integration.texi

math library from gnu

@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})

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})

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