Provided by: pdl_2.020-3_amd64 
NAME
PDL::GSL::INTEG - PDL interface to numerical integration routines in GSL
DESCRIPTION
This is an interface to the numerical integration package present in the GNU Scientific Library, which is
an implementation of QUADPACK.
Functions are named gslinteg_{algorithm} where {algorithm} is the QUADPACK naming convention. The
available functions are:
gslinteg_qng: Non-adaptive Gauss-Kronrod integration
gslinteg_qag: Adaptive integration
gslinteg_qags: Adaptive integration with singularities
gslinteg_qagp: Adaptive integration with known singular points
gslinteg_qagi: Adaptive integration on infinite interval of the form (-\infty,\infty)
gslinteg_qagiu: Adaptive integration on infinite interval of the form (la,\infty)
gslinteg_qagil: Adaptive integration on infinite interval of the form (-\infty,lb)
gslinteg_qawc: Adaptive integration for Cauchy principal values
gslinteg_qaws: Adaptive integration for singular functions
gslinteg_qawo: Adaptive integration for oscillatory functions
gslinteg_qawf: Adaptive integration for Fourier integrals
Each algorithm computes an approximation to the integral, I, of the function f(x)w(x), 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|)
The routines will fail to converge if the error bounds are too stringent, but always return the best
approximation obtained up to that stage
All functions return the result, and estimate of the absolute error and an error flag (which is zero if
there were no problems). You are responsible for checking for any errors, no warnings are issued unless
the option {Warn => 'y'} is specified in which case the reason of failure will be printed.
You can nest integrals up to 20 levels. If you find yourself in the unlikely situation that you need
more, you can change the value of 'max_nested_integrals' in the first line of the file 'FUNC.c' and
recompile.
NOMENCLATURE
Throughout this documentation we strive to use the same variables that are present in the original GSL
documentation (see See Also). Oftentimes those variables are called "a" and "b". Since good Perl coding
practices discourage the use of Perl variables $a and $b, here we refer to Parameters "a" and "b" as $pa
and $pb, respectively, and Limits (of domain or integration) as $la and $lb.
Please check the GSL documentation for more information.
SYNOPSIS
use PDL;
use PDL::GSL::INTEG;
my $la = 1.2;
my $lb = 3.7;
my $epsrel = 0;
my $epsabs = 1e-6;
# Non adaptive integration
my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\&myf,$la,$lb,$epsrel,$epsabs);
# Warnings on
my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\&myf,$la,$lb,$epsrel,$epsabs,{Warn=>'y'});
# Adaptive integration with warnings on
my $limit = 1000;
my $key = 5;
my ($res,$abserr,$ierr) = gslinteg_qag(\&myf,$la,$lb,$epsrel,
$epsabs,$limit,$key,{Warn=>'y'});
sub myf{
my ($x) = @_;
return exp(-$x**2);
}
FUNCTIONS
qng_meat
Signature: (double a(); double b(); double epsabs();
double epsrel(); double [o] result(); double [o] abserr();
int [o] neval(); int [o] ierr(); int gslwarn(); SV* function)
info not available
qng_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qag_meat
Signature: (double a(); double b(); double epsabs();double epsrel(); int limit();
int key(); double [o] result(); double [o] abserr();int n();int [o] ierr();int gslwarn();; SV* function)
info not available
qag_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qags_meat
Signature: (double a(); double b(); double epsabs();double epsrel(); int limit();
double [o] result(); double [o] abserr();int n();int [o] ierr();int gslwarn();; SV* function)
info not available
qags_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qagp_meat
Signature: (double pts(l); double epsabs();double epsrel();int limit();
double [o] result(); double [o] abserr();int n();int [o] ierr();int gslwarn();; SV* function)
info not available
qagp_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qagi_meat
Signature: (double epsabs();double epsrel(); int limit();
double [o] result(); double [o] abserr(); int n(); int [o] ierr();int gslwarn();; SV* function)
info not available
qagi_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qagiu_meat
Signature: (double a(); double epsabs();double epsrel();int limit();
double [o] result(); double [o] abserr();int n();int [o] ierr();int gslwarn();; SV* function)
info not available
qagiu_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qagil_meat
Signature: (double b(); double epsabs();double epsrel();int limit();
double [o] result(); double [o] abserr();int n();int [o] ierr();int gslwarn();; SV* function)
info not available
qagil_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qawc_meat
Signature: (double a(); double b(); double c(); double epsabs();double epsrel();int limit();
double [o] result(); double [o] abserr();int n();int [o] ierr();int gslwarn();; SV* function)
info not available
qawc_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qaws_meat
Signature: (double a(); double b();double epsabs();double epsrel();int limit();
double [o] result(); double [o] abserr();int n();
double alpha(); double beta(); int mu(); int nu();int [o] ierr();int gslwarn();; SV* function)
info not available
qaws_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qawo_meat
Signature: (double a(); double b();double epsabs();double epsrel();int limit();
double [o] result(); double [o] abserr();int n();
int sincosopt(); double omega(); double L(); int nlevels();int [o] ierr();int gslwarn();; SV* function)
info not available
qawo_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
qawf_meat
Signature: (double a(); double epsabs();int limit();
double [o] result(); double [o] abserr();int n();
int sincosopt(); double omega(); int nlevels();int [o] ierr();int gslwarn();; SV* function)
info not available
qawf_meat does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
gslinteg_qng
Non-adaptive Gauss-Kronrod integration
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 ($la,$lb) is achieved within the desired absolute
and relative error limits, $epsabs and $epsrel. It is meant for fast integration of smooth functions. It
returns an array with the result, an estimate of the absolute error, an error flag and the number of
function evaluations performed.
Usage:
($res,$abserr,$ierr,$neval) = gslinteg_qng($function_ref,$la,$lb,
$epsrel,$epsabs,[{Warn => $warn}]);
Example:
my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\&f,0,1,0,1e-9);
# with warnings on
my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\&f,0,1,0,1e-9,{Warn => 'y'});
sub f{
my ($x) = @_;
return ($x**2.6)*log(1.0/$x);
}
gslinteg_qag
Adaptive integration
This function applies an integration rule adaptively until an estimate of the integral of f over
($la,$lb) is achieved within the desired absolute and relative error limits, $epsabs and $epsrel. On each
iteration the adaptive integration strategy bisects the interval with the largest error estimate; the
maximum number of allowed subdivisions is given by the parameter $limit. The integration rule is
determined by the value of $key, which has to be one of (1,2,3,4,5,6) and correspond to the 15, 21, 31,
41, 51 and 61 point Gauss-Kronrod rules respectively. It returns an array with the result, an estimate
of the absolute error and an error flag.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qag($function_ref,$la,$lb,$epsrel,
$epsabs,$limit,$key,[{Warn => $warn}]);
Example:
my ($res,$abserr,$ierr) = gslinteg_qag(\&f,0,1,0,1e-10,1000,1);
# with warnings on
my ($res,$abserr,$ierr) = gslinteg_qag(\&f,0,1,0,1e-10,1000,1,{Warn => 'y'});
sub f{
my ($x) = @_;
return ($x**2.6)*log(1.0/$x);
}
gslinteg_qags
Adaptive integration with singularities
This function applies the Gauss-Kronrod 21-point integration rule adaptively until an estimate of the
integral of f over ($la,$lb) is achieved within the desired absolute and relative error limits, $epsabs
and $epsrel. The algorithm is such that it accelerates the convergence of the integral in the presence of
discontinuities and integrable singularities. The maximum number of allowed subdivisions done by the
adaptive algorithm must be supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qags($function_ref,$la,$lb,$epsrel,
$epsabs,$limit,[{Warn => $warn}]);
Example:
my ($res,$abserr,$ierr) = gslinteg_qags(\&f,0,1,0,1e-10,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qags(\&f,0,1,0,1e-10,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
return ($x)*log(1.0/$x);
}
gslinteg_qagp
Adaptive integration with known singular points
This function applies the adaptive integration algorithm used by gslinteg_qags taking into account the
location of singular points until an estimate of the integral of f over ($la,$lb) is achieved within the
desired absolute and relative error limits, $epsabs and $epsrel. Singular points are supplied in the
piddle $points, whose endpoints determine the integration range. So, for example, if the function has
singular points at x_1 and x_2 and the integral is desired from a to b (a < x_1 < x_2 < b), $points =
pdl(a,x_1,x_2,b). The maximum number of allowed subdivisions done by the adaptive algorithm must be
supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qagp($function_ref,$points,$epsabs,
$epsrel,$limit,[{Warn => $warn}])
Example:
my $points = pdl(0,1,sqrt(2),3);
my ($res,$abserr,$ierr) = gslinteg_qagp(\&f,$points,0,1e-3,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qagp(\&f,$points,0,1e-3,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
my $x2 = $x**2;
my $x3 = $x**3;
return $x3 * log(abs(($x2-1.0)*($x2-2.0)));
}
gslinteg_qagi
Adaptive integration on infinite interval
This function estimates the integral of the function f over the infinite interval (-\infty,+\infty)
within the desired absolute and relative error limits, $epsabs and $epsrel. After a transformation, the
algorithm of gslinteg_qags with a 15-point Gauss-Kronrod rule is used. The maximum number of allowed
subdivisions done by the adaptive algorithm must be supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qagi($function_ref,$epsabs,
$epsrel,$limit,[{Warn => $warn}]);
Example:
my ($res,$abserr,$ierr) = gslinteg_qagi(\&myfn,1e-7,0,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qagi(\&myfn,1e-7,0,1000,{Warn => 'y'});
sub myfn{
my ($x) = @_;
return exp(-$x - $x*$x) ;
}
gslinteg_qagiu
Adaptive integration on infinite interval
This function estimates the integral of the function f over the infinite interval (la,+\infty) within the
desired absolute and relative error limits, $epsabs and $epsrel. After a transformation, the algorithm
of gslinteg_qags with a 15-point Gauss-Kronrod rule is used. The maximum number of allowed subdivisions
done by the adaptive algorithm must be supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qagiu($function_ref,$la,$epsabs,
$epsrel,$limit,[{Warn => $warn}]);
Example:
my $alfa = 1;
my ($res,$abserr,$ierr) = gslinteg_qagiu(\&f,99.9,1e-7,0,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qagiu(\&f,99.9,1e-7,0,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
if (($x==0) && ($alfa == 1)) {return 1;}
if (($x==0) && ($alfa > 1)) {return 0;}
return ($x**($alfa-1))/((1+10*$x)**2);
}
gslinteg_qagil
Adaptive integration on infinite interval
This function estimates the integral of the function f over the infinite interval (-\infty,lb) within the
desired absolute and relative error limits, $epsabs and $epsrel. After a transformation, the algorithm
of gslinteg_qags with a 15-point Gauss-Kronrod rule is used. The maximum number of allowed subdivisions
done by the adaptive algorithm must be supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qagl($function_ref,$lb,$epsabs,
$epsrel,$limit,[{Warn => $warn}]);
Example:
my ($res,$abserr,$ierr) = gslinteg_qagil(\&myfn,1.0,1e-7,0,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qagil(\&myfn,1.0,1e-7,0,1000,{Warn => 'y'});
sub myfn{
my ($x) = @_;
return exp($x);
}
gslinteg_qawc
Adaptive integration for Cauchy principal values
This function computes the Cauchy principal value of the integral of f over (la,lb), with a singularity
at c, I = \int_{la}^{lb} dx f(x)/(x - c). The integral is estimated within the desired absolute and
relative error limits, $epsabs and $epsrel. The maximum number of allowed subdivisions done by the
adaptive algorithm must be supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qawc($function_ref,$la,$lb,$c,$epsabs,$epsrel,$limit)
Example:
my ($res,$abserr,$ierr) = gslinteg_qawc(\&f,-1,5,0,0,1e-3,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qawc(\&f,-1,5,0,0,1e-3,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
return 1.0 / (5.0 * $x * $x * $x + 6.0) ;
}
gslinteg_qaws
Adaptive integration for singular functions
The algorithm in gslinteg_qaws is designed for integrands with algebraic-logarithmic singularities at the
end-points of an integration region. Specifically, this function computes the integral given by I =
\int_{la}^{lb} dx f(x) (x-la)^alpha (lb-x)^beta log^mu (x-la) log^nu (lb-x). The integral is estimated
within the desired absolute and relative error limits, $epsabs and $epsrel. The maximum number of
allowed subdivisions done by the adaptive algorithm must be supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) =
gslinteg_qawc($function_ref,$alpha,$beta,$mu,$nu,$la,$lb,
$epsabs,$epsrel,$limit,[{Warn => $warn}]);
Example:
my ($res,$abserr,$ierr) = gslinteg_qaws(\&f,0,0,1,0,0,1,0,1e-7,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qaws(\&f,0,0,1,0,0,1,0,1e-7,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
if($x==0){return 0;}
else{
my $u = log($x);
my $v = 1 + $u*$u;
return 1.0/($v*$v);
}
}
gslinteg_qawo
Adaptive integration for oscillatory functions
This function uses an adaptive algorithm to compute the integral of f over (la,lb) with the weight
function sin(omega*x) or cos(omega*x) -- which of sine or cosine is used is determined by the parameter
$opt ('cos' or 'sin'). The integral is estimated within the desired absolute and relative error limits,
$epsabs and $epsrel. The maximum number of allowed subdivisions done by the adaptive algorithm must be
supplied in the parameter $limit.
Please check the GSL documentation for more information.
Usage:
($res,$abserr,$ierr) = gslinteg_qawo($function_ref,$omega,$sin_or_cos,
$la,$lb,$epsabs,$epsrel,$limit,[opt])
Example:
my $PI = 3.14159265358979323846264338328;
my ($res,$abserr,$ierr) = PDL::GSL::INTEG::gslinteg_qawo(\&f,10*$PI,'sin',0,1,0,1e-7,1000);
# with warnings on
($res,$abserr,$ierr) = PDL::GSL::INTEG::gslinteg_qawo(\&f,10*$PI,'sin',0,1,0,1e-7,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
if($x==0){return 0;}
else{ return log($x);}
}
gslinteg_qawf
Adaptive integration for Fourier integrals
This function attempts to compute a Fourier integral of the function f over the semi-infinite interval
[la,+\infty). Specifically, it attempts tp compute I = \int_{la}^{+\infty} dx f(x)w(x), where w(x) is
sin(omega*x) or cos(omega*x) -- which of sine or cosine is used is determined by the parameter $opt
('cos' or 'sin'). The integral is estimated within the desired absolute error limit $epsabs. The
maximum number of allowed subdivisions done by the adaptive algorithm must be supplied in the parameter
$limit.
Please check the GSL documentation for more information.
Usage:
gslinteg_qawf($function_ref,$omega,$sin_or_cos,$la,$epsabs,$limit,[opt])
Example:
my ($res,$abserr,$ierr) = gslinteg_qawf(\&f,$PI/2.0,'cos',0,1e-7,1000);
# with warnings on
($res,$abserr,$ierr) = gslinteg_qawf(\&f,$PI/2.0,'cos',0,1e-7,1000,{Warn => 'y'});
sub f{
my ($x) = @_;
if ($x == 0){return 0;}
return 1.0/sqrt($x)
}
BUGS
Feedback is welcome. Log bugs in the PDL bug database (the database is always linked from
<http://pdl.perl.org>).
SEE ALSO
PDL
The GSL documentation for numerical integration is online at
<https://www.gnu.org/software/gsl/doc/html/integration.html>
AUTHOR
This file copyright (C) 2003,2005 Andres Jordan <ajordan@eso.org> All rights reserved. There is no
warranty. You are allowed to redistribute this software documentation under certain conditions. For
details, see the file COPYING in the PDL distribution. If this file is separated from the PDL
distribution, the copyright notice should be included in the file.
The GSL integration routines were written by Brian Gough. QUADPACK was written by Piessens, Doncker-
Kapenga, Uberhuber and Kahaner.
perl v5.30.0 2020-01-18 INTEG(3pm)