@cindex quadrature
@cindex numerical integration (quadrature)
@cindex integration, numerical (quadrature)
@cindex QUADPACK

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
@sc{quadpack}, a numerical integration package written by Piessens,
de Doncker-Kapenga, Ueberhuber and Kahaner.  Fortran code for @sc{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
@file{gsl_integration.h}.

@menu
* Numerical Integration Introduction::  
* QNG non-adaptive Gauss-Kronrod integration::  
* QAG adaptive integration::    
* QAGS adaptive integration with singularities::  
* QAGP adaptive integration with known singular points::  
* QAGI adaptive integration on infinite intervals::  
* QAWC adaptive integration for Cauchy principal values::  
* QAWS adaptive integration for singular functions::  
* QAWO adaptive integration for oscillatory functions::  
* QAWF adaptive integration for Fourier integrals::  
* CQUAD doubly-adaptive integration::  
* Fixed order Gauss-Legendre integration::  
* Numerical integration error codes::  
* Numerical integration examples::  
* Numerical integration References and Further Reading::  
@end menu

@node Numerical Integration Introduction
@section Introduction

Each algorithm computes an approximation to a definite integral of the
form,
@tex
\beforedisplay
$$
I = \int_a^b f(x) w(x) \,dx
$$
\afterdisplay
@end tex
@ifinfo

@example
I = \int_a^b f(x) w(x) dx
@end example

@end ifinfo
@noindent
where @math{w(x)} is a weight function (for general integrands @math{w(x)=1}).
The user provides absolute and relative error bounds 
@c{$(\hbox{\it epsabs}, \hbox{\it epsrel}\,)$}
@math{(epsabs, epsrel)} which specify the following accuracy requirement,
@tex
\beforedisplay
$$
|\hbox{\it RESULT} - I|  \leq \max(\hbox{\it epsabs}, \hbox{\it epsrel}\, |I|)
$$
\afterdisplay
@end tex
@ifinfo

@example
|RESULT - I|  <= max(epsabs, epsrel |I|)
@end example

@end ifinfo
@noindent
where 
@c{$\hbox{\it RESULT}$}
@math{RESULT} is the numerical approximation obtained by the
algorithm.  The algorithms attempt to estimate the absolute error
@c{$\hbox{\it ABSERR} = |\hbox{\it RESULT} - I|$}
@math{ABSERR = |RESULT - I|} in such a way that the following inequality
holds,
@tex
\beforedisplay
$$
|\hbox{\it RESULT} - I| \leq \hbox{\it ABSERR} \leq \max(\hbox{\it epsabs}, \hbox{\it epsrel}\,|I|)
$$
\afterdisplay
@end tex
@ifinfo

@example
|RESULT - I| <= ABSERR <= max(epsabs, epsrel |I|)
@end example

@end ifinfo
@noindent
In short, the routines return the first approximation 
which has an absolute error smaller than @c{$\hbox{\it epsabs}$}
@math{epsabs} or a relative error smaller than @c{$\hbox{\it epsrel}$}
@math{epsrel}.   

Note that this is an @i{either-or} constraint, 
not simultaneous.  To compute to a specified absolute error, set @c{$\hbox{\it epsrel}$}
@math{epsrel} to zero.  To compute to a specified relative error,
set @c{$\hbox{\it epsabs}$}
@math{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 @sc{quadpack} use a naming convention based on the
following letters,

@display 
@code{Q} - quadrature routine

@code{N} - non-adaptive integrator
@code{A} - adaptive integrator

@code{G} - general integrand (user-defined)
@code{W} - weight function with integrand

@code{S} - singularities can be more readily integrated
@code{P} - points of special difficulty can be supplied
@code{I} - infinite range of integration
@code{O} - oscillatory weight function, cos or sin
@code{F} - Fourier integral
@code{C} - Cauchy principal value
@end display

@noindent
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.

@menu
* Integrands without weight functions::  
* Integrands with weight functions::  
* Integrands with singular weight functions::  
@end menu

@node Integrands without weight functions
@subsection Integrands without weight functions
@cindex Gauss-Kronrod quadrature
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 @math{m}.  This is extended with additional points between each of
the abscissae to give a higher order Kronrod rule of order @math{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.

@node Integrands with weight functions
@subsection Integrands with weight functions
@cindex Clenshaw-Curtis quadrature
@cindex Modified Clenshaw-Curtis quadrature
For integrands with weight functions the algorithms use Clenshaw-Curtis
quadrature rules.  

A Clenshaw-Curtis rule begins with an @math{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.

@node Integrands with singular weight functions
@subsection Integrands with singular weight functions

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 @sc{quadpack} separate out several common
weight functions which cause slow convergence.  

These weight functions are integrated analytically against the Chebyshev
polynomials to precompute @dfn{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.


@node QNG non-adaptive Gauss-Kronrod integration
@section QNG non-adaptive Gauss-Kronrod integration
@cindex QNG quadrature algorithm

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.

@deftypefun int gsl_integration_qng (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, double * @var{result}, double * @var{abserr}, size_t * @var{neval})

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 @math{f} over @math{(a,b)} is achieved within the desired
absolute and relative error limits, @var{epsabs} and @var{epsrel}.  The
function returns the final approximation, @var{result}, an estimate of
the absolute error, @var{abserr} and the number of function evaluations
used, @var{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.
@end deftypefun


@node QAG adaptive integration
@section QAG adaptive integration
@cindex QAG quadrature algorithm

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 @code{gsl_integration_workspace} struct,
which handles the memory for the subinterval ranges, results and error
estimates.

@deftypefun {gsl_integration_workspace *} gsl_integration_workspace_alloc (size_t @var{n}) 
@tindex gsl_integration_workspace
This function allocates a workspace sufficient to hold @var{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.
@end deftypefun

@deftypefun void gsl_integration_workspace_free (gsl_integration_workspace * @var{w})
This function frees the memory associated with the workspace @var{w}.
@end deftypefun

@deftypefun int gsl_integration_qag (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, int @var{key}, gsl_integration_workspace * @var{workspace},  double * @var{result}, double * @var{abserr})

This function applies an integration rule adaptively until an estimate
of the integral of @math{f} over @math{(a,b)} is achieved within the
desired absolute and relative error limits, @var{epsabs} and
@var{epsrel}.  The function returns the final approximation,
@var{result}, and an estimate of the absolute error, @var{abserr}.  The
integration rule is determined by the value of @var{key}, which should
be chosen from the following symbolic names,

@example
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)
@end example

@noindent
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 @var{workspace}.  The maximum number of
subintervals is given by @var{limit}, which may not exceed the allocated
size of the workspace.
@end deftypefun


@node QAGS adaptive integration with singularities
@section QAGS adaptive integration with singularities
@cindex QAGS quadrature algorithm

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.

@deftypefun int gsl_integration_qags (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})

This function applies the Gauss-Kronrod 21-point integration rule
adaptively until an estimate of the integral of @math{f} over
@math{(a,b)} is achieved within the desired absolute and relative error
limits, @var{epsabs} and @var{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, @var{result}, and an estimate of the absolute error,
@var{abserr}.  The subintervals and their results are stored in the
memory provided by @var{workspace}.  The maximum number of subintervals
is given by @var{limit}, which may not exceed the allocated size of the
workspace.

@end deftypefun

@node QAGP adaptive integration with known singular points
@section QAGP adaptive integration with known singular points
@cindex QAGP quadrature algorithm
@cindex singular points, specifying positions in quadrature
@deftypefun int gsl_integration_qagp (const gsl_function * @var{f}, double * @var{pts}, size_t @var{npts}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})

This function applies the adaptive integration algorithm QAGS taking
account of the user-supplied locations of singular points.  The array
@var{pts} of length @var{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
@math{(a,b)} with break-points at @math{x_1, x_2, x_3} (where 
@math{a < x_1 < x_2 < x_3 < b}) the following @var{pts} array should be used

@example
pts[0] = a
pts[1] = x_1
pts[2] = x_2
pts[3] = x_3
pts[4] = b
@end example

@noindent
with @var{npts} = 5.

@noindent
If you know the locations of the singular points in the integration
region then this routine will be faster than @code{QAGS}.

@end deftypefun

@node QAGI adaptive integration on infinite intervals
@section QAGI adaptive integration on infinite intervals
@cindex QAGI quadrature algorithm

@deftypefun int gsl_integration_qagi (gsl_function * @var{f}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})

This function computes the integral of the function @var{f} over the
infinite interval @math{(-\infty,+\infty)}.  The integral is mapped onto the
semi-open interval @math{(0,1]} using the transformation @math{x = (1-t)/t},
@tex
\beforedisplay
$$
\int_{-\infty}^{+\infty} dx \, f(x) 
  = \int_0^1 dt \, (f((1-t)/t) + f(-(1-t)/t))/t^2.
$$
\afterdisplay
@end tex
@ifinfo

@example
\int_@{-\infty@}^@{+\infty@} dx f(x) = 
     \int_0^1 dt (f((1-t)/t) + f((-1+t)/t))/t^2.
@end example

@end ifinfo
@noindent
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.
@end deftypefun

@deftypefun int gsl_integration_qagiu (gsl_function * @var{f}, double @var{a}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})

This function computes the integral of the function @var{f} over the
semi-infinite interval @math{(a,+\infty)}.  The integral is mapped onto the
semi-open interval @math{(0,1]} using the transformation @math{x = a + (1-t)/t},
@tex
\beforedisplay
$$
\int_{a}^{+\infty} dx \, f(x) 
  = \int_0^1 dt \, f(a + (1-t)/t)/t^2
$$
\afterdisplay
@end tex
@ifinfo

@example
\int_@{a@}^@{+\infty@} dx f(x) = 
     \int_0^1 dt f(a + (1-t)/t)/t^2
@end example

@end ifinfo
@noindent
and then integrated using the QAGS algorithm.
@end deftypefun

@deftypefun int gsl_integration_qagil (gsl_function * @var{f}, double @var{b}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})
This function computes the integral of the function @var{f} over the
semi-infinite interval @math{(-\infty,b)}.  The integral is mapped onto the
semi-open interval @math{(0,1]} using the transformation @math{x = b - (1-t)/t},
@tex
\beforedisplay
$$
\int_{-\infty}^{b} dx \, f(x) 
  = \int_0^1 dt \, f(b - (1-t)/t)/t^2
$$
\afterdisplay
@end tex
@ifinfo

@example
\int_@{-\infty@}^@{b@} dx f(x) = 
     \int_0^1 dt f(b - (1-t)/t)/t^2
@end example

@end ifinfo
@noindent
and then integrated using the QAGS algorithm.
@end deftypefun

@node  QAWC adaptive integration for Cauchy principal values
@section QAWC adaptive integration for Cauchy principal values
@cindex QAWC quadrature algorithm
@cindex Cauchy principal value, by numerical quadrature
@deftypefun int gsl_integration_qawc (gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{c}, double @var{epsabs}, double @var{epsrel}, size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})

This function computes the Cauchy principal value of the integral of
@math{f} over @math{(a,b)}, with a singularity at @var{c},
@tex
\beforedisplay
$$
I = \int_a^b dx\, {f(x) \over x - c}
  = \lim_{\epsilon \to 0} 
\left\{
\int_a^{c-\epsilon} dx\, {f(x) \over x - c}
+
\int_{c+\epsilon}^b dx\, {f(x) \over x - c}
\right\}
$$
\afterdisplay
@end tex
@ifinfo

@example
I = \int_a^b dx f(x) / (x - c)
@end example

@end ifinfo
@noindent
The adaptive bisection algorithm of QAG is used, with modifications to
ensure that subdivisions do not occur at the singular point @math{x = c}.
When a subinterval contains the point @math{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.

@end deftypefun

@node  QAWS adaptive integration for singular functions
@section QAWS adaptive integration for singular functions
@cindex QAWS quadrature algorithm
@cindex singular functions, numerical integration of
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.

@deftypefun {gsl_integration_qaws_table *} gsl_integration_qaws_table_alloc (double @var{alpha}, double @var{beta}, int @var{mu}, int @var{nu})
@tindex gsl_integration_qaws_table
This function allocates space for a @code{gsl_integration_qaws_table}
struct describing a singular weight function
@math{W(x)} with the parameters @math{(\alpha, \beta, \mu, \nu)},
@tex
\beforedisplay
$$
W(x) = (x - a)^\alpha (b - x)^\beta \log^\mu (x - a) \log^\nu (b - x)
$$
\afterdisplay
@end tex
@ifinfo

@example
W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
@end example

@end ifinfo
@noindent
where @math{\alpha > -1}, @math{\beta > -1}, and @math{\mu = 0, 1},
@math{\nu = 0, 1}.  The weight function can take four different forms
depending on the values of @math{\mu} and @math{\nu},
@tex
\beforedisplay
$$
\matrix{
W(x) = (x - a)^\alpha (b - x)^\beta  
                                                \hfill~ (\mu = 0, \nu = 0) \cr
W(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) 
                                                \hfill~ (\mu = 1, \nu = 0) \cr
W(x) = (x - a)^\alpha (b - x)^\beta \log(b - x) 
                                                \hfill~ (\mu = 0, \nu = 1) \cr
W(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) \log(b - x) 
                                                \hfill~ (\mu = 1, \nu = 1)
}
$$
\afterdisplay
@end tex
@ifinfo

@example
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)
@end example

@end ifinfo
@noindent
The singular points @math{(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
@code{gsl_integration_qaws_table} if no errors were detected, and 0 in
the case of error.
@end deftypefun

@deftypefun int gsl_integration_qaws_table_set (gsl_integration_qaws_table * @var{t}, double @var{alpha}, double @var{beta}, int @var{mu}, int @var{nu})
This function modifies the parameters @math{(\alpha, \beta, \mu, \nu)} of
an existing @code{gsl_integration_qaws_table} struct @var{t}.
@end deftypefun

@deftypefun void gsl_integration_qaws_table_free (gsl_integration_qaws_table * @var{t})
This function frees all the memory associated with the
@code{gsl_integration_qaws_table} struct @var{t}.
@end deftypefun

@deftypefun int gsl_integration_qaws (gsl_function * @var{f}, const double @var{a}, const double @var{b}, gsl_integration_qaws_table * @var{t}, const double @var{epsabs}, const double @var{epsrel}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, double * @var{result}, double * @var{abserr})

This function computes the integral of the function @math{f(x)} over the
interval @math{(a,b)} with the singular weight function
@math{(x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x)}.  The parameters 
of the weight function @math{(\alpha, \beta, \mu, \nu)} are taken from the
table @var{t}.  The integral is,
@tex
\beforedisplay
$$
I = \int_a^b dx\, f(x) (x - a)^\alpha (b - x)^\beta 
        \log^\mu (x - a) \log^\nu (b - x).
$$
\afterdisplay
@end tex
@ifinfo

@example
I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).
@end example

@end ifinfo
@noindent
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.

@end deftypefun

@node  QAWO adaptive integration for oscillatory functions
@section QAWO adaptive integration for oscillatory functions
@cindex QAWO quadrature algorithm
@cindex oscillatory functions, numerical integration of
The QAWO algorithm is designed for integrands with an oscillatory
factor, @math{\sin(\omega x)} or @math{\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.

@deftypefun {gsl_integration_qawo_table *} gsl_integration_qawo_table_alloc (double @var{omega}, double @var{L}, enum gsl_integration_qawo_enum @var{sine}, size_t @var{n})
@tindex gsl_integration_qawo_table
This function allocates space for a @code{gsl_integration_qawo_table}
struct and its associated workspace describing a sine or cosine weight
function @math{W(x)} with the parameters @math{(\omega, L)},
@tex
\beforedisplay
$$
\eqalign{
W(x) & = \left\{\matrix{\sin(\omega x) \cr \cos(\omega x)} \right\}
}
$$
\afterdisplay
@end tex
@ifinfo

@example
W(x) = sin(omega x)
W(x) = cos(omega x)
@end example

@end ifinfo
@noindent
The parameter @var{L} must be the length of the interval over which the
function will be integrated @math{L = b - a}.  The choice of sine or
cosine is made with the parameter @var{sine} which should be chosen from
one of the two following symbolic values:

@example
GSL_INTEG_COSINE
GSL_INTEG_SINE
@end example

@noindent
The @code{gsl_integration_qawo_table} is a table of the trigonometric
coefficients required in the integration process.  The parameter @var{n}
determines the number of levels of coefficients that are computed.  Each
level corresponds to one bisection of the interval @math{L}, so that
@var{n} levels are sufficient for subintervals down to the length
@math{L/2^n}.  The integration routine @code{gsl_integration_qawo}
returns the error @code{GSL_ETABLE} if the number of levels is
insufficient for the requested accuracy.

@end deftypefun

@deftypefun int gsl_integration_qawo_table_set (gsl_integration_qawo_table * @var{t}, double @var{omega}, double @var{L}, enum gsl_integration_qawo_enum @var{sine})
This function changes the parameters @var{omega}, @var{L} and @var{sine}
of the existing workspace @var{t}.
@end deftypefun

@deftypefun int gsl_integration_qawo_table_set_length (gsl_integration_qawo_table * @var{t}, double @var{L})
This function allows the length parameter @var{L} of the workspace
@var{t} to be changed.
@end deftypefun

@deftypefun void gsl_integration_qawo_table_free (gsl_integration_qawo_table * @var{t})
This function frees all the memory associated with the workspace @var{t}.
@end deftypefun

@deftypefun int gsl_integration_qawo (gsl_function * @var{f}, const double @var{a}, const double @var{epsabs}, const double @var{epsrel}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, gsl_integration_qawo_table * @var{wf}, double * @var{result}, double * @var{abserr})

This function uses an adaptive algorithm to compute the integral of
@math{f} over @math{(a,b)} with the weight function 
@math{\sin(\omega x)} or @math{\cos(\omega x)} defined 
by the table @var{wf},
@tex
\beforedisplay
$$
\eqalign{
I & = \int_a^b dx\, f(x) \left\{ \matrix{\sin(\omega x) \cr \cos(\omega x)}\right\}
}
$$
\afterdisplay
@end tex
@ifinfo

@example
I = \int_a^b dx f(x) sin(omega x)
I = \int_a^b dx f(x) cos(omega x)
@end example

@end ifinfo
@noindent
The results are extrapolated using the epsilon-algorithm to accelerate
the convergence of the integral.  The function returns the final
approximation from the extrapolation, @var{result}, and an estimate of
the absolute error, @var{abserr}.  The subintervals and their results are
stored in the memory provided by @var{workspace}.  The maximum number of
subintervals is given by @var{limit}, which may not exceed the allocated
size of the workspace.

Those subintervals with ``large'' widths @math{d} where @math{d\omega > 4} are
computed using a 25-point Clenshaw-Curtis integration rule, which handles the
oscillatory behavior.  Subintervals with a ``small'' widths where
@math{d\omega < 4} are computed using a 15-point Gauss-Kronrod integration.

@end deftypefun

@node  QAWF adaptive integration for Fourier integrals
@section QAWF adaptive integration for Fourier integrals
@cindex QAWF quadrature algorithm
@cindex Fourier integrals, numerical

@deftypefun int gsl_integration_qawf (gsl_function * @var{f}, const double @var{a}, const double @var{epsabs}, const size_t @var{limit}, gsl_integration_workspace * @var{workspace}, gsl_integration_workspace * @var{cycle_workspace}, gsl_integration_qawo_table * @var{wf}, double * @var{result}, double * @var{abserr})

This function attempts to compute a Fourier integral of the function
@var{f} over the semi-infinite interval @math{[a,+\infty)}.
@tex
\beforedisplay
$$
\eqalign{
I & = \int_a^{+\infty} dx\, f(x) \left\{ \matrix{ \sin(\omega x) \cr
                                                 \cos(\omega x) } \right\}
}
$$
\afterdisplay
@end tex
@ifinfo

@example
I = \int_a^@{+\infty@} dx f(x) sin(omega x)
I = \int_a^@{+\infty@} dx f(x) cos(omega x)
@end example
@end ifinfo

The parameter @math{\omega} and choice of @math{\sin} or @math{\cos} is
taken from the table @var{wf} (the length @var{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,
@tex
\beforedisplay
$$
\eqalign{
C_1 & = [a, a + c] \cr
C_2 & = [a + c, a + 2c] \cr
\dots & = \dots \cr
C_k & = [a + (k-1) c, a + k c]
}
$$
\afterdisplay
@end tex
@ifinfo

@example
C_1 = [a, a + c]
C_2 = [a + c, a + 2 c]
... = ...
C_k = [a + (k-1) c, a + k c]
@end example

@end ifinfo
@noindent
where 
@c{$c = (2 \,\hbox{floor}(|\omega|) + 1) \pi/|\omega|$}
@math{c = (2 floor(|\omega|) + 1) \pi/|\omega|}.  The width @math{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
@var{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
@var{abserr}.  The following strategy is used: on each interval
@math{C_k} the algorithm tries to achieve the tolerance
@tex
\beforedisplay
$$
TOL_k = u_k \hbox{\it abserr}
$$
\afterdisplay
@end tex
@ifinfo

@example
TOL_k = u_k abserr
@end example

@end ifinfo
@noindent
where 
@c{$u_k = (1 - p)p^{k-1}$}
@math{u_k = (1 - p)p^@{k-1@}} and @math{p = 9/10}.  
The sum of the geometric series of contributions from each interval
gives an overall tolerance of @var{abserr}.

If the integration of a subinterval leads to difficulties then the
accuracy requirement for subsequent intervals is relaxed,
@tex
\beforedisplay
$$
TOL_k = u_k \max(\hbox{\it abserr}, \max_{i<k}\{E_i\})
$$
\afterdisplay
@end tex
@ifinfo

@example
TOL_k = u_k max(abserr, max_@{i<k@}@{E_i@})
@end example

@end ifinfo
@noindent
where @math{E_k} is the estimated error on the interval @math{C_k}.

The subintervals and their results are stored in the memory provided by
@var{workspace}.  The maximum number of subintervals is given by
@var{limit}, which may not exceed the allocated size of the workspace.
The integration over each subinterval uses the memory provided by
@var{cycle_workspace} as workspace for the QAWO algorithm.

@end deftypefun

@node CQUAD doubly-adaptive integration
@section CQUAD doubly-adaptive integration
@cindex cquad, doubly-adaptive integration

@sc{cquad} is a new doubly-adaptive general-purpose quadrature
routine which can handle most types of singularities,
non-numerical function values such as @code{Inf} or @code{NaN},
as well as some divergent integrals. It generally requires more
function evaluations than the integration routines in
@sc{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 @math{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.
     
@deftypefun {gsl_integration_cquad_workspace *} gsl_integration_cquad_workspace_alloc (size_t @var{n}) 
@tindex gsl_integration_cquad_workspace
This function allocates a workspace sufficient to hold the data for
@var{n} intervals. The number @var{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.
@end deftypefun

@deftypefun void gsl_integration_cquad_workspace_free (gsl_integration_cquad_workspace * @var{w})
This function frees the memory associated with the workspace @var{w}.
@end deftypefun

@deftypefun int gsl_integration_cquad (const gsl_function * @var{f}, double @var{a}, double @var{b}, double @var{epsabs}, double @var{epsrel}, gsl_integration_cquad_workspace * @var{workspace},  double * @var{result}, double * @var{abserr}, size_t * @var{nevals})
This function computes the integral of @math{f} over @math{(a,b)}
within the desired absolute and relative error limits, @var{epsabs}
and @var{epsrel} using the @sc{cquad} algorithm.  The function returns
the final approximation, @var{result}, an estimate of the absolute
error, @var{abserr}, and the number of function evaluations required,
@var{nevals}.

The @sc{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
@math{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 @var{workspace}. If the error estimate or the number of 
function evaluations is not needed, the pointers @var{abserr} and @var{nevals}
can be set to @code{NULL}.
@end deftypefun

@node Fixed order Gauss-Legendre integration
@section Gauss-Legendre integration

The fixed-order Gauss-Legendre integration routines are provided for fast
integration of smooth functions with known polynomial order.  The
@math{n}-point Gauss-Legendre rule is exact for polynomials of order
@math{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.

@deftypefun {gsl_integration_glfixed_table *} gsl_integration_glfixed_table_alloc (size_t @var{n})
@tindex gsl_integration_glfixed_table
This function determines the Gauss-Legendre abscissae and weights necessary for
an @math{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.
@end deftypefun

@deftypefun double gsl_integration_glfixed ({const gsl_function *} @var{f}, double @var{a}, double @var{b}, {const gsl_integration_glfixed_table *} @var{t})
This function applies the Gauss-Legendre integration rule
contained in table @var{t} and returns the result.
@end deftypefun

@deftypefun int gsl_integration_glfixed_point (double @var{a}, double @var{b}, size_t @var{i}, {double *} @var{xi}, {double *} @var{wi}, {const gsl_integration_glfixed_table *} @var{t})
For @var{i} in @math{[0, @dots{}, t->n - 1]}, this function obtains the
@var{i}-th Gauss-Legendre point @var{xi} and weight @var{wi} on the interval
[@var{a},@var{b}].  The points and weights are ordered by increasing point
value.  A function @math{f} may be integrated on [@var{a},@var{b}] by summing
@math{wi * f(xi)} over @var{i}.
@end deftypefun

@deftypefun void gsl_integration_glfixed_table_free ({gsl_integration_glfixed_table *} @var{t})
@tindex gsl_integration_glfixed_table
This function frees the memory associated with the table @var{t}.
@end deftypefun

@node Numerical integration error codes
@section Error codes

In addition to the standard error codes for invalid arguments the
functions can return the following values,

@table @code
@item GSL_EMAXITER
the maximum number of subdivisions was exceeded.
@item GSL_EROUND
cannot reach tolerance because of roundoff error,
or roundoff error was detected in the extrapolation table.
@item GSL_ESING  
a non-integrable singularity or other bad integrand behavior was found
in the integration interval.
@item GSL_EDIVERGE
the integral is divergent, or too slowly convergent to be integrated
numerically.
@end table

@node Numerical integration examples
@section Examples

The integrator @code{QAGS} will handle a large class of definite
integrals.  For example, consider the following integral, which has an
algebraic-logarithmic singularity at the origin,
@tex
\beforedisplay
$$
\int_0^1 x^{-1/2} \log(x) \,dx = -4
$$
\afterdisplay
@end tex
@ifinfo

@example
\int_0^1 x^@{-1/2@} log(x) dx = -4
@end example

@end ifinfo
@noindent
The program below computes this integral to a relative accuracy bound of
@code{1e-7}.

@example
@verbatiminclude examples/integration.c
@end example

@noindent
The results below show that the desired accuracy is achieved after 8
subdivisions. 

@example
$ ./a.out 
@verbatiminclude examples/integration.txt
@end example

@noindent
In fact, the extrapolation procedure used by @code{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.


@node Numerical integration References and Further Reading
@section References and Further Reading

The following book is the definitive reference for @sc{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 @sc{quadpack}.

@itemize @w{}
@item
R. Piessens, E. de Doncker-Kapenga, C.W. Ueberhuber, D.K. Kahaner.
@cite{@sc{quadpack} A subroutine package for automatic integration}
Springer Verlag, 1983.
@end itemize

@noindent
The @sc{cquad} integration algorithm is described in the following paper:

@itemize @w{}
@item
P. Gonnet, ``Increasing the Reliability of Adaptive Quadrature Using
Explicit Interpolants'', @cite{ACM Transactions on Mathematical
Software}, Volume 37 (2010), Issue 3, Article 26.
@end itemize






