# NAME

Math::GSL::Integration - Routines for performing numerical integration (quadrature) of a function in one dimension

# SYNOPSIS

``````    use Math::GSL::Integration qw /:all/;

my \$function = sub { \$_**2 } ;
my (\$lower, \$upper ) = (0,1);
my (\$relerr,\$abserr) = (0,1e-7);

my (\$status, \$result, \$abserr, \$num_evals) = gsl_integration_qng ( \$function,
\$lower, \$upper, \$relerr, \$abserr
);``````

# DESCRIPTION

This module allows you to numerically integrate a Perl subroutine. Depending on the properties of your function (singularities, smoothness) and the type of integration range (finite, infinite, semi-infinite), you will need to choose a quadrature routine that fits your needs.

• `gsl_integration_workspace_alloc(\$n)`

This function allocates a workspace sufficient to hold \$n double precision intervals, their integration results and error estimates.

• `gsl_integration_workspace_free(\$w)`

`` This function frees the memory associated with the workspace \$w.``
• `gsl_integration_qaws_table_alloc(\$alpha, \$beta, \$mu, \$nu)`

`````` This function allocates space for a gsl_integration_qaws_table struct
describing a singular weight function W(x) with the parameters (\$alpha, \$beta,
\$mu, \$nu), W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x) where
\$alpha > -1, \$beta > -1, and \$mu = 0, 1, \$nu = 0, 1. The weight function can
take four different forms depending on the values of \$mu and \$nu,

W(x) = (x-a)^alpha (b-x)^beta                   (mu = 0, nu = 0)
W(x) = (x-a)^alpha (b-x)^beta log(x-a)          (mu = 1, nu = 0)
W(x) = (x-a)^alpha (b-x)^beta log(b-x)          (mu = 0, nu = 1)
W(x) = (x-a)^alpha (b-x)^beta log(x-a) log(b-x) (mu = 1, nu = 1)``````

The singular points (a,b) do not have to be specified until the integral is computed, where they are the endpoints of the integration range. The function returns a pointer to the newly allocated table gsl_integration_qaws_table if no errors were detected, and 0 in the case of error.

• `gsl_integration_qaws_table_set(\$t, \$alpha, \$beta, \$mu, \$nu)`

`````` This function modifies the parameters (\$alpha, \$beta, \$mu, \$nu) of an existing
gsl_integration_qaws_table struct \$t.``````
• `gsl_integration_qaws_table_free(\$t)`

`````` This function frees all the memory associated with the
gsl_integration_qaws_table struct \$t.``````
• `gsl_integration_qawo_table_alloc(\$omega, \$L, \$sine, \$n)`

• `gsl_integration_qawo_table_set(\$t, \$omega, \$L, \$sine, \$n)`

`````` This function changes the parameters omega, L and sine of the existing
workspace \$t.``````
• `gsl_integration_qawo_table_set_length(\$t, \$L)`

`````` This function allows the length parameter \$L of the workspace \$t to be
changed.``````
• `gsl_integration_qawo_table_free(\$t)`

`` This function frees all the memory associated with the workspace \$t.``
• `gsl_integration_qk15(\$function,\$a,\$b,\$resabs,\$resasc) `

• `gsl_integration_qk21(\$function,\$a,\$b,\$resabs,\$resasc) `

• `gsl_integration_qk31(\$function,\$a,\$b,\$resabs,\$resasc) `

• `gsl_integration_qk41(\$function,\$a,\$b,\$resabs,\$resasc) `

• `gsl_integration_qk51(\$function,\$a,\$b,\$resabs,\$resasc) `

• `gsl_integration_qk61(\$function,\$a,\$b,\$resabs,\$resasc) `

• `gsl_integration_qcheb(\$function, \$a, \$b, \$cheb12, \$cheb24) `

• `gsl_integration_qk `

• `gsl_integration_qng(\$function,\$a,\$b,\$epsabs,\$epsrel,\$num_evals) `

This routine QNG (Quadrature Non-Adaptive Gaussian) is inexpensive is the sense that it will evaluate the function much fewer times than the adaptive routines. Because of this it does not need any workspaces, so it is also more memory efficient. It should be perfectly fine for well-behaved functions (smooth and nonsingular), but will not be able to get the required accuracy or may not converge for more complicated functions.

• `gsl_integration_qag(\$function,\$a,\$b,\$epsabs,\$epsrel,\$limit,\$key,\$workspace) `

• `gsl_integration_qagi(\$function,\$epsabs,\$epsrel,\$limit,\$workspace) `

• `gsl_integration_qagiu(\$function,\$a,\$epsabs,\$epsrel,\$limit,\$workspace) `

• `gsl_integration_qagil(\$function,\$b,\$epsabs,\$epsrel,\$limit,\$workspace) `

• `gsl_integration_qags(\$func,\$a,\$b,\$epsabs,\$epsrel,\$limit,\$workspace)`

``````    (\$status, \$result, \$abserr) = gsl_integration_qags (
sub { 1/\$_} ,
1, 10, 0, 1e-7, 1000,
\$workspace,
);

This function applies the Gauss-Kronrod 21-point integration rule
adaptively until an estimate of the integral of \$func over (\$a,\$b) is
achieved within the desired absolute and relative error limits,
\$epsabs and \$epsrel.``````
• `gsl_integration_qagp(\$function, \$pts, \$npts, \$epsbs, \$epsrel, \$limit, \$workspace) `

• `gsl_integration_qawc(\$function, \$a, \$b, \$c, \$epsabs, \$epsrel, \$limit, \$workspace) `

• `gsl_integration_qaws(\$function, \$a, \$b, \$qaws_table, \$epsabs, \$epsrel, \$limit, \$workspace) `

• `gsl_integration_qawo(\$function, \$a, \$epsabs, \$epsrel, \$limit, \$workspace, \$qawo_table) `

• `gsl_integration_qawf(\$function, \$a, \$epsabs, \$limit, \$workspace, \$cycle_workspace, \$qawo_table) `

This module also includes the following constants :

• \$GSL_INTEG_COSINE

• \$GSL_INTEG_SINE

• \$GSL_INTEG_GAUSS15

• \$GSL_INTEG_GAUSS21

• \$GSL_INTEG_GAUSS31

• \$GSL_INTEG_GAUSS41

• \$GSL_INTEG_GAUSS51

• \$GSL_INTEG_GAUSS61

The following error constants are part of the Math::GSL::Errno module and can be returned by the gsl_integration_* functions :

• \$GSL_EMAXITER

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.