# 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_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 (a,\infty)
gslinteg_qagil: Adaptive integration on infinite interval of the form (-\infty,b)
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.

# SYNOPSIS

   use PDL;
use PDL::GSL::INTEG;
my $a = 1.2; my$b = 3.7;
my $epsrel = 0; my$epsabs = 1e-6;
my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\&myf,$a,$b,$epsrel,$epsabs);
# Warnings on
my ($res,$abserr,$ierr,$neval) = gslinteg_qng(\&myf,$a,$b,$epsrel,$epsabs,{Warn=>'y'});
# Adaptive integration with warnings on
my $limit = 1000; my$key = 5;
my ($res,$abserr,$ierr) = gslinteg_qag(\&myf,$a,$b,$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 warn(); SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 warn();; SV* funcion) 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 ($a,$b) is achieved within the desired absolute and relative error limits,$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,$a,$b,
$epsrel,$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

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. 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.

Usage:

  ($res,$abserr,$ierr) = gslinteg_qag($function_ref,$a,$b,$epsrel,$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

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 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($function_ref,$a,$b,$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 ($a,$b) 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.

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

This function estimates the integral of the function f over the infinite interval (a,+\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($function_ref,$a,$epsabs,$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*$x)**2); } ## gslinteg_qagil Adaptive integration on infinite interval This function estimates the integral of the function f over the infinite interval (-\infty,b) within the desired absolute and relative error limits,$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.

Usage:

  ($res,$abserr,$ierr) = gslinteg_qagl($function_ref,$b,$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($x); } ## gslinteg_qawc Adaptive integration for Cauchy principal values This function computes the Cauchy principal value of the integral of f over (a,b), with a singularity at c, I = \int_a^b dx f(x)/(x - c). The integral is estimated within the desired absolute and relative error limits,$epsabs and $epsrel. The maximum number of allowed subdivisions done by the adaptive algorithm must be supplied in the parameter$limit.

Usage:

  ($res,$abserr,$ierr) = gslinteg_qawc($function_ref,$a,$b,$c,$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

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_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-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,$nu,$a,$b,$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

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) -- 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.

Usage:

  ($res,$abserr,$ierr) = gslinteg_qawo($function_ref,$omega,$sin_or_cos,
$a,$b,$epsabs,$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($x);} } ## gslinteg_qawf Adaptive integration for Fourier integrals This function attempts to compute a Fourier integral of the function f over the semi-infinite interval [a,+\infty). Specifically, it attempts tp compute I = \int_a^{+\infty} dx f(x)w(x), where w(x) is sin(omega*x) or cos(omega*x) -- which of sine or cosine is used is determined by the parameter$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.

Usage:

  gslinteg_qawf($function_ref,$omega,$sin_or_cos,$a,$epsabs,$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).

  http://www.gnu.org/software/gsl/manual/