric rank-2 update A = \alpha x
y^T + \alpha y x^T + A of the symmetric matrix A. Since the
matrix A is symmetric only its upper half or lower half need to be
stored. When Uplo is CblasUpper
then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower
then the lower triangle and diagonal of A are
used.
These functions compute the hermitian rank-2 update A = \alpha x
y^H + \alpha^* y x^H + A of the hermitian matrix A. Since the
matrix A is hermitian only its upper half or lower half need to be
stored. When Uplo is CblasUpper
then the upper triangle
and diagonal of A are used, and when Uplo is
CblasLower
then the lower triangle and diagonal of A are
used. The imaginary elements of the diagonal are automatically set to zero.
Previous: Level 2 GSL BLAS Interface, Up: GSL BLAS Interface [Index]
These functions compute the matrix-matrix product and sum C =
\alpha op(A) op(B) + \beta C where op(A) = A, A^T,
A^H for TransA = CblasNoTrans
, CblasTrans
,
CblasConjTrans
and similarly for the parameter TransB.
These functions compute the matrix-matrix product and sum C =
\alpha A B + \beta C for Side is CblasLeft
and C =
\alpha B A + \beta C for Side is CblasRight
, where the
matrix A is symmetric. When Uplo is CblasUpper
then
the upper triangle and diagonal of A are used, and when Uplo
is CblasLower
then the lower triangle and diagonal of A are
used.
These functions compute the matrix-matrix product and sum C =
\alpha A B + \beta C for Side is CblasLeft
and C =
\alpha B A + \beta C for Side is CblasRight
, where the
matrix A is hermitian. When Uplo is CblasUpper
then
the upper triangle and diagonal of A are used, and when Uplo
is CblasLower
then the lower triangle and diagonal of A are
used. The imaginary elements of the diagonal are automatically set to
zero.
These functions compute the matrix-matrix product B = \alpha op(A)
B for Side is CblasLeft
and B = \alpha B op(A) for
Side is CblasRight
. The matrix A is triangular and
op(A) = A, A^T, A^H for TransA =
CblasNoTrans
, CblasTrans
, CblasConjTrans
. When
Uplo is CblasUpper
then the upper triangle of A is
used, and when Uplo is CblasLower
then the lower triangle
of A is used. If Diag is CblasNonUnit
then the
diagonal of A is used, but if Diag is CblasUnit
then
the diagonal elements of the matrix A are taken as unity and are
not referenced.
These functions compute the inverse-matrix matrix product
B = \alpha op(inv(A))B for Side is
CblasLeft
and B = \alpha B op(inv(A)) for
Side is CblasRight
. The matrix A is triangular and
op(A) = A, A^T, A^H for TransA =
CblasNoTrans
, CblasTrans
, CblasConjTrans
. When
Uplo is CblasUpper
then the upper triangle of A is
used, and when Uplo is CblasLower
then the lower triangle
of A is used. If Diag is CblasNonUnit
then the
diagonal of A is used, but if Diag is CblasUnit
then
the diagonal elements of the matrix A are taken as unity and are
not referenced.
These functions compute a rank-k update of the symmetric matrix C,
C = \alpha A A^T + \beta C when Trans is
CblasNoTrans
and C = \alpha A^T A + \beta C when
Trans is CblasTrans
. Since the matrix C is symmetric
only its upper half or lower half need to be stored. When Uplo is
CblasUpper
then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower
then the lower triangle
and diagonal of C are used.
These functions compute a rank-k update of the hermitian matrix C,
C = \alpha A A^H + \beta C when Trans is
CblasNoTrans
and C = \alpha A^H A + \beta C when
Trans is CblasConjTrans
. Since the matrix C is hermitian
only its upper half or lower half need to be stored. When Uplo is
CblasUpper
then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower
then the lower triangle
and diagonal of C are used. The imaginary elements of the
diagonal are automatically set to zero.
These functions compute a rank-2k update of the symmetric matrix C,
C = \alpha A B^T + \alpha B A^T + \beta C when Trans is
CblasNoTrans
and C = \alpha A^T B + \alpha B^T A + \beta C when
Trans is CblasTrans
. Since the matrix C is symmetric
only its upper half or lower half need to be stored. When Uplo is
CblasUpper
then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower
then the lower triangle
and diagonal of C are used.
These functions compute a rank-2k update of the hermitian matrix C,
C = \alpha A B^H + \alpha^* B A^H + \beta C when Trans is
CblasNoTrans
and C = \alpha A^H B + \alpha^* B^H A + \beta C when
Trans is CblasConjTrans
. Since the matrix C is hermitian
only its upper half or lower half need to be stored. When Uplo is
CblasUpper
then the upper triangle and diagonal of C are
used, and when Uplo is CblasLower
then the lower triangle
and diagonal of C are used. The imaginary elements of the
diagonal are automatically set to zero.
Next: BLAS References and Further Reading, Previous: GSL BLAS Interface, Up: BLAS Support [Index]
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 ]
Previous: BLAS Examples, Up: BLAS Support [Index]
Information on the BLAS standards, including both the legacy and updated interface standards, is available online from the BLAS Homepage and BLAS Technical Forum web-site.
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.
Next: Eigensystems, Previous: BLAS Support, Up: Top [Index]
This chapter describes functions for solving linear systems. The
library provides linear algebra operations which operate directly on
the gsl_vector
and gsl_matrix
objects. These routines
use the standard algorithms from Golub & Van Loan’s Matrix
Computations with Level-1 and Level-2 BLAS calls for efficiency.
The functions described in this chapter are declared in the header file gsl_linalg.h.
Next: QR Decomposition, Up: Linear Algebra [Index]
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 on output. The j-th column of the matrix P is given by the k-th column of the identity matrix, where k = p_j the j-th element of the permutation vector. The sign of the permutation is given by signum. It has the value (-1)^n, where n is the number of interchanges in the permutation.
The algorithm used in the decomposition is Gaussian Elimination with partial pivoting (Golub & Van Loan, Matrix Computations, Algorithm 3.4.1).
These functions solve the square system A x = b using the LU
decomposition of A into (LU, p) given by
gsl_linalg_LU_decomp
or gsl_linalg_complex_LU_decomp
as input.
These functions solve the square system A x = b in-place using the precomputed LU decomposition of A into (LU,p). On input x should contain the right-hand side b, which is replaced by the solution on output.
These functions apply an iterative improvement to x, the solution of A x = b, from the precomputed 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.
Next: QR Decomposition with Column Pivoting, Previous: LU Decomposition, Up: Linear Algebra [Index]
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 held in (QR, tau) which must
have been computed previously with gsl_linalg_QR_decomp
.
The least-squares solution for
rectangular systems can be found using gsl_linalg_QR_lssolve
.
This function solves the square system A x = b in-place using
the QR decomposition of A held in (QR,tau)
which must have been computed previously by
gsl_linalg_QR_decomp
. On input x should contain the
right-hand side b, which is replaced by the solution on output.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns. The least squares solution minimizes the Euclidean norm of the
residual, ||Ax - b||.The routine requires as input
the QR decomposition
of A into (QR, tau) given by
gsl_linalg_QR_decomp
. The solution is returned in x. The
residual is computed as a by-product and stored in residual.
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 applies the matrix Q^T encoded in the decomposition (QR,tau) to the matrix A, storing the result Q^T A in A. The matrix multiplication is carried out directly using the encoding of the Householder vectors without needing to form the full matrix Q^T.
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.
Next: Complete Orthogonal Decomposition, Previous: QR Decomposition, Up: Linear Algebra [Index]
The QR decomposition of an M-by-N matrix A can be extended to the rank deficient case by introducing a column permutation P,
A P = Q R
The first r columns of Q form an orthonormal basis for the range of A for a matrix with column rank r. This decomposition can also be used to convert the linear system A x = b into the triangular system R y = Q^T b, x = P y, which can be solved by back-substitution and permutation. We denote the QR decomposition with column pivoting by QRP^T since A = Q R P^T. When A is rank deficient with r = {\rm rank}(A), the matrix R can be partitioned as
R = [ R11 R12; 0 R22 ] =~ [ R11 R12; 0 0 ]
where R_{11} is r-by-r and nonsingular. In this case, a “basic” least squares solution for the overdetermined system A x = b can be obtained as
x = P [ R11^-1 c1 ; 0 ]
where c_1 consists of the first r elements of Q^T b. This basic solution is not guaranteed to be the minimum norm solution unless R_{12} = 0 (see Complete Orthogonal Decomposition).
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 held in (QR, tau, p) which must
have been computed previously by gsl_linalg_QRPT_decomp
.
This function solves the square system A x = b in-place using the QRP^T decomposition of A held in (QR,tau,p). On input x should contain the right-hand side b, which is replaced by the solution on output.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns and is assumed to have full rank. The least squares solution minimizes
the Euclidean norm of the residual, ||b - A x||. The routine requires as input
the QR decomposition of A into (QR, tau, p) given by
gsl_linalg_QRPT_decomp
. The solution is returned in x. The
residual is computed as a by-product and stored in residual. For rank
deficient matrices, gsl_linalg_QRPT_lssolve2
should be used instead.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns and has rank given by the input rank. If the user does not
know the rank of A, the routine gsl_linalg_QRPT_rank
can be
called to estimate it. The least squares solution is
the so-called “basic” solution discussed above and may not be the minimum
norm solution. The routine requires as input
the QR decomposition of A into (QR, tau, p) given by
gsl_linalg_QRPT_decomp
. The solution is returned in x. The
residual is computed as a by-product and stored in residual.
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 P) where the output matrices Q' and R' are also orthogonal and right triangular. Note that w is destroyed by the update. The permutation p is not changed.
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.
This function estimates the rank of the triangular matrix R contained in QR. The algorithm simply counts the number of diagonal elements of R whose absolute value is greater than the specified tolerance tol. If the input tol is negative, a default value of 20 (M + N) eps(max(|diag(R)|)) is used.
This function estimates the reciprocal condition number (using the 1-norm) of the R factor, stored in the upper triangle of QR. The reciprocal condition number estimate, defined as 1 / (||R||_1 \cdot ||R^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Singular Value Decomposition, Previous: QR Decomposition with Column Pivoting, Up: Linear Algebra [Index]
The complete orthogonal decomposition of a M-by-N matrix A is a generalization of the QR decomposition with column pivoting, given by
A P = Q [ R11 0 ] Z [ 0 0 ]
where P is a N-by-N permutation matrix, Q is M-by-M orthogonal, R_{11} is r-by-r upper triangular, with r = {\rm rank}(A), and Z is N-by-N orthogonal. If A has full rank, then R_{11} = R, Z = I and this reduces to the QR decomposition with column pivoting. The advantage of using the complete orthogonal decomposition for rank deficient matrices is the ability to compute the minimum norm solution to the linear least squares problem Ax = b, which is given by
x = P Z^T [ R11^-1 c1 ] [ 0 ]
and the vector c_1 is the first r elements of Q^T b.
These functions factor the M-by-N matrix A into the decomposition A = Q R Z P^T. The rank of A
is computed as the number of diagonal elements of R greater than the tolerance tol and output in rank.
If tol is not specified, a default value is used (see gsl_linalg_QRPT_rank
). On output, the permutation
matrix P is stored in p. The matrix R_{11} is stored in the upper rank-by-rank block of A.
The matrices Q and Z are encoded in packed storage in A on output. The vectors tau_Q and tau_Z
contain the Householder scalars corresponding to the matrices Q and Z respectively and must be
of length k = \min(M,N). The vector work is additional workspace of length N.
This function finds the least squares solution to the overdetermined
system A x = b where the matrix A has more rows than
columns. The least squares solution minimizes the Euclidean norm of the
residual, ||b - A x||. The routine requires as input
the QRZ decomposition of A into (QRZ, tau_Q, tau_Z, p, rank)
given by gsl_linalg_COD_decomp
. The solution is returned in x. The
residual is computed as a by-product and stored in residual.
This function unpacks the encoded QRZ decomposition (QRZ, tau_Q, tau_Z, rank) into the matrices Q, R, and Z, where Q is M-by-M, R is M-by-N, and Z is N-by-N.
This function multiplies the input matrix A on the right by Z, A' = A Z using the encoded QRZ decomposition (QRZ, tau_Z, rank). A must have N columns but may have any number of rows. Additional workspace of length M is provided in work.
Next: Cholesky Decomposition, Previous: Complete Orthogonal Decomposition, Up: Linear Algebra [Index]
A general rectangular M-by-N matrix A has a singular value decomposition (SVD) into the product of an M-by-N orthogonal matrix U, an N-by-N diagonal matrix of singular values S and the transpose of an N-by-N orthogonal square matrix V,
A = U S V^T
The singular values \sigma_i = S_{ii} are all non-negative and are generally chosen to form a non-increasing sequence \sigma_1 >= \sigma_2 >= ... >= \sigma_N >= 0.
The singular value decomposition of a matrix has many practical uses. The condition number of the matrix is given by the ratio of the largest singular value to the smallest singular value. The presence of a zero singular value indicates that the matrix is singular. The number of non-zero singular values indicates the rank of the matrix. In practice singular value decomposition of a rank-deficient matrix will not produce exact zeroes for singular values, due to finite numerical precision. Small singular values should be edited by choosing a suitable tolerance.
For a rank-deficient matrix, the null space of A is given by the columns of V corresponding to the zero singular values. Similarly, the range of A is given by columns of U corresponding to the non-zero singular values.
Note that the routines here compute the “thin” version of the SVD with U as M-by-N orthogonal matrix. This allows in-place computation and is the most commonly-used form in practice. Mathematically, the “full” SVD is defined with U as an M-by-M orthogonal matrix and S as an M-by-N diagonal matrix (with additional rows of zeros).
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 which must
have been computed previously with gsl_linalg_SV_decomp
.
Only non-zero singular values are used in computing the solution. The parts of the solution corresponding to singular values of zero are ignored. Other singular values can be edited out by setting them to zero before calling this function.
In the over-determined case where A has more rows than columns the system is solved in the least squares sense, returning the solution x which minimizes ||A x - b||_2.
This function computes the statistical leverage values h_i of a matrix A
using its singular value decomposition (U, S, V) previously computed
with gsl_linalg_SV_decomp
. h_i are the diagonal values of the matrix
A (A^T A)^{-1} A^T and depend only on the matrix U which is the input to
this function.
Next: Pivoted Cholesky Decomposition, Previous: Singular Value Decomposition, Up: Linear Algebra [Index]
A symmetric, positive definite square matrix A has a Cholesky decomposition into a product of a lower triangular matrix L and its transpose L^T,
A = L L^T
This is sometimes referred to as taking the square-root of a matrix. The Cholesky decomposition can only be carried out when all the eigenvalues of the matrix are positive. This decomposition can be used to convert the linear system A x = b into a pair of triangular systems (L y = b, L^T x = y), which can be solved by forward and back-substitution.
If the matrix A is near singular, it is sometimes possible to reduce the condition number and recover a more accurate solution vector x by scaling as
( S A S ) ( S^(-1) x ) = S b
where S is a diagonal matrix whose elements are given by S_{ii} = 1/\sqrt{A_{ii}}. This scaling is also known as Jacobi preconditioning. There are routines below to solve both the scaled and unscaled systems.
These functions factorize the symmetric, positive-definite square matrix
A into the Cholesky decomposition A = L L^T (or
A = L L^H
for the complex case). On input, the values from the diagonal and lower-triangular
part of the matrix A are used (the upper triangular part is ignored). On output
the diagonal and lower triangular part of the input matrix A contain the matrix
L, while the upper triangular part is unmodified. If the matrix is not
positive-definite then the decomposition will fail, returning the
error code GSL_EDOM
.
When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
This function is now deprecated and is provided only for backward compatibility.
These functions solve the system A x = b using the Cholesky
decomposition of A held in the matrix cholesky which must
have been previously computed by gsl_linalg_cholesky_decomp
or
gsl_linalg_complex_cholesky_decomp
.
These functions solve the system A x = b in-place using the
Cholesky decomposition of A held in the matrix cholesky
which must have been previously computed by
gsl_linalg_cholesky_decomp
or
gsl_linalg_complex_cholesky_decomp
. On input x should
contain the right-hand side b, which is replaced by the
solution on output.
These functions compute the inverse of a matrix from its Cholesky
decomposition cholesky, which must have been previously computed
by gsl_linalg_cholesky_decomp
or
gsl_linalg_complex_cholesky_decomp
. On output, the inverse is
stored in-place in cholesky.
This function calculates a diagonal scaling transformation S for
the symmetric, positive-definite square matrix A, and then
computes the Cholesky decomposition S A S = L L^T.
On input, the values from the diagonal and lower-triangular part of the matrix A are
used (the upper triangular part is ignored). On output the diagonal and lower triangular part
of the input matrix A contain the matrix L, while the upper triangular part
of the input matrix is overwritten with L^T (the diagonal terms being
identical for both L and L^T). If the matrix is not
positive-definite then the decomposition will fail, returning the
error code GSL_EDOM
. The diagonal scale factors are stored in S
on output.
When testing whether a matrix is positive-definite, disable the error handler first to avoid triggering an error.
This function solves the system (S A S) (S^{-1} x) = S b using the Cholesky
decomposition of S A S held in the matrix cholesky which must
have been previously computed by gsl_linalg_cholesky_decomp2
.
This function solves the system (S A S) (S^{-1} x) = S b in-place using the
Cholesky decomposition of S A S held in the matrix cholesky
which must have been previously computed by
gsl_linalg_cholesky_decomp2
. On input x should
contain the right-hand side b, which is replaced by the
solution on output.
This function calculates a diagonal scaling transformation of the symmetric, positive definite matrix A, such that S A S has a condition number within a factor of N of the matrix of smallest possible condition number over all possible diagonal scalings. On output, S contains the scale factors, given by S_i = 1/\sqrt{A_{ii}}. For any A_{ii} \le 0, the corresponding scale factor S_i is set to 1.
This function applies the scaling transformation S to the matrix A. On output, A is replaced by S A S.
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite matrix A, using its Cholesky decomposition provided in cholesky. The reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot ||A^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Modified Cholesky Decomposition, Previous: Cholesky Decomposition, Up: Linear Algebra [Index]
A symmetric, positive definite square matrix A has an alternate Cholesky decomposition into a product of a lower unit triangular matrix L, a diagonal matrix D and L^T, given by L D L^T. This is equivalent to the Cholesky formulation discussed above, with the standard Cholesky lower triangular factor given by L D^{1 \over 2}. For ill-conditioned matrices, it can help to use a pivoting strategy to prevent the entries of D and L from growing too large, and also ensure D_1 \ge D_2 \ge \cdots \ge D_n > 0, where D_i are the diagonal entries of D. The final decomposition is given by
P A P^T = L D L^T
where P is a permutation matrix.
This function factors the symmetric, positive-definite square matrix A into the Pivoted Cholesky decomposition P A P^T = L D L^T. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on output.
This function solves the system A x = b using the Pivoted Cholesky
decomposition of A held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_pcholesky_decomp
.
This function solves the system A x = b in-place using the Pivoted Cholesky
decomposition of A held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_pcholesky_decomp
.
On input, x contains the right hand side vector b which is
replaced by the solution vector on output.
This function computes the pivoted Cholesky factorization of the matrix S A S, where the input matrix A is symmetric and positive definite, and the diagonal scaling matrix S is computed to reduce the condition number of A as much as possible. See Cholesky Decomposition for more information on the matrix S. The Pivoted Cholesky decomposition satisfies P S A S P^T = L D L^T. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on output. The diagonal scaling transformation is stored in S on output.
This function solves the system (S A S) (S^{-1} x) = S b using the Pivoted Cholesky
decomposition of S A S held in the matrix LDLT, permutation
p, and vector S, which must have been previously computed by
gsl_linalg_pcholesky_decomp2
.
This function solves the system (S A S) (S^{-1} x) = S b in-place using the Pivoted Cholesky
decomposition of S A S held in the matrix LDLT, permutation
p and vector S, which must have been previously computed by
gsl_linalg_pcholesky_decomp2
.
On input, x contains the right hand side vector b which is
replaced by the solution vector on output.
This function computes the inverse of the matrix A, using the Pivoted Cholesky decomposition stored in LDLT and p. On output, the matrix Ainv contains A^{-1}.
This function estimates the reciprocal condition number (using the 1-norm) of the symmetric positive definite matrix A, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, defined as 1 / (||A||_1 \cdot ||A^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Tridiagonal Decomposition of Real Symmetric Matrices, Previous: Pivoted Cholesky Decomposition, Up: Linear Algebra [Index]
The modified Cholesky decomposition is suitable for solving systems A x = b where A is a symmetric indefinite matrix. Such matrices arise in nonlinear optimization algorithms. The standard Cholesky decomposition requires a positive definite matrix and would fail in this case. Instead of resorting to a method like QR or SVD, which do not take into account the symmetry of the matrix, we can instead introduce a small perturbation to the matrix A to make it positive definite, and then use a Cholesky decomposition on the perturbed matrix. The resulting decomposition satisfies
P (A + E) P^T = L D L^T
where P is a permutation matrix, E is a diagonal perturbation matrix, L is unit lower triangular, and D is diagonal. If A is sufficiently positive definite, then the perturbation matrix E will be zero and this method is equivalent to the pivoted Cholesky algorithm. For indefinite matrices, the perturbation matrix E is computed to ensure that A + E is positive definite and well conditioned.
This function factors the symmetric, indefinite square matrix A into the Modified Cholesky decomposition P (A + E) P^T = L D L^T. On input, the values from the diagonal and lower-triangular part of the matrix A are used to construct the factorization. On output the diagonal of the input matrix A stores the diagonal elements of D, and the lower triangular portion of A contains the matrix L. Since L has ones on its diagonal these do not need to be explicitely stored. The upper triangular portion of A is unmodified. The permutation matrix P is stored in p on output. The diagonal perturbation matrix is stored in E on output. The parameter E may be set to NULL if it is not required.
This function solves the perturbed system (A + E) x = b using the Cholesky
decomposition of A + E held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_mcholesky_decomp
.
This function solves the perturbed system (A + E) x = b in-place using the Cholesky
decomposition of A + E held in the matrix LDLT and permutation
p which must have been previously computed by gsl_linalg_mcholesky_decomp
.
On input, x contains the right hand side vector b which is
replaced by the solution vector on output.
This function estimates the reciprocal condition number (using the 1-norm) of the perturbed matrix A + E, using its pivoted Cholesky decomposition provided in LDLT. The reciprocal condition number estimate, defined as 1 / (||A + E||_1 \cdot ||(A + E)^{-1}||_1), is stored in rcond. Additional workspace of size 3 N is required in work.
Next: Tridiagonal Decomposition of Hermitian Matrices, Previous: Modified Cholesky Decomposition, Up: Linear Algebra [Index]
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_decomp
into
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_decomp
into the vectors diag and subdiag.
Next: Hessenberg Decomposition of Real Matrices, Previous: Tridiagonal Decomposition of Real Symmetric Matrices, Up: Linear Algebra [Index]
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 unitary matrix U. This storage scheme is the same as used by LAPACK. The upper triangular part of A and imaginary parts of the diagonal are not referenced.
This function unpacks the encoded tridiagonal decomposition (A,
tau) obtained from gsl_linalg_hermtd_decomp
into the
unitary matrix U, the real vector of diagonal elements diag and
the real vector of subdiagonal elements subdiag.
This function unpacks the diagonal and subdiagonal of the encoded
tridiagonal decomposition (A, tau) obtained from the
gsl_linalg_hermtd_decomp
into the real vectors
diag and subdiag.
Next: Hessenberg-Triangular Decomposition of Real Matrices, Previous: Tridiagonal Decomposition of Hermitian Matrices, Up: Linear Algebra [Index]
A general real matrix A can be decomposed by orthogonal similarity transformations into the form
A = U H U^T
where U is orthogonal and H is an upper Hessenberg matrix, meaning that it has zeros below the first subdiagonal. The Hessenberg reduction is the first step in the Schur decomposition for the nonsymmetric eigenvalue problem, but has applications in other areas as well.
This function computes the Hessenberg decomposition of the matrix A by applying the similarity transformation H = U^T A U. On output, H is stored in the upper portion of A. The information required to construct the matrix U is stored in the lower triangular portion of A. U is a product of N - 2 Householder matrices. The Householder vectors are stored in the lower portion of A (below the subdiagonal) and the Householder coefficients are stored in the vector tau. tau must be of length N.
This function constructs the orthogonal matrix U from the
information stored in the Hessenberg matrix H along with the
vector tau. H and tau are outputs from
gsl_linalg_hessenberg_decomp
.
This function is similar to gsl_linalg_hessenberg_unpack
, except
it accumulates the matrix U into V, so that V' = VU.
The matrix V must be initialized prior to calling this function.
Setting V to the identity matrix provides the same result as
gsl_linalg_hessenberg_unpack
. If H is order N, then
V must have N columns but may have any number of rows.
This function sets the lower triangular portion of H, below
the subdiagonal, to zero. It is useful for clearing out the
Householder vectors after calling gsl_linalg_hessenberg_decomp
.
Next: Bidiagonalization, Previous: Hessenberg Decomposition of Real Matrices, Up: Linear Algebra [Index]
A general real matrix pair (A, B) can be decomposed by orthogonal similarity transformations into the form
A = U H V^T B = U R V^T
where U and V are orthogonal, H is an upper Hessenberg matrix, and R is upper triangular. The Hessenberg-Triangular reduction is the first step in the generalized Schur decomposition for the generalized eigenvalue problem.
This function computes the Hessenberg-Triangular decomposition of the matrix pair (A, B). On output, H is stored in A, and R is stored in B. If U and V are provided (they may be null), the similarity transformations are stored in them. Additional workspace of length N is needed in work.
Next: Givens Rotations, Previous: Hessenberg-Triangular Decomposition of Real Matrices, Up: Linear Algebra [Index]
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 produced by
gsl_linalg_bidiag_decomp
, (A, tau_U, tau_V)
into the separate orthogonal matrices U, V and the diagonal
vector diag and superdiagonal superdiag. Note that U
is stored as a compact M-by-N orthogonal matrix satisfying
U^T U = I for efficiency.
This function unpacks the bidiagonal decomposition of A produced by
gsl_linalg_bidiag_decomp
, (A, tau_U, tau_V)
into the separate orthogonal matrices U, V and the diagonal
vector diag and superdiagonal superdiag. The matrix U
is stored in-place in A.
This function unpacks the diagonal and superdiagonal of the bidiagonal
decomposition of A from gsl_linalg_bidiag_decomp
, into
the diagonal vector diag and superdiagonal vector superdiag.
Next: Householder Transformations, Previous: Bidiagonalization, Up: Linear Algebra [Index]
A Givens rotation is a rotation in the plane acting on two elements of a given vector. It can be represented in matrix form as
where the \cos{\theta} and \sin{\theta} appear at the intersection of the ith and jth rows and columns. When acting on a vector x, G(i,j,\theta) x performs a rotation of the (i,j) elements of x. Givens rotations are typically used to introduce zeros in vectors, such as during the QR decomposition of a matrix. In this case, it is typically desired to find c and s such that
with r = \sqrt{a^2 + b^2}.
This function computes c = \cos{\theta} and s = \sin{\theta} so that the Givens matrix G(\theta) acting on the vector (a,b) produces (r, 0), with r = \sqrt{a^2 + b^2}.
This function applies the Givens rotation defined by c = \cos{\theta} and s = \sin{\theta} to the i and j elements of v. On output, (v(i),v(j)) \leftarrow G(\theta) (v(i),v(j)).
Next: Householder solver for linear systems, Previous: Givens Rotations, Up: Linear Algebra [Index]
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 w except the first. On output the Householder vector v is stored in w and the scalar \tau is returned. The householder vector v is normalized so that v[0] = 1, however this 1 is not stored in the output vector. Instead, w[0] is set to the first element of the transformed vector, so that if u = P w, w[0] = u[0] on output and the remainder of u is zero.
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.
Next: Tridiagonal Systems, Previous: Householder Transformations, Up: Linear Algebra [Index]
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.
Next: Triangular Systems, Previous: Householder solver for linear systems, Up: Linear Algebra [Index]
The functions described in this section efficiently solve symmetric,
non-symmetric and cyclic tridiagonal systems with minimal storage.
Note that the current implementations of these functions use a variant
of Cholesky decomposition, so the tridiagonal matrix must be positive
definite. For non-positive definite matrices, the functions return
the error code GSL_ESING
.
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 )
Next: Balancing, Previous: Tridiagonal Systems, Up: Linear Algebra [Index]
These functions calculate the in-place inverse of the triangular matrix T. When
the upper
prefix is specified, then the upper triangle of T is used, and when
the lower
prefix is specified, the lower triangle is used. If the unit
prefix is specified, then the diagonal elements of the matrix T are taken as
unity and are not referenced. Otherwise the diagonal elements are used in the inversion.
These functions estimate the reciprocal condition number, in the 1-norm, of the upper or lower N-by-N triangular matrix T. The reciprocal condition number is stored in rcond on output, and is defined by 1 / (||T||_1 \cdot ||T^{-1}||_1). Additional workspace of size 3 N is required in work.
Next: Linear Algebra Examples, Previous: Triangular Systems, Up: Linear Algebra [Index]
The process of balancing a matrix applies similarity transformations to make the rows and columns have comparable norms. This is useful, for example, to reduce roundoff errors in the solution of eigenvalue problems. Balancing a matrix A consists of replacing A with a similar matrix
A' = D^(-1) A D
where D is a diagonal matrix whose entries are powers of the floating point radix.
This function replaces the matrix A with its balanced counterpart and stores the diagonal elements of the similarity transformation into the vector D.
Next: Linear Algebra References and Further Reading, Previous: Balancing, Up: Linear Algebra [Index]
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); gsl_vector_free (x); return 0; }
Here is the output from the program,
x = -4.05205 -12.6056 1.66091 8.69377
This can be verified by multiplying the solution x by the original matrix A using GNU OCTAVE,
octave> A = [ 0.18, 0.60, 0.57, 0.96; 0.41, 0.24, 0.99, 0.58; 0.14, 0.30, 0.97, 0.66; 0.51, 0.13, 0.19, 0.85 ]; octave> x = [ -4.05205; -12.6056; 1.66091; 8.69377]; octave> A * x ans = 1.0000 2.0000 3.0000 4.0000
This reproduces the original right-hand side vector, b, in accordance with the equation A x = b.
Previous: Linear Algebra Examples, Up: Linear Algebra [Index]
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.
The algorithm for estimating a matrix condition number is described in the following paper,
Next: Fast Fourier Transforms, Previous: Linear Algebra, Up: Top [Index]
This chapter describes functions for computing eigenvalues and eigenvectors of matrices. There are routines for real symmetric, real nonsymmetric, complex hermitian, real generalized symmetric-definite, complex generalized hermitian-definite, and real generalized nonsymmetric eigensystems. Eigenvalues can be computed with or without eigenvectors. The hermitian and real symmetric matrix algorithms are symmetric bidiagonalization followed by QR reduction. The nonsymmetric algorithm is the Francis QR double-shift. The generalized nonsymmetric algorithm is the QZ method due to Moler and Stewart.
The functions described in this chapter are declared in the header file gsl_eigen.h.
Next: Complex Hermitian Matrices, Up: Eigensystems [Index]
For real symmetric matrices, the library uses the symmetric bidiagonalization and QR reduction method. This is described in Golub & van Loan, section 8.3. The computed eigenvalues are accurate to an absolute accuracy of \epsilon ||A||_2, where \epsilon is the machine precision.
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.
Next: Real Nonsymmetric Matrices, Previous: Real Symmetric Matrices, Up: Eigensystems [Index]
For hermitian matrices, the library uses the complex form of the symmetric bidiagonalization and QR reduction method.
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.
Next: Real Generalized Symmetric-Definite Eigensystems, Previous: Complex Hermitian Matrices, Up: Eigensystems [Index]
The solution of the real nonsymmetric eigensystem problem for a matrix A involves computing the Schur decomposition
A = Z T Z^T
where Z is an orthogonal matrix of Schur vectors and T, the Schur form, is quasi upper triangular with diagonal 1-by-1 blocks which are real eigenvalues of A, and diagonal 2-by-2 blocks whose eigenvalues are complex conjugate eigenvalues of A. The algorithm used is the double-shift Francis method.
This function allocates a workspace for computing eigenvalues of n-by-n real nonsymmetric matrices. The size of the workspace is O(2n).
This function frees the memory associated with the workspace w.
This function sets some parameters which determine how the eigenvalue
problem is solved in subsequent calls to gsl_eigen_nonsymm
.
If compute_t is set to 1, the full Schur form T will be
computed by gsl_eigen_nonsymm
. If it is set to 0,
T will not be computed (this is the default setting). Computing
the full Schur form T requires approximately 1.5–2 times the
number of flops.
If balance is set to 1, a balancing transformation is applied
to the matrix prior to computing eigenvalues. This transformation is
designed to make the rows and columns of the matrix have comparable
norms, and can result in more accurate eigenvalues for matrices
whose entries vary widely in magnitude. See Balancing for more
information. Note that the balancing transformation does not preserve
the orthogonality of the Schur vectors, so if you wish to compute the
Schur vectors with gsl_eigen_nonsymm_Z
you will obtain the Schur
vectors of the balanced matrix instead of the original matrix. The
relationship will be
T = Q^t D^(-1) A D Q
where Q is the matrix of Schur vectors for the balanced matrix, and
D is the balancing transformation. Then gsl_eigen_nonsymm_Z
will compute a matrix Z which satisfies
T = Z^(-1) A Z
with Z = D Q. Note that Z will not be orthogonal. For this reason, balancing is not performed by default.
This function computes the eigenvalues of the real nonsymmetric matrix
A and stores them in the vector eval. If T is
desired, it is stored in the upper portion of A on output.
Otherwise, on output, the diagonal of A will contain the
1-by-1 real eigenvalues and 2-by-2
complex conjugate eigenvalue systems, and the rest of A is
destroyed. In rare cases, this function may fail to find all
eigenvalues. If this happens, an error code is returned
and the number of converged eigenvalues is stored in w->n_evals
.
The converged eigenvalues are stored in the beginning of eval.
This function is identical to gsl_eigen_nonsymm
except that it also
computes the Schur vectors and stores them into Z.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real nonsymmetric matrices. The size of the workspace is O(5n).
This function frees the memory associated with the workspace w.
This function sets parameters which determine how the eigenvalue
problem is solved in subsequent calls to gsl_eigen_nonsymmv
.
If balance is set to 1, a balancing transformation is applied
to the matrix. See gsl_eigen_nonsymm_params
for more information.
Balancing is turned off by default since it does not preserve the
orthogonality of the Schur vectors.
This function computes eigenvalues and right eigenvectors of the
n-by-n real nonsymmetric matrix A. It first calls
gsl_eigen_nonsymm
to compute the eigenvalues, Schur form T, and
Schur vectors. Then it finds eigenvectors of T and backtransforms
them using the Schur vectors. The Schur vectors are destroyed in the
process, but can be saved by using gsl_eigen_nonsymmv_Z
. The
computed eigenvectors are normalized to have unit magnitude. On
output, the upper portion of A contains the Schur form
T. If gsl_eigen_nonsymm
fails, no eigenvectors are
computed, and an error code is returned.
This function is identical to gsl_eigen_nonsymmv
except that it also saves
the Schur vectors into Z.
Next: Complex Generalized Hermitian-Definite Eigensystems, Previous: Real Nonsymmetric Matrices, Up: Eigensystems [Index]
The real generalized symmetric-definite eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
where A and B are symmetric matrices, and B is positive-definite. This problem reduces to the standard symmetric eigenvalue problem by applying the Cholesky decomposition to B:
A x = \lambda B x A x = \lambda L L^t x ( L^{-1} A L^{-t} ) L^t x = \lambda L^t x
Therefore, the problem becomes C y = \lambda y where C = L^{-1} A L^{-t} is symmetric, and y = L^t x. The standard symmetric eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues and eigenvectors of the generalized symmetric-definite eigenproblem are always real.
This function allocates a workspace for computing eigenvalues of n-by-n real generalized symmetric-definite eigensystems. The size of the workspace is O(2n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues of the real generalized symmetric-definite matrix pair (A, B), and stores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A is destroyed.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized symmetric-definite eigensystems. The size of the workspace is O(4n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues and eigenvectors of the real generalized symmetric-definite matrix pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.
Next: Real Generalized Nonsymmetric Eigensystems, Previous: Real Generalized Symmetric-Definite Eigensystems, Up: Eigensystems [Index]
The complex generalized hermitian-definite eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
where A and B are hermitian matrices, and B is positive-definite. Similarly to the real case, this can be reduced to C y = \lambda y where C = L^{-1} A L^{-H} is hermitian, and y = L^H x. The standard hermitian eigensolver can be applied to the matrix C. The resulting eigenvectors are backtransformed to find the vectors of the original problem. The eigenvalues of the generalized hermitian-definite eigenproblem are always real.
This function allocates a workspace for computing eigenvalues of n-by-n complex generalized hermitian-definite eigensystems. The size of the workspace is O(3n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues of the complex generalized hermitian-definite matrix pair (A, B), and stores them in eval, using the method outlined above. On output, B contains its Cholesky decomposition and A is destroyed.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n complex generalized hermitian-definite eigensystems. The size of the workspace is O(5n).
This function frees the memory associated with the workspace w.
This function computes the eigenvalues and eigenvectors of the complex generalized hermitian-definite matrix pair (A, B), and stores them in eval and evec respectively. The computed eigenvectors are normalized to have unit magnitude. On output, B contains its Cholesky decomposition and A is destroyed.
Next: Sorting Eigenvalues and Eigenvectors, Previous: Complex Generalized Hermitian-Definite Eigensystems, Up: Eigensystems [Index]
Given two square matrices (A, B), the generalized nonsymmetric eigenvalue problem is to find eigenvalues \lambda and eigenvectors x such that
A x = \lambda B x
We may also define the problem as finding eigenvalues \mu and eigenvectors y such that
\mu A y = B y
Note that these two problems are equivalent (with \lambda = 1/\mu) if neither \lambda nor \mu is zero. If say, \lambda is zero, then it is still a well defined eigenproblem, but its alternate problem involving \mu is not. Therefore, to allow for zero (and infinite) eigenvalues, the problem which is actually solved is
\beta A x = \alpha B x
The eigensolver routines below will return two values \alpha and \beta and leave it to the user to perform the divisions \lambda = \alpha / \beta and \mu = \beta / \alpha.
If the determinant of the matrix pencil A - \lambda B is zero for all \lambda, the problem is said to be singular; otherwise it is called regular. Singularity normally leads to some \alpha = \beta = 0 which means the eigenproblem is ill-conditioned and generally does not have well defined eigenvalue solutions. The routines below are intended for regular matrix pencils and could yield unpredictable results when applied to singular pencils.
The solution of the real generalized nonsymmetric eigensystem problem for a matrix pair (A, B) involves computing the generalized Schur decomposition
A = Q S Z^T B = Q T Z^T
where Q and Z are orthogonal matrices of left and right Schur vectors respectively, and (S, T) is the generalized Schur form whose diagonal elements give the \alpha and \beta values. The algorithm used is the QZ method due to Moler and Stewart (see references).
This function allocates a workspace for computing eigenvalues of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(n).
This function frees the memory associated with the workspace w.
This function sets some parameters which determine how the eigenvalue
problem is solved in subsequent calls to gsl_eigen_gen
.
If compute_s is set to 1, the full Schur form S will be
computed by gsl_eigen_gen
. If it is set to 0,
S will not be computed (this is the default setting). S
is a quasi upper triangular matrix with 1-by-1 and 2-by-2 blocks
on its diagonal. 1-by-1 blocks correspond to real eigenvalues, and
2-by-2 blocks correspond to complex eigenvalues.
If compute_t is set to 1, the full Schur form T will be
computed by gsl_eigen_gen
. If it is set to 0,
T will not be computed (this is the default setting). T
is an upper triangular matrix with non-negative elements on its diagonal.
Any 2-by-2 blocks in S will correspond to a 2-by-2 diagonal
block in T.
The balance parameter is currently ignored, since generalized balancing is not yet implemented.
This function computes the eigenvalues of the real generalized nonsymmetric matrix pair (A, B), and stores them as pairs in (alpha, beta), where alpha is complex and beta is real. If \beta_i is non-zero, then \lambda = \alpha_i / \beta_i is an eigenvalue. Likewise, if \alpha_i is non-zero, then \mu = \beta_i / \alpha_i is an eigenvalue of the alternate problem \mu A y = B y. The elements of beta are normalized to be non-negative.
If S is desired, it is stored in A on output. If T is desired, it is stored in B on output. The ordering of eigenvalues in (alpha, beta) follows the ordering of the diagonal blocks in the Schur forms S and T. In rare cases, this function may fail to find all eigenvalues. If this occurs, an error code is returned.
This function is identical to gsl_eigen_gen
except that it also
computes the left and right Schur vectors and stores them into Q
and Z respectively.
This function allocates a workspace for computing eigenvalues and eigenvectors of n-by-n real generalized nonsymmetric eigensystems. The size of the workspace is O(7n).
This function frees the memory associated with the workspace w.
This function computes eigenvalues and right eigenvectors of the
n-by-n real generalized nonsymmetric matrix pair
(A, B). The eigenvalues are stored in (alpha, beta)
and the eigenvectors are stored in evec. It first calls
gsl_eigen_gen
to compute the eigenvalues, Schur forms, and
Schur vectors. Then it finds eigenvectors of the Schur forms and
backtransforms them using the Schur vectors. The Schur vectors are
destroyed in the process, but can be saved by using
gsl_eigen_genv_QZ
. The computed eigenvectors are normalized
to have unit magnitude. On output, (A, B) contains
the generalized Schur form (S, T). If gsl_eigen_gen
fails, no eigenvectors are computed, and an error code is returned.
This function is identical to gsl_eigen_genv
except that it also
computes the left and right Schur vectors and stores them into Q
and Z respectively.
Next: Eigenvalue and Eigenvector Examples, Previous: Real Generalized Nonsymmetric Eigensystems, Up: Eigensystems [Index]
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.
This function simultaneously sorts the eigenvalues stored in the vector
eval and the corresponding complex eigenvectors stored in the
columns of the matrix evec into ascending or descending order
according to the value of the parameter sort_type as shown above.
Only GSL_EIGEN_SORT_ABS_ASC
and GSL_EIGEN_SORT_ABS_DESC
are
supported due to the eigenvalues being complex.
This function simultaneously sorts the eigenvalues stored in the vector eval and the corresponding real eigenvectors stored in the columns of the matrix evec into ascending or descending order according to the value of the parameter sort_type as shown above.
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.
This function simultaneously sorts the eigenvalues stored in the vectors
(alpha, beta) and the corresponding complex eigenvectors
stored in the columns of the matrix evec into ascending or
descending order according to the value of the parameter sort_type
as shown above. Only GSL_EIGEN_SORT_ABS_ASC
and
GSL_EIGEN_SORT_ABS_DESC
are supported due to the eigenvalues being
complex.
Next: Eigenvalue and Eigenvector References, Previous: Sorting Eigenvalues and Eigenvectors, Up: Eigensystems [Index]
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"); } } gsl_vector_free (eval); gsl_matrix_free (evec); return 0; }
Here is the beginning of the output from the program,
$ ./a.out eigenvalue = 9.67023e-05 eigenvector = -0.0291933 0.328712 -0.791411 0.514553 ...
This can be compared with the corresponding output from GNU OCTAVE,
octave> [v,d] = eig(hilb(4)); octave> diag(d) ans = 9.6702e-05 6.7383e-03 1.6914e-01 1.5002e+00 octave> v v = 0.029193 0.179186 -0.582076 0.792608 -0.328712 -0.741918 0.370502 0.451923 0.791411 0.100228 0.509579 0.322416 -0.514553 0.638283 0.514048 0.252161
Note that the eigenvectors can differ by a change of sign, since the sign of an eigenvector is arbitrary.
The following program illustrates the use of the nonsymmetric eigensolver, by computing the eigenvalues and eigenvectors of the Vandermonde matrix V(x;i,j) = x_i^{n - j} with x = (-1,-2,3,4).
#include <stdio.h> #include <gsl/gsl_math.h> #include <gsl/gsl_eigen.h> int main (void) { double data[] = { -1.0, 1.0, -1.0, 1.0, -8.0, 4.0, -2.0, 1.0, 27.0, 9.0, 3.0, 1.0, 64.0, 16.0, 4.0, 1.0 }; gsl_matrix_view m = gsl_matrix_view_array (data, 4, 4); gsl_vector_complex *eval = gsl_vector_complex_alloc (4); gsl_matrix_complex *evec = gsl_matrix_complex_alloc (4, 4); gsl_eigen_nonsymmv_workspace * w = gsl_eigen_nonsymmv_alloc (4); gsl_eigen_nonsymmv (&m.matrix, eval, evec, w); gsl_eigen_nonsymmv_free (w); gsl_eigen_nonsymmv_sort (eval, evec, GSL_EIGEN_SORT_ABS_DESC); { int i, j; for (i = 0; i < 4; i++) { gsl_complex eval_i = gsl_vector_complex_get (eval, i); gsl_vector_complex_view evec_i = gsl_matrix_complex_column (evec, i); printf ("eigenvalue = %g + %gi\n", GSL_REAL(eval_i), GSL_IMAG(eval_i)); printf ("eigenvector = \n"); for (j = 0; j < 4; ++j) { gsl_complex z = gsl_vector_complex_get(&evec_i.vector, j); printf("%g + %gi\n", GSL_REAL(z), GSL_IMAG(z)); } } } gsl_vector_complex_free(eval); gsl_matrix_complex_free(evec); return 0; }
Here is the beginning of the output from the program,
$ ./a.out eigenvalue = -6.41391 + 0i eigenvector = -0.0998822 + 0i -0.111251 + 0i 0.292501 + 0i 0.944505 + 0i eigenvalue = 5.54555 + 3.08545i eigenvector = -0.043487 + -0.0076308i 0.0642377 + -0.142127i -0.515253 + 0.0405118i -0.840592 + -0.00148565i ...
This can be compared with the corresponding output from GNU OCTAVE,
octave> [v,d] = eig(vander([-1 -2 3 4])); octave> diag(d) ans = -6.4139 + 0.0000i 5.5456 + 3.0854i 5.5456 - 3.0854i 2.3228 + 0.0000i octave> v v = Columns 1 through 3: -0.09988 + 0.00000i -0.04350 - 0.00755i -0.04350 + 0.00755i -0.11125 + 0.00000i 0.06399 - 0.14224i 0.06399 + 0.14224i 0.29250 + 0.00000i -0.51518 + 0.04142i -0.51518 - 0.04142i 0.94451 + 0.00000i -0.84059 + 0.00000i -0.84059 - 0.00000i Column 4: -0.14493 + 0.00000i 0.35660 + 0.00000i 0.91937 + 0.00000i 0.08118 + 0.00000i
Note that the eigenvectors corresponding to the eigenvalue 5.54555 + 3.08545i differ by the multiplicative constant 0.9999984 + 0.0017674i which is an arbitrary phase factor of magnitude 1.
Previous: Eigenvalue and Eigenvector Examples, Up: Eigensystems [Index]
Further information on the algorithms described in this section can be found in the following book,
Further information on the generalized eigensystems QZ algorithm can be found in this paper,
Eigensystem routines for very large matrices can be found in the Fortran library LAPACK. 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.
Next: Numerical Integration, Previous: Eigensystems, Up: Top [Index]
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)
Next: Overview of complex data FFTs, Up: Fast Fourier Transforms [Index]
Fast Fourier Transforms are efficient algorithms for calculating the discrete Fourier transform (DFT),
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
The DFT usually arises as an approximation to the continuous Fourier transform when functions are sampled at discrete intervals in space or time. The naive evaluation of the discrete Fourier transform is a matrix-vector multiplication W\vec{z}. A general matrix-vector multiplication takes O(n^2) operations for n data-points. Fast Fourier transform algorithms use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, corresponding to the integer factors of the length n. If n can be factorized into a product of integers f_1 f_2 ... f_m then the DFT can be computed in O(n \sum f_i) operations. For a radix-2 FFT this gives an operation count of O(n \log_2 n).
All the FFT functions offer three types of transform: forwards, inverse and backwards, based on the same mathematical definitions. The definition of the forward Fourier transform, x = FFT(z), is,
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
and the definition of the inverse Fourier transform, x = IFFT(z), is,
z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
The factor of 1/n makes this a true inverse. For example, a call
to gsl_fft_complex_forward
followed by a call to
gsl_fft_complex_inverse
should return the original data (within
numerical errors).
In general there are two possible choices for the sign of the exponential in the transform/ inverse-transform pair. GSL follows the same convention as FFTPACK, using a negative exponential for the forward transform. The advantage of this convention is that the inverse transform recreates the original function with simple Fourier synthesis. Numerical Recipes uses the opposite convention, a positive exponential in the forward transform.
The backwards FFT is simply our terminology for an unscaled version of the inverse FFT,
z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
When the overall scale of the result is unimportant it is often convenient to use the backwards FFT instead of the inverse to save unnecessary divisions.
Next: Radix-2 FFT routines for complex data, Previous: Mathematical Definitions, Up: Fast Fourier Transforms [Index]
The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, the following definition of a packed array of length 6,
double x[3*2]; gsl_complex_packed_array data = x;
can be used to hold an array of three complex numbers, z[3]
, in
the following way,
data[0] = Re(z[0]) data[1] = Im(z[0]) data[2] = Re(z[1]) data[3] = Im(z[1]) data[4] = Re(z[2]) data[5] = Im(z[2])
The array indices for the data have the same ordering as those in the definition of the DFT—i.e. there are no index transformations or permutations of the data.
A stride parameter allows the user to perform transforms on the
elements z[stride*i]
instead of z[i]
. A stride greater
than 1 can be used to take an in-place FFT of the column of a matrix. A
stride of 1 accesses the array without any additional spacing between
elements.
To perform an FFT on a vector argument, such as gsl_vector_complex
* v
, use the following definitions (or their equivalents) when calling
the functions described in this chapter:
gsl_complex_packed_array data = v->data; size_t stride = v->stride; size_t n = v->size;
For physical applications it is important to remember that the index appearing in the DFT does not correspond directly to a physical frequency. If the time-step of the DFT is \Delta then the frequency-domain includes both positive and negative frequencies, ranging from -1/(2\Delta) through 0 to +1/(2\Delta). The positive frequencies are stored from the beginning of the array up to the middle, and the negative frequencies are stored backwards from the end of the array.
Here is a table which shows the layout of the array data, and the correspondence between the time-domain data z, and the frequency-domain data x.
index z x = FFT(z) 0 z(t = 0) x(f = 0) 1 z(t = 1) x(f = 1/(n Delta)) 2 z(t = 2) x(f = 2/(n Delta)) . ........ .................. n/2 z(t = n/2) x(f = +1/(2 Delta), -1/(2 Delta)) . ........ .................. n-3 z(t = n-3) x(f = -3/(n Delta)) n-2 z(t = n-2) x(f = -2/(n Delta)) n-1 z(t = n-1) x(f = -1/(n Delta))
When n is even the location n/2 contains the most positive and negative frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent. If n is odd then general structure of the table above still applies, but n/2 does not appear.
Next: Mixed-radix FFT routines for complex data, Previous: Overview of complex data FFTs, Up: Fast Fourier Transforms [Index]
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
transform
version of the function the sign argument can be
either forward
(-1) or backward
(+1).
The functions return a value of GSL_SUCCESS
if no errors were
detected, or GSL_EDOM
if the length of the data n is not a
power of two.
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.
Next: Overview of real data FFTs, Previous: Radix-2 FFT routines for complex data, Up: Fast Fourier Transforms [Index]
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_wavetable
if no errors were detected, and a null
pointer in the case of error. The length n is factorized into a
product of subtransforms, and the factors and their trigonometric
coefficients are stored in the wavetable. The trigonometric coefficients
are computed using direct calls to sin
and cos
, for
accuracy. Recursion relations could be used to compute the lookup table
faster, but if an application performs many FFTs of the same length then
this computation is a one-off overhead which does not affect the final
throughput.
The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.
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 n
was decomposed into.
size_t factor[64]
This is the array of factors. Only the first nf
elements are
used.
gsl_complex * trig
This is a pointer to a preallocated trigonometric lookup table of
n
complex elements.
gsl_complex * twiddle[64]
This is an array of pointers into trig
, giving the twiddle
factors for each pass.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.
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
transform
version of the function the sign argument can be
either forward
(-1) or backward
(+1).
The functions return a value of 0
if no errors were detected. The
following gsl_errno
conditions are defined for these functions:
GSL_EDOM
The length of the data n is not a positive integer (i.e. n is zero).
GSL_EINVAL
The length of the data n and the length used to compute the given wavetable do not match.
Here is an example program which computes the FFT of a short pulse in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.
#include <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 < (int) wavetable->nf; i++) { printf ("# factor %d: %zu\n", i, wavetable->factor[i]); } gsl_fft_complex_forward (data, 1, n, wavetable, workspace); for (i = 0; i < n; i++) { printf ("%d: %e %e\n", i, REAL(data,i), IMAG(data,i)); } gsl_fft_complex_wavetable_free (wavetable); gsl_fft_complex_workspace_free (workspace); return 0; }
Note that we have assumed that the program is using the default
gsl
error handler (which calls abort
for any errors). If
you are not using a safe error handler you would need to check the
return status of all the gsl
routines.
Next: Radix-2 FFT routines for real data, Previous: Mixed-radix FFT routines for complex data, Up: Fast Fourier Transforms [Index]
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).
Next: Mixed-radix FFT routines for real data, Previous: Overview of real data FFTs, Up: Fast Fourier Transforms [Index]
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 (assuming stride=1),
complex[0].real = data[0] complex[0].imag = 0 complex[1].real = data[1] complex[1].imag = data[n-1] ............... ................ complex[k].real = data[k] complex[k].imag = data[n-k] ............... ................ complex[n/2].real = data[n/2] complex[n/2].imag = 0 ............... ................ complex[k'].real = data[k] k' = n - k complex[k'].imag = -data[n-k] ............... ................ complex[n-1].real = data[1] complex[n-1].imag = -data[n-1]
Note that the output data can be converted into the full complex
sequence using the function gsl_fft_halfcomplex_radix2_unpack
described below.
The radix-2 FFT functions for halfcomplex data are declared in the header file gsl_fft_halfcomplex.h.
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 function converts halfcomplex_coefficient, an array of
half-complex coefficients as returned by gsl_fft_real_radix2_transform
, into an ordinary complex array, complex_coefficient. It fills in the
complex array using the symmetry
z_k = z_{n-k}^*
to reconstruct the redundant elements. The algorithm for the conversion
is,
complex_coefficient[0].real = halfcomplex_coefficient[0]; complex_coefficient[0].imag = 0.0; for (i = 1; i < n - i; i++) { double hc_real = halfcomplex_coefficient[i*stride]; double hc_imag = halfcomplex_coefficient[(n-i)*stride]; complex_coefficient[i*stride].real = hc_real; complex_coefficient[i*stride].imag = hc_imag; complex_coefficient[(n - i)*stride].real = hc_real; complex_coefficient[(n - i)*stride].imag = -hc_imag; } if (i == n - i) { complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride]; complex_coefficient[i*stride].imag = 0.0; }
Next: FFT References and Further Reading, Previous: Radix-2 FFT routines for real data, Up: Fast Fourier Transforms [Index]
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 sin
and cos
, for accuracy.
Recursion relations could be used to compute the lookup table faster,
but if an application performs many FFTs of the same length then
computing the wavetable is a one-off overhead which does not affect the
final throughput.
The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or inverse half-complex transforms.
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_transform
data is an array of
time-ordered real data. For gsl_fft_halfcomplex_transform
data contains Fourier coefficients in the half-complex ordering
described above. There is no restriction on the length n.
Efficient modules are provided for subtransforms of length 2, 3, 4 and
5. Any remaining factors are computed with a slow, O(n^2),
general-n module. The caller must supply a wavetable containing
trigonometric lookup tables and a workspace work.
This function converts a single real array, real_coefficient into
an equivalent complex array, complex_coefficient, (with imaginary
part set to zero), suitable for gsl_fft_complex
routines. The
algorithm for the conversion is simply,
for (i = 0; i < n; i++) { complex_coefficient[i*stride].real = real_coefficient[i*stride]; complex_coefficient[i*stride].imag = 0.0; }
This function converts halfcomplex_coefficient, an array of
half-complex coefficients as returned by gsl_fft_real_transform
, into an
ordinary complex array, complex_coefficient. It fills in the
complex array using the symmetry
z_k = z_{n-k}^*
to reconstruct the redundant elements. The algorithm for the conversion
is,
complex_coefficient[0].real = halfcomplex_coefficient[0]; complex_coefficient[0].imag = 0.0; for (i = 1; i < n - i; i++) { double hc_real = halfcomplex_coefficient[(2 * i - 1)*stride]; double hc_imag = halfcomplex_coefficient[(2 * i)*stride]; complex_coefficient[i*stride].real = hc_real; complex_coefficient[i*stride].imag = hc_imag; complex_coefficient[(n - i)*stride].real = hc_real; complex_coefficient[(n - i)*stride].imag = -hc_imag; } if (i == n - i) { complex_coefficient[i*stride].real = halfcomplex_coefficient[(n - 1)*stride]; complex_coefficient[i*stride].imag = 0.0; }
Here is an example program using gsl_fft_real_transform
and
gsl_fft_halfcomplex_inverse
. It generates a real signal in the
shape of a square pulse. The pulse is Fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of Fourier coefficients returned by
gsl_fft_real_transform
.
The remaining Fourier coefficients are transformed back to the time-domain, to give a filtered version of the square pulse. Since Fourier coefficients are stored using the half-complex symmetry both positive and negative frequencies are removed and the final filtered signal is also real.
#include <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; }
Previous: Mixed-radix FFT routines for real data, Up: Fast Fourier Transforms [Index]
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,
Next: Random Number Generation, Previous: Fast Fourier Transforms, Up: Top [Index]
This chapter describes routines for performing numerical integration (quadrature) of a function in one dimension. There are routines for adaptive and non-adaptive integration of general functions, with specialised routines for specific cases. These include integration over infinite and semi-infinite ranges, singular integrals, including logarithmic singularities, computation of Cauchy principal values and oscillatory integrals. The library reimplements the algorithms used in QUADPACK, a numerical integration package written by Piessens, de Doncker-Kapenga, Ueberhuber and Kahaner. Fortran code for QUADPACK is available on Netlib. Also included are non-adaptive, fixed-order Gauss-Legendre integration routines with high precision coefficients by Pavel Holoborodko.
The functions described in this chapter are declared in the header file gsl_integration.h.
Each algorithm computes an approximation to a definite integral of the form,
I = \int_a^b f(x) w(x) dx
where w(x) is a weight function (for general integrands w(x)=1). The user provides absolute and relative error bounds (epsabs, epsrel) which specify the following accuracy requirement,
|RESULT - I| <= max(epsabs, epsrel |I|)
where RESULT is the numerical approximation obtained by the algorithm. The algorithms attempt to estimate the absolute error ABSERR = |RESULT - I| in such a way that the following inequality holds,
|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
In short, the routines return the first approximation which has an absolute error smaller than epsabs or a relative error smaller than epsrel.
Note that this is an either-or constraint, not simultaneous. To compute to a specified absolute error, set epsrel to zero. To compute to a specified relative error, set epsabs to zero. The routines will fail to converge if the error bounds are too stringent, but always return the best approximation obtained up to that stage.
The algorithms in QUADPACK use a naming convention based on the following letters,
Q
- quadrature 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.
• Integrands without weight functions: | ||
• Integrands with weight functions: | ||
• Integrands with singular weight functions: |
The algorithms for general functions (without a weight function) are based on Gauss-Kronrod rules.
A Gauss-Kronrod rule begins with a classical Gaussian quadrature rule of order m. This is extended with additional points between each of the abscissae to give a higher order Kronrod rule of order 2m+1. The Kronrod rule is efficient because it reuses existing function evaluations from the Gaussian rule.
The higher order Kronrod rule is used as the best approximation to the integral, and the difference between the two rules is used as an estimate of the error in the approximation.
Next: Integrands with singular weight functions, Previous: Integrands without weight functions, Up: Numerical Integration Introduction [Index]
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.
Previous: Integrands with weight functions, Up: Numerical Integration Introduction [Index]
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.
Next: QAG adaptive integration, Previous: Numerical Integration Introduction, Up: Numerical Integration [Index]
The QNG algorithm is a non-adaptive procedure which uses fixed Gauss-Kronrod-Patterson abscissae to sample the integrand at a maximum of 87 points. It is provided for fast integration of smooth functions.
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.
Next: QAGS adaptive integration with singularities, Previous: QNG non-adaptive Gauss-Kronrod integration, Up: Numerical Integration [Index]
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. One workspace may be used multiple times as all necessary reinitialization is performed automatically by the integration routines.
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.
Next: QAGP adaptive integration with known singular points, Previous: QAG adaptive integration, Up: Numerical Integration [Index]
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.
Next: QAGI adaptive integration on infinite intervals, Previous: QAGS adaptive integration with singularities, Up: Numerical Integration [Index]
This function applies the adaptive integration algorithm QAGS taking account of the user-supplied locations of singular points. The array pts of length npts should contain the endpoints of the integration ranges defined by the integration region and locations of the singularities. For example, to integrate over the region (a,b) with break-points at x_1, x_2, x_3 (where a < x_1 < x_2 < x_3 < b) the following pts array should be used
pts[0] = a pts[1] = x_1 pts[2] = x_2 pts[3] = x_3 pts[4] = b
with npts = 5.
If you know the locations of the singular points in the integration
region then this routine will be faster than QAGS
.
Next: QAWC adaptive integration for Cauchy principal values, Previous: QAGP adaptive integration with known singular points, Up: Numerical Integration [Index]
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^2
and 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^2
and then integrated using the QAGS algorithm.
Next: QAWS adaptive integration for singular functions, Previous: QAGI adaptive integration on infinite intervals, Up: Numerical Integration [Index]
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.
Next: QAWO adaptive integration for oscillatory functions, Previous: QAWC adaptive integration for Cauchy principal values, Up: Numerical Integration [Index]
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_table
struct describing a singular weight function
W(x) with the parameters (\alpha, \beta, \mu, \nu),
W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
where \alpha > -1, \beta > -1, and \mu = 0, 1, \nu = 0, 1. The weight function can take four different forms depending on the values of \mu and \nu,
W(x) = (x-a)^alpha (b-x)^beta (mu = 0, nu = 0) W(x) = (x-a)^alpha (b-x)^beta log(x-a) (mu = 1, nu = 0) W(x) = (x-a)^alpha (b-x)^beta log(b-x) (mu = 0, nu = 1) W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)
The singular points (a,b) do not have to be specified until the integral is computed, where they are the endpoints of the integration range.
The function returns a pointer to the newly allocated table
gsl_integration_qaws_table
if no errors were detected, and 0 in
the case of error.
This function modifies the parameters (\alpha, \beta, \mu, \nu) of
an existing gsl_integration_qaws_table
struct t.
This function frees all the memory associated with the
gsl_integration_qaws_table
struct 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.
Next: QAWF adaptive integration for Fourier integrals, Previous: QAWS adaptive integration for singular functions, Up: Numerical Integration [Index]
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_table
struct and its associated workspace describing a sine or cosine weight
function W(x) with the parameters (\omega, L),
W(x) = sin(omega x) W(x) = cos(omega x)
The parameter L must be the length of the interval over which the function will be integrated L = b - a. The choice of sine or cosine is made with the parameter sine which should be chosen from one of the two following symbolic values:
GSL_INTEG_COSINE GSL_INTEG_SINE
The gsl_integration_qawo_table
is a table of the trigonometric
coefficients required in the integration process. The parameter n
determines the number of levels of coefficients that are computed. Each
level corresponds to one bisection of the interval L, so that
n levels are sufficient for subintervals down to the length
L/2^n. The integration routine gsl_integration_qawo
returns the error GSL_ETABLE
if the number of levels is
insufficient for the requested accuracy.
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.
Next: CQUAD doubly-adaptive integration, Previous: QAWO adaptive integration for oscillatory functions, Up: Numerical Integration [Index]
This function attempts to compute a Fourier integral of the function f over the semi-infinite interval [a,+\infty).
I = \int_a^{+\infty} dx f(x) sin(omega x) I = \int_a^{+\infty} dx f(x) cos(omega x)
The parameter \omega and choice of \sin or \cos is taken from the table wf (the length L can take any value, since it is overridden by this function to a value appropriate for the Fourier integration). The integral is computed using the QAWO algorithm over each of the subintervals,
C_1 = [a, a + c] C_2 = [a + c, a + 2 c] ... = ... C_k = [a + (k-1) c, a + k c]
where c = (2 floor(|\omega|) + 1) \pi/|\omega|. The width c is chosen to cover an odd number of periods so that the contributions from the intervals alternate in sign and are monotonically decreasing when f is positive and monotonically decreasing. The sum of this sequence of contributions is accelerated using the epsilon-algorithm.
This function works to an overall absolute tolerance of abserr. The following strategy is used: on each interval C_k the algorithm tries to achieve the tolerance
TOL_k = u_k abserr
where u_k = (1 - p)p^{k-1} and p = 9/10. The sum of the geometric series of contributions from each interval gives an overall tolerance of abserr.
If the integration of a subinterval leads to difficulties then the accuracy requirement for subsequent intervals is relaxed,
TOL_k = u_k max(abserr, max_{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.
Next: Fixed order Gauss-Legendre integration, Previous: QAWF adaptive integration for Fourier integrals, Up: Numerical Integration [Index]
CQUAD is a new doubly-adaptive general-purpose quadrature
routine which can handle most types of singularities,
non-numerical function values such as Inf
or NaN
,
as well as some divergent integrals. It generally requires more
function evaluations than the integration routines in
QUADPACK, yet fails less often for difficult integrands.
The underlying algorithm uses a doubly-adaptive scheme in which Clenshaw-Curtis quadrature rules of increasing degree are used to compute the integral in each interval. The L_2-norm of the difference between the underlying interpolatory polynomials of two successive rules is used as an error estimate. The interval is subdivided if the difference between two successive rules is too large or a rule of maximum degree has been reached.
This function allocates a workspace sufficient to hold the data for n intervals. The number n is not the maximum number of intervals that will be evaluated. If the workspace is full, intervals with smaller error estimates will be discarded. A minimum of 3 intervals is required and for most functions, a workspace of size 100 is sufficient.
This function frees the memory associated with the workspace w.
This function computes the integral of f over (a,b) within the desired absolute and relative error limits, epsabs and epsrel using the CQUAD algorithm. The function returns the final approximation, result, an estimate of the absolute error, abserr, and the number of function evaluations required, nevals.
The CQUAD algorithm divides the integration region into subintervals, and in each iteration, the subinterval with the largest estimated error is processed. The algorithm uses Clenshaw-Curits quadrature rules of degree 4, 8, 16 and 32 over 5, 9, 17 and 33 nodes respectively. Each interval is initialized with the lowest-degree rule. When an interval is processed, the next-higher degree rule is evaluated and an error estimate is computed based on the L_2-norm of the difference between the underlying interpolating polynomials of both rules. If the highest-degree rule has already been used, or the interpolatory polynomials differ significantly, the interval is bisected.
The subintervals and their results are stored in the memory
provided by workspace. If the error estimate or the number of
function evaluations is not needed, the pointers abserr and nevals
can be set to NULL
.
Next: Numerical integration error codes, Previous: CQUAD doubly-adaptive integration, Up: Numerical Integration [Index]
The fixed-order Gauss-Legendre integration routines are provided for fast integration of smooth functions with known polynomial order. The n-point Gauss-Legendre rule is exact for polynomials of order 2*n-1 or less. For example, these rules are useful when integrating basis functions to form mass matrices for the Galerkin method. Unlike other numerical integration routines within the library, these routines do not accept absolute or relative error bounds.
This function determines the Gauss-Legendre abscissae and weights necessary for an n-point fixed order integration scheme. If possible, high precision precomputed coefficients are used. If precomputed weights are not available, lower precision coefficients are computed on the fly.
This function applies the Gauss-Legendre integration rule contained in table t and returns the result.
For i in [0, …, t->n - 1], this function obtains the i-th Gauss-Legendre point xi and weight wi on the interval [a,b]. The points and weights are ordered by increasing point value. A function f may be integrated on [a,b] by summing wi * f(xi) over i.
This function frees the memory associated with the table t.
Next: Numerical integration examples, Previous: Fixed order Gauss-Legendre integration, Up: Numerical Integration [Index]
In addition to the standard error codes for invalid arguments the functions can return the following values,
GSL_EMAXITER
the maximum number of subdivisions was exceeded.
GSL_EROUND
cannot reach tolerance because of roundoff error, or roundoff error was detected in the extrapolation table.
GSL_ESING
a non-integrable singularity or other bad integrand behavior was found in the integration interval.
GSL_EDIVERGE
the integral is divergent, or too slowly convergent to be integrated numerically.
Next: Numerical integration References and Further Reading, Previous: Numerical integration error codes, Up: Numerical Integration [Index]
The integrator QAGS
will handle a large class of definite
integrals. For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,
\int_0^1 x^{-1/2} log(x) dx = -4
The program below computes this integral to a relative accuracy bound of
1e-7
.
#include <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 = %zu\n", w->size); gsl_integration_workspace_free (w); return 0; }
The results below show that the desired accuracy is achieved after 8 subdivisions.
$ ./a.out
result = -4.000000000000085265 exact result = -4.000000000000000000 estimated error = 0.000000000000135447 actual error = -0.000000000000085265 intervals = 8
In fact, the extrapolation procedure used by QAGS
produces an
accuracy of almost twice as many digits. The error estimate returned by
the extrapolation procedure is larger than the actual error, giving a
margin of safety of one order of magnitude.
Previous: Numerical integration examples, Up: Numerical Integration [Index]
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 CQUAD integration algorithm is described in the following paper:
Next: Quasi-Random Sequences, Previous: Numerical Integration, Up: Top [Index]
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”.
Next: Random number generator initialization, Previous: General comments on random numbers, Up: Random Number Generation [Index]
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.
Next: Sampling from a random number generator, Previous: The Random Number Generator Interface, Up: Random Number Generation [Index]
This function returns a pointer to a newly-created instance of a random number generator of type T. For example, the following code creates an instance of the Tausworthe generator,
gsl_rng * r = gsl_rng_alloc (gsl_rng_taus);
If there is insufficient memory to create the generator then the
function returns a null pointer and the error handler is invoked with an
error code of GSL_ENOMEM
.
The generator is automatically initialized with the default seed,
gsl_rng_default_seed
. This is zero by default but can be changed
either directly or by using the environment variable GSL_RNG_SEED
(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 >= 1 are supplied, then the generated streams of random
numbers should be completely different. If the seed s is zero
then the standard seed from the original implementation is used
instead. For example, the original Fortran source code for the
ranlux
generator used a seed of 314159265, and so choosing
s equal to zero reproduces this when using
gsl_rng_ranlux
.
When using multiple seeds with the same generator, choose seed values greater than zero to avoid collisions with the default setting.
Note that the most generators only accept 32-bit seeds, with higher values being reduced modulo 2^32. For generators with smaller ranges the maximum seed value will typically be lower.
This function frees all the memory associated with the generator r.
Next: Auxiliary random number generator functions, Previous: Random number generator initialization, Up: Random Number Generation [Index]
The following functions return uniformly distributed random numbers,
either as integers or double precision floating point numbers. Inline versions of these functions are used when HAVE_INLINE
is defined.
To obtain non-uniform distributions 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 be determined using the auxiliary
functions gsl_rng_max (r)
and gsl_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)
by gsl_rng_max(r) + 1.0
in double
precision. Some generators compute this ratio internally so that they
can provide floating point numbers with more than 32 bits of randomness
(the maximum number of bits that can be portably represented in a single
unsigned long int
).
This function returns a positive double precision floating point number
uniformly distributed in the range (0,1), excluding both 0.0 and 1.0.
The number is obtained by sampling the generator with the algorithm of
gsl_rng_uniform
until a non-zero value is obtained. You can use
this function if you need to avoid a singularity at 0.0.
This function returns a random integer from 0 to n-1 inclusive by scaling down and/or discarding samples from the generator r. All integers in the range [0,n-1] are produced with equal probability. For generators with a non-zero minimum value an offset is applied so that zero is returned with the correct probability.
Note that this function is designed for sampling from ranges smaller
than the range of the underlying generator. The parameter n
must be less than or equal to the range of the generator r.
If n is larger than the range of the generator then the function
calls the error handler with an error code of GSL_EINVAL
and
returns zero.
In particular, this function is not intended for generating the full range of
unsigned integer values [0,2^32-1]. Instead
choose a generator with the maximal integer range and zero minimum
value, such as gsl_rng_ranlxd1
, gsl_rng_mt19937
or
gsl_rng_taus
, and sample it directly using
gsl_rng_get
. The range of each generator can be found using
the auxiliary functions described in the next section.
Next: Random number environment variables, Previous: Sampling from a random number generator, Up: Random Number Generation [Index]
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_max
returns the largest value that gsl_rng_get
can return.
gsl_rng_min
returns the smallest value that gsl_rng_get
can return. Usually this value is zero. There are some generators with
algorithms that cannot return zero, and for these generators the minimum
value is 1.
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); }
Next: Copying random number generator state, Previous: Auxiliary random number generator functions, Up: Random Number Generation [Index]
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_TYPE
and
GSL_RNG_SEED
and uses their values to set the corresponding
library variables gsl_rng_default
and
gsl_rng_default_seed
. These global variables are defined as
follows,
extern const gsl_rng_type *gsl_rng_default extern unsigned long int gsl_rng_default_seed
The environment variable GSL_RNG_TYPE
should be the name of a
generator, such as taus
or mt19937
. The environment
variable GSL_RNG_SEED
should contain the desired seed value. It
is converted to an unsigned long int
using the C library function
strtoul
.
If you don’t specify a generator for GSL_RNG_TYPE
then
gsl_rng_mt19937
is used as the default. The initial value of
gsl_rng_default_seed
is zero.
Here is a short program which shows how to create a global
generator using the environment variables GSL_RNG_TYPE
and
GSL_RNG_SEED
,
#include <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)); gsl_rng_free (r); return 0; }
Running the program without any environment variables uses the initial
defaults, an mt19937
generator with a seed of 0,
$ ./a.out
generator type: mt19937 seed = 0 first value = 4293858116
By setting the two variables on the command line we can change the default generator and the seed,
$ GSL_RNG_TYPE="taus" GSL_RNG_SEED=123 ./a.out GSL_RNG_TYPE=taus GSL_RNG_SEED=123 generator type: taus seed = 123 first value = 2720986350
Next: Reading and writing random number generator state, Previous: Random number environment variables, Up: Random Number Generation [Index]
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.
Next: Random number generator algorithms, Previous: Copying random number generator state, Up: Random Number Generation [Index]
The library provides functions for reading and writing the random number state to a file as binary data.
This function writes the random number state of the random number
generator r to the stream stream in binary format. The
return value is 0 for success and GSL_EFAILED
if there was a
problem writing to the file. Since the data is written in the native
binary format it may not be portable between different architectures.
This function reads the random number state into the random number
generator r from the open stream stream in binary format.
The random number generator r must be preinitialized with the
correct random number generator type since type information is not
saved. The return value is 0 for success and GSL_EFAILED
if
there was a problem reading from the file. The data is assumed to
have been written in the native binary format on the same
architecture.
Next: Unix random number generators, Previous: Reading and writing random number generator state, Up: Random Number Generation [Index]
The functions described above make no reference to the actual algorithm used. This is deliberate so that you can switch algorithms without having to change any of your application source code. The library provides a large number of generators of different types, including simulation quality generators, generators provided for compatibility with other libraries and historical generators from the past.
The following generators are recommended for use in simulation. They have extremely long periods, low correlation and pass most statistical tests. For the most reliable source of uncorrelated numbers, the second-generation RANLUX generators have the strongest proof of randomness.
The MT19937 generator of Makoto Matsumoto and Takuji Nishimura is a
variant of the twisted generalized feedback shift-register algorithm,
and is known as the “Mersenne Twister” generator. It has a Mersenne
prime period of
2^19937 - 1 (about
10^6000) and is
equi-distributed in 623 dimensions. It has passed the DIEHARD
statistical tests. It uses 624 words of state per generator and is
comparable in speed to the other generators. The original generator used
a default seed of 4357 and choosing s equal to zero in
gsl_rng_set
reproduces this. Later versions switched to 5489
as the default seed, you can choose this explicitly via gsl_rng_set
instead if you require it.
For more information see,
The generator gsl_rng_mt19937
uses the second revision of the
seeding procedure published by the two authors above in 2002. The
original seeding procedures could cause spurious artifacts for some seed
values. They are still available through the alternative generators
gsl_rng_mt19937_1999
and gsl_rng_mt19937_1998
.
The generator ranlxs0
is 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 levels ranlxs0
, ranlxs1
and ranlxs2
,
in increasing order of strength.
It uses double-precision floating point arithmetic internally and can be
significantly faster than the integer version of ranlux
,
particularly on 64-bit architectures. The period of the generator is
about 10^171. The algorithm has mathematically proven properties and
can provide truly decorrelated numbers at a known level of randomness.
The higher luxury levels provide increased decorrelation between samples
as an additional safety margin.
Note that the range of allowed seeds for this generator is [0,2^31-1]. Higher seed values are wrapped modulo 2^31.
These generators produce double precision output (48 bits) from the
RANLXS generator. The library provides two luxury levels
ranlxd1
and ranlxd2
, in increasing order of strength.
The ranlux
generator 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 by gsl_rng_ranlux
, while gsl_rng_ranlux389
gives the highest level of randomness, with all 24 bits decorrelated.
Both types of generator use 24 words of state per generator.
For more information see,
This is a combined multiple recursive generator by L’Ecuyer. Its sequence is,
z_n = (x_n - y_n) mod m_1
where the two underlying generators x_n and y_n are,
x_n = (a_1 x_{n-1} + a_2 x_{n-2} + a_3 x_{n-3}) mod m_1 y_n = (b_1 y_{n-1} + b_2 y_{n-2} + b_3 y_{n-3}) mod m_2
with coefficients a_1 = 0, a_2 = 63308, a_3 = -183326, b_1 = 86098, b_2 = 0, b_3 = -539608, and moduli m_1 = 2^31 - 1 = 2147483647 and m_2 = 2145483479.
The period of this generator is lcm(m_1^3-1, m_2^3-1), which is approximately 2^185 (about 10^56). It uses 6 words of state per generator. For more information see,
This is a fifth-order multiple recursive generator by L’Ecuyer, Blouin and Coutre. Its sequence is,
x_n = (a_1 x_{n-1} + a_5 x_{n-5}) mod m
with a_1 = 107374182, a_2 = a_3 = a_4 = 0, a_5 = 104480 and m = 2^31 - 1.
The period of this generator is about 10^46. It uses 5 words of state per generator. More information can be found in the following paper,
This is a maximally equidistributed combined Tausworthe generator by L’Ecuyer. The sequence is,
x_n = (s1_n ^^ s2_n ^^ s3_n)
where,
s1_{n+1} = (((s1_n&4294967294)<<12)^^(((s1_n<<13)^^s1_n)>>19)) s2_{n+1} = (((s2_n&4294967288)<< 4)^^(((s2_n<< 2)^^s2_n)>>25)) s3_{n+1} = (((s3_n&4294967280)<<17)^^(((s3_n<< 3)^^s3_n)>>11))
computed modulo
2^32. In the formulas above
^^
denotes “exclusive-or”. Note that the algorithm relies on the properties
of 32-bit unsigned integers and has been implemented using a bitmask
of 0xFFFFFFFF
to make it work on 64 bit machines.
The period of this generator is 2^88 (about 10^26). It uses 3 words of state per generator. For more information see,
The generator gsl_rng_taus2
uses the same algorithm as
gsl_rng_taus
but with an improved seeding procedure described in
the paper,
The generator gsl_rng_taus2
should now be used in preference to
gsl_rng_taus
.
The gfsr4
generator is like a lagged-fibonacci generator, and
produces each number as an xor
’d sum of four previous values.
r_n = r_{n-A} ^^ r_{n-B} ^^ r_{n-C} ^^ r_{n-D}
Ziff (ref below) notes that “it is now widely known” that two-tap registers (such as R250, which is described below) have serious flaws, the most obvious one being the three-point correlation that comes from the definition of the generator. Nice mathematical properties can be derived for GFSR’s, and numerics bears out the claim that 4-tap GFSR’s with appropriately chosen offsets are as random as can be measured, using the author’s test.
This implementation uses the values suggested the example on p392 of Ziff’s article: A=471, B=1586, C=6988, D=9689.
If the offsets are appropriately chosen (such as the one ones in this
implementation), then the sequence is said to be maximal; that means
that the period is 2^D - 1, where D is the longest lag.
(It is one less than 2^D because it is not permitted to have all
zeros in the ra[]
array.) For this implementation with
D=9689 that works out to about 10^2917.
Note that the implementation of this generator using a 32-bit integer amounts to 32 parallel implementations of one-bit generators. One consequence of this is that the period of this 32-bit generator is the same as for the one-bit generator. Moreover, this independence means that all 32-bit patterns are equally likely, and in particular that 0 is an allowed random value. (We are grateful to Heiko Bauke for clarifying for us these properties of GFSR random number generators.)
For more information see,
Next: Other random number generators, Previous: Random number generator algorithms, Up: Random Number Generation [Index]
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 is
x_{n+1} = (a x_n + c) mod m
with a = 1103515245, c = 12345 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator is 2^31, and it uses 1 word of storage per generator.
These generators implement the random
family of functions, a
set of linear feedback shift register generators originally used in BSD
Unix. There are several versions of random
in use today: the
original BSD version (e.g. on SunOS4), a libc5 version (found on
older GNU/Linux systems) and a glibc2 version. Each version uses a
different seeding procedure, and thus produces different sequences.
The original BSD routines accepted a variable length buffer for the
generator state, with longer buffers providing higher-quality
randomness. The random
function implemented algorithms for
buffer lengths of 8, 32, 64, 128 and 256 bytes, and the algorithm with
the largest length that would fit into the user-supplied buffer was
used. To support these algorithms additional generators are available
with the following names,
gsl_rng_random8_bsd gsl_rng_random32_bsd gsl_rng_random64_bsd gsl_rng_random128_bsd gsl_rng_random256_bsd
where the numeric suffix indicates the buffer length. The original BSD
random
function used a 128-byte default buffer and so
gsl_rng_random_bsd
has been made equivalent to
gsl_rng_random128_bsd
. Corresponding versions of the libc5
and glibc2
generators are also available, with the names
gsl_rng_random8_libc5
, gsl_rng_random8_glibc2
, etc.
This is the Unix rand48
generator. Its sequence is
x_{n+1} = (a x_n + c) mod m
defined on 48-bit unsigned integers with
a = 25214903917,
c = 11 and
m = 2^48.
The seed specifies the upper 32 bits of the initial value, x_1,
with the lower 16 bits set to 0x330E
. The function
gsl_rng_get
returns the upper 32 bits from each term of the
sequence. This does not have a direct parallel in the original
rand48
functions, but forcing the result to type long int
reproduces the output of mrand48
. The function
gsl_rng_uniform
uses the full 48 bits of internal state to return
the double precision number x_n/m, which is equivalent to the
function drand48
. Note that some versions of the GNU C Library
contained a bug in mrand48
function which caused it to produce
different results (only the lower 16-bits of the return value were set).
Next: Random Number Generator Performance, Previous: Unix random number generators, Up: Random Number Generation [Index]
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 is
x_{n+1} = (a x_n) mod m
defined on 48-bit unsigned integers with a = 44485709377909 and m = 2^48. The seed specifies the lower 32 bits of the initial value, x_1, with the lowest bit set to prevent the seed taking an even value. The upper 16 bits of x_1 are set to 0. A consequence of this procedure is that the pairs of seeds 2 and 3, 4 and 5, etc. produce the same sequences.
The generator compatible with the CRAY MATHLIB routine RANF. It produces double precision floating point numbers which should be identical to those from the original RANF.
There is a subtlety in the implementation of the seeding. The initial state is reversed through one step, by multiplying by the modular inverse of a mod m. This is done for compatibility with the original CRAY implementation.
Note that you can only seed the generator with integers up to 2^32, while the original CRAY implementation uses non-portable wide integers which can cover all 2^48 states of the generator.
The function gsl_rng_get
returns the upper 32 bits from each term
of the sequence. The function gsl_rng_uniform
uses the full 48
bits to return the double precision number x_n/m.
The period of this generator is 2^46.
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,
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,
This is the VAX generator MTH$RANDOM
. Its sequence is,
x_{n+1} = (a x_n + c) mod m
with a = 69069, c = 1 and m = 2^32. The seed specifies the initial value, x_1. The period of this generator is 2^32 and it uses 1 word of storage per generator.
This is the random number generator from the INMOS Transputer Development system. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 1664525 and m = 2^32. The seed specifies the initial value, x_1.
This is the IBM RANDU
generator. Its sequence is
x_{n+1} = (a x_n) mod m
with a = 65539 and m = 2^31. The seed specifies the initial value, x_1. The period of this generator was only 2^29. It has become a textbook example of a poor generator.
This is Park and Miller’s “minimal standard” MINSTD generator, a simple linear congruence which takes care to avoid the major pitfalls of such algorithms. Its sequence is,
x_{n+1} = (a x_n) mod m
with a = 16807 and m = 2^31 - 1 = 2147483647. The seed specifies the initial value, x_1. The period of this generator is about 2^31.
This generator was used in the IMSL Library (subroutine RNUN) and in MATLAB (the RAND function) in the past. It is also sometimes known by the acronym “GGL” (I’m not sure what that stands for).
For more information see,
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,
This is a second-order multiple recursive generator described by Knuth in Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
x_n = (a_1 x_{n-1} + a_2 x_{n-2}) mod m
with a_1 = 271828183, a_2 = 314159269, and m = 2^31 - 1.
This is a second-order multiple recursive generator described by Knuth
in Seminumerical Algorithms, 3rd Ed., Section 3.6. Knuth
provides its C code. The updated routine gsl_rng_knuthran2002
is from the revised 9th printing and corrects some weaknesses in the
earlier version, which is implemented as gsl_rng_knuthran
.
These multiplicative generators are taken from Knuth’s Seminumerical Algorithms, 3rd Ed., pages 106–108. Their sequence is,
x_{n+1} = (a x_n) mod m
where the seed specifies the initial value, x_1. The parameters a and m are as follows, Borosh-Niederreiter: a = 1812433253, m = 2^32, Fishman18: a = 62089911, m = 2^31 - 1, Fishman20: a = 48271, m = 2^31 - 1, L’Ecuyer: a = 40692, m = 2^31 - 249, Waterman: a = 1566083941, m = 2^32.
This is the L’Ecuyer–Fishman random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., page 108. Its sequence is,
z_{n+1} = (x_n - y_n) mod m
with m = 2^31 - 1.
x_n and y_n are given by the fishman20
and lecuyer21
algorithms.
The seed specifies the initial value,
x_1.
This is the Coveyou random number generator. It is taken from Knuth’s Seminumerical Algorithms, 3rd Ed., Section 3.2.2. Its sequence is,
x_{n+1} = (x_n (x_n + 1)) mod m
with m = 2^32. The seed specifies the initial value, x_1.
Next: Random Number Generator Examples, Previous: Other random number generators, Up: Random Number Generation [Index]
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
Next: Random Number References and Further Reading, Previous: Random Number Generator Performance, Up: Random Number Generation [Index]
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
0.33050 0.86631 0.32982 0.67620 0.53391 0.06457 0.16847 0.70229 0.04371 0.86374
Next: Random Number Acknowledgements, Previous: Random Number Generator Examples, Up: Random Number Generation [Index]
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,
Previous: Random Number References and Further Reading, Up: Random Number Generation [Index]
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.
Next: Random Number Distributions, Previous: Random Number Generation, Up: Top [Index]
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.
Next: Auxiliary quasi-random number generator functions, Previous: Quasi-random number generator initialization, Up: Quasi-Random Sequences [Index]
This function stores the next point from the sequence generator q
in the array x. The space available for x must match the
dimension of the generator. The point x will lie in the range
0 < x_i < 1 for each x_i. An inline version of this function is used when HAVE_INLINE
is defined.
Next: Saving and restoring quasi-random number generator state, Previous: Sampling from a quasi-random number generator, Up: Quasi-Random Sequences [Index]
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);
Next: Quasi-random number generator algorithms, Previous: Auxiliary quasi-random number generator functions, Up: Quasi-Random Sequences [Index]
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.
Next: Quasi-random number generator examples, Previous: Saving and restoring quasi-random number generator state, Up: Quasi-Random Sequences [Index]
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.
These generators use the Halton and reverse Halton sequences described in J.H. Halton, Numerische Mathematik 2, 84-90 (1960) and B. Vandewoestyne and R. Cools Computational and Applied Mathematics 189, 1&2, 341-361 (2006). They are valid up to 1229 dimensions.
Next: Quasi-random number references, Previous: Quasi-random number generator algorithms, Up: Quasi-Random Sequences [Index]
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.
Previous: Quasi-random number generator examples, Up: Quasi-Random Sequences [Index]
The implementations of the quasi-random sequence routines are based on the algorithms described in the following paper,
Next: Statistics, Previous: Quasi-Random Sequences, Up: Top [Index]
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.
Next: The Gaussian Distribution, Up: Random Number Distributions [Index]
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).
Next: The Gaussian Tail Distribution, Previous: Random Number Distribution Introduction, Up: Random Number Distributions [Index]
This function returns a Gaussian random variate, with mean zero and standard deviation sigma. The probability distribution for Gaussian random variates is,
p(x) dx = {1 \over \sqrt{2 \pi \sigma^2}} \exp (-x^2 / 2\sigma^2) dx
for x in the range -\infty to +\infty. Use the
transformation z = \mu + x on the numbers returned by
gsl_ran_gaussian
to obtain a Gaussian distribution with mean
\mu. This function uses the Box-Muller algorithm which requires two
calls to the random number generator r.
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.
Next: The Bivariate Gaussian Distribution, Previous: The Gaussian Distribution, Up: Random Number Distributions [Index]
This function provides random variates from the upper tail of a Gaussian distribution with standard deviation sigma. The values returned are larger than the lower limit a, which must be positive. The method is based on Marsaglia’s famous rectangle-wedge-tail algorithm (Ann. Math. Stat. 32, 894–899 (1961)), with this aspect explained in Knuth, v2, 3rd ed, p139,586 (exercise 11).
The probability distribution for Gaussian tail random variates is,
p(x) dx = {1 \over N(a;\sigma) \sqrt{2 \pi \sigma^2}} \exp (- x^2/(2 \sigma^2)) dx
for x > a where N(a;\sigma) is the normalization constant,
N(a;\sigma) = (1/2) erfc(a / sqrt(2 sigma^2)).
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.
Next: The Multivariate Gaussian Distribution, Previous: The Gaussian Tail Distribution, Up: Random Number Distributions [Index]
This function generates a pair of correlated Gaussian variates, with mean zero, correlation coefficient rho and standard deviations sigma_x and sigma_y in the x and y directions. The probability distribution for bivariate Gaussian random variates is,
p(x,y) dx dy = {1 \over 2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} \exp (-(x^2/\sigma_x^2 + y^2/\sigma_y^2 - 2 \rho x y/(\sigma_x\sigma_y))/2(1-\rho^2)) dx dy
for x,y in the range -\infty to +\infty. The correlation coefficient rho should lie between 1 and -1.
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.
Next: The Exponential Distribution, Previous: The Bivariate Gaussian Distribution, Up: Random Number Distributions [Index]
This function generates a random vector satisfying the k-dimensional multivariate Gaussian
distribution with mean \mu and variance-covariance matrix
\Sigma. On input, the k-vector \mu is given in mu, and
the Cholesky factor of the k-by-k matrix \Sigma = L L^T is
given in the lower triangle of L, as output from gsl_linalg_cholesky_decomp
.
The random vector is stored in result on output. The probability distribution
for multivariate Gaussian random variates is
p(x_1,...,x_k) dx_1 ... dx_k = {1 \over \sqrt{(2 \pi)^k |\Sigma|} \exp \left(-{1 \over 2} (x - \mu)^T \Sigma^{-1} (x - \mu)\right) dx_1 \dots dx_k
These functions compute p(x) or \log{p(x)} at the point x, using mean vector mu and variance-covariance matrix specified by its Cholesky factor L using the formula above. Additional workspace of length k is required in work.
Given a set of n samples X_j from a k-dimensional multivariate Gaussian distribution, this function computes the maximum likelihood estimate of the mean of the distribution, given by
\Hat{\mu} = {1 \over n} \sum_{j=1}^n X_j
The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix X, and the maximum likelihood estimate of the mean is stored in mu_hat on output.
Given a set of n samples X_j from a k-dimensional multivariate Gaussian distribution, this function computes the maximum likelihood estimate of the variance-covariance matrix of the distribution, given by
\Hat{\Sigma} = {1 \over n} \sum_{j=1}^n \left( X_j - \Hat{\mu} \right) \left( X_j - \Hat{\mu} \right)^T
The samples X_1,X_2,\dots,X_n are given in the n-by-k matrix X and the maximum likelihood estimate of the variance-covariance matrix is stored in sigma_hat on output.
Next: The Laplace Distribution, Previous: The Multivariate Gaussian Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the exponential distribution with mean mu. The distribution is,
p(x) dx = {1 \over \mu} \exp(-x/\mu) dx
for x >= 0.
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.
Next: The Exponential Power Distribution, Previous: The Exponential Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Laplace distribution with width a. The distribution is,
p(x) dx = {1 \over 2 a} \exp(-|x/a|) dx
for -\infty < x < \infty.
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.
Next: The Cauchy Distribution, Previous: The Laplace Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the exponential power distribution with scale parameter a and exponent b. The distribution is,
p(x) dx = {1 \over 2 a \Gamma(1+1/b)} \exp(-|x/a|^b) dx
for x >= 0. For b = 1 this reduces to the Laplace distribution. For b = 2 it has the same form as a Gaussian distribution, but with a = \sqrt{2} \sigma.
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.
Next: The Rayleigh Distribution, Previous: The Exponential Power Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Cauchy distribution with scale parameter a. The probability distribution for Cauchy random variates is,
p(x) dx = {1 \over a\pi (1 + (x/a)^2) } dx
for x in the range -\infty to +\infty. The Cauchy distribution is also known as the Lorentz distribution.
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.
Next: The Rayleigh Tail Distribution, Previous: The Cauchy Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Rayleigh distribution with scale parameter sigma. The distribution is,
p(x) dx = {x \over \sigma^2} \exp(- x^2/(2 \sigma^2)) dx
for x > 0.
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.
Next: The Landau Distribution, Previous: The Rayleigh Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the tail of the Rayleigh distribution with scale parameter sigma and a lower limit of a. The distribution is,
p(x) dx = {x \over \sigma^2} \exp ((a^2 - x^2) /(2 \sigma^2)) dx
for x > a.
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.
Next: The Levy alpha-Stable Distributions, Previous: The Rayleigh Tail Distribution, Up: Random Number Distributions [Index]
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.
Next: The Levy skew alpha-Stable Distribution, Previous: The Landau Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the Levy symmetric stable distribution with scale c and exponent alpha. The symmetric stable probability distribution is defined by a Fourier transform,
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha)
There is no explicit solution for the form of p(x) and the
library does not define a corresponding pdf
function. For
\alpha = 1 the distribution reduces to the Cauchy distribution. For
\alpha = 2 it is a Gaussian distribution with \sigma = \sqrt{2} c. For \alpha < 1 the tails of the
distribution become extremely wide.
The algorithm only works for 0 < alpha <= 2.
Next: The Gamma Distribution, Previous: The Levy alpha-Stable Distributions, Up: Random Number Distributions [Index]
This function returns a random variate from the Levy skew stable distribution with scale c, exponent alpha and skewness parameter beta. The skewness parameter must lie in the range [-1,1]. The Levy skew stable probability distribution is defined by a Fourier transform,
p(x) = {1 \over 2 \pi} \int_{-\infty}^{+\infty} dt \exp(-it x - |c t|^alpha (1-i beta sign(t) tan(pi alpha/2)))
When \alpha = 1 the term \tan(\pi \alpha/2) is replaced by
-(2/\pi)\log|t|. There is no explicit solution for the form of
p(x) and the library does not define a corresponding pdf
function. For \alpha = 2 the distribution reduces to a Gaussian
distribution with \sigma = \sqrt{2} c and the skewness parameter has no effect.
For \alpha < 1 the tails of the distribution become extremely
wide. The symmetric distribution corresponds to \beta =
0.
The algorithm only works for 0 < alpha <= 2.
The Levy alpha-stable distributions have the property that if N alpha-stable variates are drawn from the distribution p(c, \alpha, \beta) then the sum Y = X_1 + X_2 + \dots + X_N will also be distributed as an alpha-stable variate, p(N^(1/\alpha) c, \alpha, \beta).
Next: The Flat (Uniform) Distribution, Previous: The Levy skew alpha-Stable Distribution, Up: Random Number Distributions [Index]
This function returns a random variate from the gamma distribution. The distribution function is,
p(x) dx = {1 \over \Gamma(a) b^a} x^{a-1} e^{-x/b} dx
for x > 0